20.7

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
20.7.E1	${\displaystyle{\displaystyle{\theta_{3}^{2}}\left(0,q\right){\theta_{3}^{2}}\left(z,q\right)={\theta_{4}^{2}}\left(0,q\right){\theta_{4}^{2}}\left(z,q\right)+{\theta_{2}^{2}}\left(0,q\right){\theta_{2}^{2}}\left(z,q\right)}}$ \Jacobithetaq{3}^{2}@{0}{q}\Jacobithetaq{3}^{2}@{z}{q} = \Jacobithetaq{4}^{2}@{0}{q}\Jacobithetaq{4}^{2}@{z}{q}+\Jacobithetaq{2}^{2}@{0}{q}\Jacobithetaq{2}^{2}@{z}{q}		(JacobiTheta3(0, q))^(2)* (JacobiTheta3(z, q))^(2) = (JacobiTheta4(0, q))^(2)* (JacobiTheta4(z, q))^(2)+ (JacobiTheta2(0, q))^(2)* (JacobiTheta2(z, q))^(2)	(EllipticTheta[3, 0, q])^(2)* (EllipticTheta[3, z, q])^(2) == (EllipticTheta[4, 0, q])^(2)* (EllipticTheta[4, z, q])^(2)+ (EllipticTheta[2, 0, q])^(2)* (EllipticTheta[2, z, q])^(2)	Failure	Failure	Error	Successful [Tested: 70]
20.7.E2	${\displaystyle{\displaystyle{\theta_{3}^{2}}\left(0,q\right){\theta_{4}^{2}}\left(z,q\right)={\theta_{2}^{2}}\left(0,q\right){\theta_{1}^{2}}\left(z,q\right)+{\theta_{4}^{2}}\left(0,q\right){\theta_{3}^{2}}\left(z,q\right)}}$ \Jacobithetaq{3}^{2}@{0}{q}\Jacobithetaq{4}^{2}@{z}{q} = \Jacobithetaq{2}^{2}@{0}{q}\Jacobithetaq{1}^{2}@{z}{q}+\Jacobithetaq{4}^{2}@{0}{q}\Jacobithetaq{3}^{2}@{z}{q}		(JacobiTheta3(0, q))^(2)* (JacobiTheta4(z, q))^(2) = (JacobiTheta2(0, q))^(2)* (JacobiTheta1(z, q))^(2)+ (JacobiTheta4(0, q))^(2)* (JacobiTheta3(z, q))^(2)	(EllipticTheta[3, 0, q])^(2)* (EllipticTheta[4, z, q])^(2) == (EllipticTheta[2, 0, q])^(2)* (EllipticTheta[1, z, q])^(2)+ (EllipticTheta[4, 0, q])^(2)* (EllipticTheta[3, z, q])^(2)	Failure	Failure	Error	Successful [Tested: 70]
20.7.E3	${\displaystyle{\displaystyle{\theta_{2}^{2}}\left(0,q\right){\theta_{4}^{2}}\left(z,q\right)={\theta_{3}^{2}}\left(0,q\right){\theta_{1}^{2}}\left(z,q\right)+{\theta_{4}^{2}}\left(0,q\right){\theta_{2}^{2}}\left(z,q\right)}}$ \Jacobithetaq{2}^{2}@{0}{q}\Jacobithetaq{4}^{2}@{z}{q} = \Jacobithetaq{3}^{2}@{0}{q}\Jacobithetaq{1}^{2}@{z}{q}+\Jacobithetaq{4}^{2}@{0}{q}\Jacobithetaq{2}^{2}@{z}{q}		(JacobiTheta2(0, q))^(2)* (JacobiTheta4(z, q))^(2) = (JacobiTheta3(0, q))^(2)* (JacobiTheta1(z, q))^(2)+ (JacobiTheta4(0, q))^(2)* (JacobiTheta2(z, q))^(2)	(EllipticTheta[2, 0, q])^(2)* (EllipticTheta[4, z, q])^(2) == (EllipticTheta[3, 0, q])^(2)* (EllipticTheta[1, z, q])^(2)+ (EllipticTheta[4, 0, q])^(2)* (EllipticTheta[2, z, q])^(2)	Failure	Failure	Error	Successful [Tested: 70]
20.7.E4	${\displaystyle{\displaystyle{\theta_{2}^{2}}\left(0,q\right){\theta_{3}^{2}}\left(z,q\right)={\theta_{4}^{2}}\left(0,q\right){\theta_{1}^{2}}\left(z,q\right)+{\theta_{3}^{2}}\left(0,q\right){\theta_{2}^{2}}\left(z,q\right)}}$ \Jacobithetaq{2}^{2}@{0}{q}\Jacobithetaq{3}^{2}@{z}{q} = \Jacobithetaq{4}^{2}@{0}{q}\Jacobithetaq{1}^{2}@{z}{q}+\Jacobithetaq{3}^{2}@{0}{q}\Jacobithetaq{2}^{2}@{z}{q}		(JacobiTheta2(0, q))^(2)* (JacobiTheta3(z, q))^(2) = (JacobiTheta4(0, q))^(2)* (JacobiTheta1(z, q))^(2)+ (JacobiTheta3(0, q))^(2)* (JacobiTheta2(z, q))^(2)	(EllipticTheta[2, 0, q])^(2)* (EllipticTheta[3, z, q])^(2) == (EllipticTheta[4, 0, q])^(2)* (EllipticTheta[1, z, q])^(2)+ (EllipticTheta[3, 0, q])^(2)* (EllipticTheta[2, z, q])^(2)	Failure	Failure	Error	Successful [Tested: 70]
20.7.E5	${\displaystyle{\displaystyle{\theta_{3}^{4}}\left(0,q\right)={\theta_{2}^{4}}\left(0,q\right)+{\theta_{4}^{4}}\left(0,q\right)}}$ \Jacobithetaq{3}^{4}@{0}{q} = \Jacobithetaq{2}^{4}@{0}{q}+\Jacobithetaq{4}^{4}@{0}{q}		(JacobiTheta3(0, q))^(4) = (JacobiTheta2(0, q))^(4)+ (JacobiTheta4(0, q))^(4)	(EllipticTheta[3, 0, q])^(4) == (EllipticTheta[2, 0, q])^(4)+ (EllipticTheta[4, 0, q])^(4)	Successful	Failure	-	Successful [Tested: 10]
20.7.E6	${\displaystyle{\displaystyle{\theta_{4}^{2}}\left(0,q\right)\theta_{1}\left(w+z,q\right)\theta_{1}\left(w-z,q\right)={\theta_{3}^{2}}\left(w,q\right){\theta_{2}^{2}}\left(z,q\right)-{\theta_{2}^{2}}\left(w,q\right){\theta_{3}^{2}}\left(z,q\right)}}$ \Jacobithetaq{4}^{2}@{0}{q}\Jacobithetaq{1}@{w+z}{q}\Jacobithetaq{1}@{w-z}{q} = \Jacobithetaq{3}^{2}@{w}{q}\Jacobithetaq{2}^{2}@{z}{q}-\Jacobithetaq{2}^{2}@{w}{q}\Jacobithetaq{3}^{2}@{z}{q}		(JacobiTheta4(0, q))^(2)* JacobiTheta1(w + z, q)*JacobiTheta1(w - z, q) = (JacobiTheta3(w, q))^(2)* (JacobiTheta2(z, q))^(2)- (JacobiTheta2(w, q))^(2)* (JacobiTheta3(z, q))^(2)	(EllipticTheta[4, 0, q])^(2)* EllipticTheta[1, w + z, q]*EllipticTheta[1, w - z, q] == (EllipticTheta[3, w, q])^(2)* (EllipticTheta[2, z, q])^(2)- (EllipticTheta[2, w, q])^(2)* (EllipticTheta[3, z, q])^(2)	Failure	Failure	Error	Successful [Tested: 300]
20.7.E7	${\displaystyle{\displaystyle{\theta_{4}^{2}}\left(0,q\right)\theta_{2}\left(w+z,q\right)\theta_{2}\left(w-z,q\right)={\theta_{4}^{2}}\left(w,q\right){\theta_{2}^{2}}\left(z,q\right)-{\theta_{1}^{2}}\left(w,q\right){\theta_{3}^{2}}\left(z,q\right)}}$ \Jacobithetaq{4}^{2}@{0}{q}\Jacobithetaq{2}@{w+z}{q}\Jacobithetaq{2}@{w-z}{q} = \Jacobithetaq{4}^{2}@{w}{q}\Jacobithetaq{2}^{2}@{z}{q}-\Jacobithetaq{1}^{2}@{w}{q}\Jacobithetaq{3}^{2}@{z}{q}		(JacobiTheta4(0, q))^(2)* JacobiTheta2(w + z, q)*JacobiTheta2(w - z, q) = (JacobiTheta4(w, q))^(2)* (JacobiTheta2(z, q))^(2)- (JacobiTheta1(w, q))^(2)* (JacobiTheta3(z, q))^(2)	(EllipticTheta[4, 0, q])^(2)* EllipticTheta[2, w + z, q]*EllipticTheta[2, w - z, q] == (EllipticTheta[4, w, q])^(2)* (EllipticTheta[2, z, q])^(2)- (EllipticTheta[1, w, q])^(2)* (EllipticTheta[3, z, q])^(2)	Failure	Failure	Error	Successful [Tested: 300]
20.7.E8	${\displaystyle{\displaystyle{\theta_{4}^{2}}\left(0,q\right)\theta_{3}\left(w+z,q\right)\theta_{3}\left(w-z,q\right)={\theta_{4}^{2}}\left(w,q\right){\theta_{3}^{2}}\left(z,q\right)-{\theta_{1}^{2}}\left(w,q\right){\theta_{2}^{2}}\left(z,q\right)}}$ \Jacobithetaq{4}^{2}@{0}{q}\Jacobithetaq{3}@{w+z}{q}\Jacobithetaq{3}@{w-z}{q} = \Jacobithetaq{4}^{2}@{w}{q}\Jacobithetaq{3}^{2}@{z}{q}-\Jacobithetaq{1}^{2}@{w}{q}\Jacobithetaq{2}^{2}@{z}{q}		(JacobiTheta4(0, q))^(2)* JacobiTheta3(w + z, q)*JacobiTheta3(w - z, q) = (JacobiTheta4(w, q))^(2)* (JacobiTheta3(z, q))^(2)- (JacobiTheta1(w, q))^(2)* (JacobiTheta2(z, q))^(2)	(EllipticTheta[4, 0, q])^(2)* EllipticTheta[3, w + z, q]*EllipticTheta[3, w - z, q] == (EllipticTheta[4, w, q])^(2)* (EllipticTheta[3, z, q])^(2)- (EllipticTheta[1, w, q])^(2)* (EllipticTheta[2, z, q])^(2)	Failure	Failure	Error	Successful [Tested: 300]
20.7.E9	${\displaystyle{\displaystyle{\theta_{4}^{2}}\left(0,q\right)\theta_{4}\left(w+z,q\right)\theta_{4}\left(w-z,q\right)={\theta_{3}^{2}}\left(w,q\right){\theta_{3}^{2}}\left(z,q\right)-{\theta_{2}^{2}}\left(w,q\right){\theta_{2}^{2}}\left(z,q\right)}}$ \Jacobithetaq{4}^{2}@{0}{q}\Jacobithetaq{4}@{w+z}{q}\Jacobithetaq{4}@{w-z}{q} = \Jacobithetaq{3}^{2}@{w}{q}\Jacobithetaq{3}^{2}@{z}{q}-\Jacobithetaq{2}^{2}@{w}{q}\Jacobithetaq{2}^{2}@{z}{q}		(JacobiTheta4(0, q))^(2)* JacobiTheta4(w + z, q)*JacobiTheta4(w - z, q) = (JacobiTheta3(w, q))^(2)* (JacobiTheta3(z, q))^(2)- (JacobiTheta2(w, q))^(2)* (JacobiTheta2(z, q))^(2)	(EllipticTheta[4, 0, q])^(2)* EllipticTheta[4, w + z, q]*EllipticTheta[4, w - z, q] == (EllipticTheta[3, w, q])^(2)* (EllipticTheta[3, z, q])^(2)- (EllipticTheta[2, w, q])^(2)* (EllipticTheta[2, z, q])^(2)	Failure	Failure	Error	Successful [Tested: 300]
20.7.E10	${\displaystyle{\displaystyle\theta_{1}\left(2z,q\right)=2\frac{\theta_{1}\left(z,q\right)\theta_{2}\left(z,q\right)\theta_{3}\left(z,q\right)\theta_{4}\left(z,q\right)}{\theta_{2}\left(0,q\right)\theta_{3}\left(0,q\right)\theta_{4}\left(0,q\right)}}}$ \Jacobithetaq{1}@{2z}{q} = 2\frac{\Jacobithetaq{1}@{z}{q}\Jacobithetaq{2}@{z}{q}\Jacobithetaq{3}@{z}{q}\Jacobithetaq{4}@{z}{q}}{\Jacobithetaq{2}@{0}{q}\Jacobithetaq{3}@{0}{q}\Jacobithetaq{4}@{0}{q}}		JacobiTheta1(2*z, q) = 2*(JacobiTheta1(z, q)*JacobiTheta2(z, q)*JacobiTheta3(z, q)*JacobiTheta4(z, q))/(JacobiTheta2(0, q)*JacobiTheta3(0, q)*JacobiTheta4(0, q))	EllipticTheta[1, 2*z, q] == 2*Divide[EllipticTheta[1, z, q]*EllipticTheta[2, z, q]*EllipticTheta[3, z, q]*EllipticTheta[4, z, q],EllipticTheta[2, 0, q]*EllipticTheta[3, 0, q]*EllipticTheta[4, 0, q]]	Failure	Failure	Error	Successful [Tested: 70]
20.7.E11	${\displaystyle{\displaystyle\frac{\theta_{1}\left(z,q\right)\theta_{2}\left(z,q\right)}{\theta_{1}\left(2z,q^{2}\right)}=\frac{\theta_{3}\left(z,q\right)\theta_{4}\left(z,q\right)}{\theta_{4}\left(2z,q^{2}\right)}}}$ \frac{\Jacobithetaq{1}@{z}{q}\Jacobithetaq{2}@{z}{q}}{\Jacobithetaq{1}@{2z}{q^{2}}} = \frac{\Jacobithetaq{3}@{z}{q}\Jacobithetaq{4}@{z}{q}}{\Jacobithetaq{4}@{2z}{q^{2}}}		(JacobiTheta1(z, q)*JacobiTheta2(z, q))/(JacobiTheta1(2*z, (q)^(2))) = (JacobiTheta3(z, q)*JacobiTheta4(z, q))/(JacobiTheta4(2*z, (q)^(2)))	Divide[EllipticTheta[1, z, q]*EllipticTheta[2, z, q],EllipticTheta[1, 2*z, (q)^(2)]] == Divide[EllipticTheta[3, z, q]*EllipticTheta[4, z, q],EllipticTheta[4, 2*z, (q)^(2)]]	Failure	Failure	Error	Failed [7 / 70] Result: Complex[-0.5078048710711283, 0.5078048710711279] Test Values: {Rule[q, -0.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.5078048710711284, 0.5078048710711281] Test Values: {Rule[q, -0.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
20.7.E11	${\displaystyle{\displaystyle\frac{\theta_{3}\left(z,q\right)\theta_{4}\left(z,q\right)}{\theta_{4}\left(2z,q^{2}\right)}=\theta_{4}\left(0,q^{2}\right)}}$ \frac{\Jacobithetaq{3}@{z}{q}\Jacobithetaq{4}@{z}{q}}{\Jacobithetaq{4}@{2z}{q^{2}}} = \Jacobithetaq{4}@{0}{q^{2}}		(JacobiTheta3(z, q)*JacobiTheta4(z, q))/(JacobiTheta4(2*z, (q)^(2))) = JacobiTheta4(0, (q)^(2))	Divide[EllipticTheta[3, z, q]*EllipticTheta[4, z, q],EllipticTheta[4, 2*z, (q)^(2)]] == EllipticTheta[4, 0, (q)^(2)]	Failure	Failure	Error	Successful [Tested: 70]
20.7.E12	${\displaystyle{\displaystyle\frac{\theta_{1}\left(z,q^{2}\right)\theta_{4}\left(z,q^{2}\right)}{\theta_{1}\left(z,q\right)}=\frac{\theta_{2}\left(z,q^{2}\right)\theta_{3}\left(z,q^{2}\right)}{\theta_{2}\left(z,q\right)}}}$ \frac{\Jacobithetaq{1}@{z}{q^{2}}\Jacobithetaq{4}@{z}{q^{2}}}{\Jacobithetaq{1}@{z}{q}} = \frac{\Jacobithetaq{2}@{z}{q^{2}}\Jacobithetaq{3}@{z}{q^{2}}}{\Jacobithetaq{2}@{z}{q}}		(JacobiTheta1(z, (q)^(2))*JacobiTheta4(z, (q)^(2)))/(JacobiTheta1(z, q)) = (JacobiTheta2(z, (q)^(2))*JacobiTheta3(z, (q)^(2)))/(JacobiTheta2(z, q))	Divide[EllipticTheta[1, z, (q)^(2)]*EllipticTheta[4, z, (q)^(2)],EllipticTheta[1, z, q]] == Divide[EllipticTheta[2, z, (q)^(2)]*EllipticTheta[3, z, (q)^(2)],EllipticTheta[2, z, q]]	Failure	Failure	Error	Successful [Tested: 70]
20.7.E12	${\displaystyle{\displaystyle\frac{\theta_{2}\left(z,q^{2}\right)\theta_{3}\left(z,q^{2}\right)}{\theta_{2}\left(z,q\right)}=\tfrac{1}{2}\theta_{2}\left(0,q\right)}}$ \frac{\Jacobithetaq{2}@{z}{q^{2}}\Jacobithetaq{3}@{z}{q^{2}}}{\Jacobithetaq{2}@{z}{q}} = \tfrac{1}{2}\Jacobithetaq{2}@{0}{q}		(JacobiTheta2(z, (q)^(2))*JacobiTheta3(z, (q)^(2)))/(JacobiTheta2(z, q)) = (1)/(2)*JacobiTheta2(0, q)	Divide[EllipticTheta[2, z, (q)^(2)]*EllipticTheta[3, z, (q)^(2)],EllipticTheta[2, z, q]] == Divide[1,2]*EllipticTheta[2, 0, q]	Failure	Failure	Error	Failed [7 / 70] Result: Complex[1.1102230246251565*^-16, -1.5053817239177183] Test Values: {Rule[q, -0.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[3.3306690738754696*^-16, -1.5053817239177185] Test Values: {Rule[q, -0.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
20.7.E13	${\displaystyle{\displaystyle\theta_{1}\left(z,q\right)\theta_{1}\left(w,q\right)=\theta_{3}\left(z+w,q^{2}\right)\theta_{2}\left(z-w,q^{2}\right)-\theta_{2}\left(z+w,q^{2}\right)\theta_{3}\left(z-w,q^{2}\right)}}$ \Jacobithetaq{1}@{z}{q}\Jacobithetaq{1}@{w}{q} = \Jacobithetaq{3}@{z+w}{q^{2}}\Jacobithetaq{2}@{z-w}{q^{2}}-\Jacobithetaq{2}@{z+w}{q^{2}}\Jacobithetaq{3}@{z-w}{q^{2}}		JacobiTheta1(z, q)*JacobiTheta1(w, q) = JacobiTheta3(z + w, (q)^(2))*JacobiTheta2(z - w, (q)^(2))- JacobiTheta2(z + w, (q)^(2))*JacobiTheta3(z - w, (q)^(2))	EllipticTheta[1, z, q]*EllipticTheta[1, w, q] == EllipticTheta[3, z + w, (q)^(2)]*EllipticTheta[2, z - w, (q)^(2)]- EllipticTheta[2, z + w, (q)^(2)]*EllipticTheta[3, z - w, (q)^(2)]	Failure	Failure	Error	Successful [Tested: 300]
20.7.E14	${\displaystyle{\displaystyle\theta_{3}\left(z,q\right)\theta_{3}\left(w,q\right)=\theta_{3}\left(z+w,q^{2}\right)\theta_{3}\left(z-w,q^{2}\right)+\theta_{2}\left(z+w,q^{2}\right)\theta_{2}\left(z-w,q^{2}\right)}}$ \Jacobithetaq{3}@{z}{q}\Jacobithetaq{3}@{w}{q} = \Jacobithetaq{3}@{z+w}{q^{2}}\Jacobithetaq{3}@{z-w}{q^{2}}+\Jacobithetaq{2}@{z+w}{q^{2}}\Jacobithetaq{2}@{z-w}{q^{2}}		JacobiTheta3(z, q)*JacobiTheta3(w, q) = JacobiTheta3(z + w, (q)^(2))*JacobiTheta3(z - w, (q)^(2))+ JacobiTheta2(z + w, (q)^(2))*JacobiTheta2(z - w, (q)^(2))	EllipticTheta[3, z, q]*EllipticTheta[3, w, q] == EllipticTheta[3, z + w, (q)^(2)]*EllipticTheta[3, z - w, (q)^(2)]+ EllipticTheta[2, z + w, (q)^(2)]*EllipticTheta[2, z - w, (q)^(2)]	Failure	Failure	Error	Successful [Tested: 300]
20.7.E16	${\displaystyle{\displaystyle\theta_{1}\left(2z\middle|2\tau\right)=A\theta_{1}\left(z\middle|\tau\right)\theta_{2}\left(z\middle|\tau\right)}}$ \Jacobithetatau{1}@{2z}{2\tau} = A\Jacobithetatau{1}@{z}{\tau}\Jacobithetatau{2}@{z}{\tau}		JacobiTheta1(2*z,exp(I*Pi*2*tau)) = A*JacobiTheta1(z,exp(I*Pi*tau))*JacobiTheta2(z,exp(I*Pi*tau))	EllipticTheta[1, 2*z, Exp[I*Pi*(2*\[Tau])]] == A*EllipticTheta[1, z, Exp[I*Pi*(\[Tau])]]*EllipticTheta[2, z, Exp[I*Pi*(\[Tau])]]	Failure	Failure	Failed [300 / 300] Result: 1.631641333-1.744983248*I Test Values: {A = 1/2*3^(1/2)+1/2*I, tau = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I} Result: -1.353330373+4.008308689*I Test Values: {A = 1/2*3^(1/2)+1/2*I, tau = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [60 / 300] Result: Complex[1.6316413333035786, -1.7449832486391479] Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.25205232655780907, -0.3227610482702816] Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
20.7.E17	${\displaystyle{\displaystyle\theta_{2}\left(2z\middle|2\tau\right)=A\theta_{1}\left(\tfrac{1}{4}\pi-z\middle|\tau\right)\theta_{1}\left(\tfrac{1}{4}\pi+z\middle|\tau\right)}}$ \Jacobithetatau{2}@{2z}{2\tau} = A\Jacobithetatau{1}@{\tfrac{1}{4}\pi-z}{\tau}\Jacobithetatau{1}@{\tfrac{1}{4}\pi+z}{\tau}		JacobiTheta2(2*z,exp(I*Pi*2*tau)) = A*JacobiTheta1((1)/(4)*Pi - z,exp(I*Pi*tau))*JacobiTheta1((1)/(4)*Pi + z,exp(I*Pi*tau))	EllipticTheta[2, 2*z, Exp[I*Pi*(2*\[Tau])]] == A*EllipticTheta[1, Divide[1,4]*Pi - z, Exp[I*Pi*(\[Tau])]]*EllipticTheta[1, Divide[1,4]*Pi + z, Exp[I*Pi*(\[Tau])]]	Error	Failure	-	Failed [60 / 300] Result: Complex[-1.4403734484961686, -1.1891981543571708] Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.23150096143650367, 0.21570115304796234] Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
20.7.E18	${\displaystyle{\displaystyle\theta_{3}\left(2z\middle|2\tau\right)=A\theta_{3}\left(\tfrac{1}{4}\pi-z\middle|\tau\right)\theta_{3}\left(\tfrac{1}{4}\pi+z\middle|\tau\right)}}$ \Jacobithetatau{3}@{2z}{2\tau} = A\Jacobithetatau{3}@{\tfrac{1}{4}\pi-z}{\tau}\Jacobithetatau{3}@{\tfrac{1}{4}\pi+z}{\tau}		JacobiTheta3(2*z,exp(I*Pi*2*tau)) = A*JacobiTheta3((1)/(4)*Pi - z,exp(I*Pi*tau))*JacobiTheta3((1)/(4)*Pi + z,exp(I*Pi*tau))	EllipticTheta[3, 2*z, Exp[I*Pi*(2*\[Tau])]] == A*EllipticTheta[3, Divide[1,4]*Pi - z, Exp[I*Pi*(\[Tau])]]*EllipticTheta[3, Divide[1,4]*Pi + z, Exp[I*Pi*(\[Tau])]]	Error	Failure	-	Failed [60 / 300] Result: Complex[0.3438479503598899, -0.39372543999621956] Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.12535543238516544, -0.5211900545642698] Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
20.7.E19	${\displaystyle{\displaystyle\theta_{4}\left(2z\middle|2\tau\right)=A\theta_{3}\left(z\middle|\tau\right)\theta_{4}\left(z\middle|\tau\right)}}$ \Jacobithetatau{4}@{2z}{2\tau} = A\Jacobithetatau{3}@{z}{\tau}\Jacobithetatau{4}@{z}{\tau}		JacobiTheta4(2*z,exp(I*Pi*2*tau)) = A*JacobiTheta3(z,exp(I*Pi*tau))*JacobiTheta4(z,exp(I*Pi*tau))	EllipticTheta[4, 2*z, Exp[I*Pi*(2*\[Tau])]] == A*EllipticTheta[3, z, Exp[I*Pi*(\[Tau])]]*EllipticTheta[4, z, Exp[I*Pi*(\[Tau])]]	Failure	Failure	Failed [300 / 300] Result: .88393938e-1-.6601554491*I Test Values: {A = 1/2*3^(1/2)+1/2*I, tau = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I} Result: -.5678871113-.5102031247*I Test Values: {A = 1/2*3^(1/2)+1/2*I, tau = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [60 / 300] Result: Complex[0.08839393747885427, -0.6601554493410663] Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.12758234205780994, -0.4874768056112989] Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
20.7.E21	${\displaystyle{\displaystyle\theta_{1}\left(4z\middle|4\tau\right)=B\theta_{1}\left(z\middle|\tau\right)\theta_{1}\left(\tfrac{1}{4}\pi-z\middle|\tau\right)\*\theta_{1}\left(\tfrac{1}{4}\pi+z\middle|\tau\right)\theta_{2}\left(z\middle|\tau\right)}}$ \Jacobithetatau{1}@{4z}{4\tau} = B\Jacobithetatau{1}@{z}{\tau}\Jacobithetatau{1}@{\tfrac{1}{4}\pi-z}{\tau}\*\Jacobithetatau{1}@{\tfrac{1}{4}\pi+z}{\tau}\Jacobithetatau{2}@{z}{\tau}		JacobiTheta1(4*z,exp(I*Pi*4*tau)) = B*JacobiTheta1(z,exp(I*Pi*tau))*JacobiTheta1((1)/(4)*Pi - z,exp(I*Pi*tau))* JacobiTheta1((1)/(4)*Pi + z,exp(I*Pi*tau))*JacobiTheta2(z,exp(I*Pi*tau))	EllipticTheta[1, 4*z, Exp[I*Pi*(4*\[Tau])]] == B*EllipticTheta[1, z, Exp[I*Pi*(\[Tau])]]*EllipticTheta[1, Divide[1,4]*Pi - z, Exp[I*Pi*(\[Tau])]]* EllipticTheta[1, Divide[1,4]*Pi + z, Exp[I*Pi*(\[Tau])]]*EllipticTheta[2, z, Exp[I*Pi*(\[Tau])]]	Error	Failure	-	Failed [60 / 300] Result: Complex[-1.1596846442931608, -2.448595776474227] Test Values: {Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.3218907084595235, -0.36082838804303224] Test Values: {Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
20.7.E22	${\displaystyle{\displaystyle\theta_{2}\left(4z\middle|4\tau\right)=B\theta_{2}\left(\tfrac{1}{8}\pi-z\middle|\tau\right)\theta_{2}\left(\tfrac{1}{8}\pi+z\middle|\tau\right)\*\theta_{2}\left(\tfrac{3}{8}\pi-z\middle|\tau\right)\theta_{2}\left(\tfrac{3}{8}\pi+z\middle|\tau\right)}}$ \Jacobithetatau{2}@{4z}{4\tau} = B\Jacobithetatau{2}@{\tfrac{1}{8}\pi-z}{\tau}\Jacobithetatau{2}@{\tfrac{1}{8}\pi+z}{\tau}\*\Jacobithetatau{2}@{\tfrac{3}{8}\pi-z}{\tau}\Jacobithetatau{2}@{\tfrac{3}{8}\pi+z}{\tau}		JacobiTheta2(4*z,exp(I*Pi*4*tau)) = B*JacobiTheta2((1)/(8)*Pi - z,exp(I*Pi*tau))*JacobiTheta2((1)/(8)*Pi + z,exp(I*Pi*tau))* JacobiTheta2((3)/(8)*Pi - z,exp(I*Pi*tau))*JacobiTheta2((3)/(8)*Pi + z,exp(I*Pi*tau))	EllipticTheta[2, 4*z, Exp[I*Pi*(4*\[Tau])]] == B*EllipticTheta[2, Divide[1,8]*Pi - z, Exp[I*Pi*(\[Tau])]]*EllipticTheta[2, Divide[1,8]*Pi + z, Exp[I*Pi*(\[Tau])]]* EllipticTheta[2, Divide[3,8]*Pi - z, Exp[I*Pi*(\[Tau])]]*EllipticTheta[2, Divide[3,8]*Pi + z, Exp[I*Pi*(\[Tau])]]	Error	Failure	-	Failed [60 / 300] Result: Complex[-2.54672123948714, 1.1372871673366372] Test Values: {Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.36415557562453404, -0.3395547407401721] Test Values: {Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
20.7.E23	${\displaystyle{\displaystyle\theta_{3}\left(4z\middle|4\tau\right)=B\theta_{3}\left(\tfrac{1}{8}\pi-z\middle|\tau\right)\theta_{3}\left(\tfrac{1}{8}\pi+z\middle|\tau\right)\*\theta_{3}\left(\tfrac{3}{8}\pi-z\middle|\tau\right)\theta_{3}\left(\tfrac{3}{8}\pi+z\middle|\tau\right)}}$ \Jacobithetatau{3}@{4z}{4\tau} = B\Jacobithetatau{3}@{\tfrac{1}{8}\pi-z}{\tau}\Jacobithetatau{3}@{\tfrac{1}{8}\pi+z}{\tau}\*\Jacobithetatau{3}@{\tfrac{3}{8}\pi-z}{\tau}\Jacobithetatau{3}@{\tfrac{3}{8}\pi+z}{\tau}		JacobiTheta3(4*z,exp(I*Pi*4*tau)) = B*JacobiTheta3((1)/(8)*Pi - z,exp(I*Pi*tau))*JacobiTheta3((1)/(8)*Pi + z,exp(I*Pi*tau))* JacobiTheta3((3)/(8)*Pi - z,exp(I*Pi*tau))*JacobiTheta3((3)/(8)*Pi + z,exp(I*Pi*tau))	EllipticTheta[3, 4*z, Exp[I*Pi*(4*\[Tau])]] == B*EllipticTheta[3, Divide[1,8]*Pi - z, Exp[I*Pi*(\[Tau])]]*EllipticTheta[3, Divide[1,8]*Pi + z, Exp[I*Pi*(\[Tau])]]* EllipticTheta[3, Divide[3,8]*Pi - z, Exp[I*Pi*(\[Tau])]]*EllipticTheta[3, Divide[3,8]*Pi + z, Exp[I*Pi*(\[Tau])]]	Error	Failure	-	Failed [60 / 300] Result: Complex[0.2353615104715142, -0.5335293147703523] Test Values: {Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.11871524589758675, -0.5091754766273449] Test Values: {Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
20.7.E24	${\displaystyle{\displaystyle\theta_{4}\left(4z\middle|4\tau\right)=B\theta_{4}\left(z\middle|\tau\right)\theta_{4}\left(\tfrac{1}{4}\pi-z\middle|\tau\right)\*\theta_{4}\left(\tfrac{1}{4}\pi+z\middle|\tau\right)\theta_{3}\left(z\middle|\tau\right)}}$ \Jacobithetatau{4}@{4z}{4\tau} = B\Jacobithetatau{4}@{z}{\tau}\Jacobithetatau{4}@{\tfrac{1}{4}\pi-z}{\tau}\*\Jacobithetatau{4}@{\tfrac{1}{4}\pi+z}{\tau}\Jacobithetatau{3}@{z}{\tau}		JacobiTheta4(4*z,exp(I*Pi*4*tau)) = B*JacobiTheta4(z,exp(I*Pi*tau))*JacobiTheta4((1)/(4)*Pi - z,exp(I*Pi*tau))* JacobiTheta4((1)/(4)*Pi + z,exp(I*Pi*tau))*JacobiTheta3(z,exp(I*Pi*tau))	EllipticTheta[4, 4*z, Exp[I*Pi*(4*\[Tau])]] == B*EllipticTheta[4, z, Exp[I*Pi*(\[Tau])]]*EllipticTheta[4, Divide[1,4]*Pi - z, Exp[I*Pi*(\[Tau])]]* EllipticTheta[4, Divide[1,4]*Pi + z, Exp[I*Pi*(\[Tau])]]*EllipticTheta[3, z, Exp[I*Pi*(\[Tau])]]	Error	Failure	-	Failed [60 / 300] Result: Complex[0.3584730563399423, -0.5666107505620169] Test Values: {Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.11914720780154586, -0.5081951100786072] Test Values: {Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
20.7.E25	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\left(\frac{\theta_{2}\left(z\middle|\tau\right)}{\theta_{4}\left(z\middle|\tau\right)}\right)=-\frac{{\theta_{3}^{2}}\left(0\middle|\tau\right)\theta_{1}\left(z\middle|\tau\right)\theta_{3}\left(z\middle|\tau\right)}{{\theta_{4}^{2}}\left(z\middle|\tau\right)}}}$ \deriv{}{z}\left(\frac{\Jacobithetatau{2}@{z}{\tau}}{\Jacobithetatau{4}@{z}{\tau}}\right) = -\frac{\Jacobithetatau{3}^{2}@{0}{\tau}\Jacobithetatau{1}@{z}{\tau}\Jacobithetatau{3}@{z}{\tau}}{\Jacobithetatau{4}^{2}@{z}{\tau}}		diff((JacobiTheta2(z,exp(I*Pi*tau)))/(JacobiTheta4(z,exp(I*Pi*tau))), z) = -((JacobiTheta3(0,exp(I*Pi*tau)))^(2)* JacobiTheta1(z,exp(I*Pi*tau))*JacobiTheta3(z,exp(I*Pi*tau)))/((JacobiTheta4(z,exp(I*Pi*tau)))^(2))	D[Divide[EllipticTheta[2, z, Exp[I*Pi*(\[Tau])]],EllipticTheta[4, z, Exp[I*Pi*(\[Tau])]]], z] == -Divide[(EllipticTheta[3, 0, Exp[I*Pi*(\[Tau])]])^(2)* EllipticTheta[1, z, Exp[I*Pi*(\[Tau])]]*EllipticTheta[3, z, Exp[I*Pi*(\[Tau])]],(EllipticTheta[4, z, Exp[I*Pi*(\[Tau])]])^(2)]	Failure	Failure	Error	Successful [Tested: 70]
20.7.E26	${\displaystyle{\displaystyle\theta_{1}\left(z\middle|\tau+1\right)=e^{i\pi/4}\theta_{1}\left(z\middle|\tau\right)}}$ \Jacobithetatau{1}@{z}{\tau+1} = e^{i\pi/4}\Jacobithetatau{1}@{z}{\tau}		JacobiTheta1(z,exp(I*Pi*tau + 1)) = exp(I*Pi/4)*JacobiTheta1(z,exp(I*Pi*tau))	EllipticTheta[1, z, Exp[I*Pi*(\[Tau]+ 1)]] == Exp[I*Pi/4]*EllipticTheta[1, z, Exp[I*Pi*(\[Tau])]]	Failure	Failure	Failed [70 / 70] Result: .7294764132+1.608567858*I Test Values: {tau = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I} Result: 1.107791050+1.561378050*I Test Values: {tau = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [7 / 70] Result: Complex[1.6985877827537141, -0.7949460182709149] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.345921896794935, 1.4881712816971224] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
20.7.E27	${\displaystyle{\displaystyle\theta_{2}\left(z\middle|\tau+1\right)=e^{i\pi/4}\theta_{2}\left(z\middle|\tau\right)}}$ \Jacobithetatau{2}@{z}{\tau+1} = e^{i\pi/4}\Jacobithetatau{2}@{z}{\tau}		JacobiTheta2(z,exp(I*Pi*tau + 1)) = exp(I*Pi/4)*JacobiTheta2(z,exp(I*Pi*tau))	EllipticTheta[2, z, Exp[I*Pi*(\[Tau]+ 1)]] == Exp[I*Pi/4]*EllipticTheta[2, z, Exp[I*Pi*(\[Tau])]]	Failure	Failure	Failed [70 / 70] Result: -.369621756e-1-.9012887423*I Test Values: {tau = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I} Result: 4.590414642+4.526034042*I Test Values: {tau = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [7 / 70] Result: Complex[0.22524015718924872, -1.3838317643459628] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[2.711359141795916, -1.3916787489924032] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
20.7.E28	${\displaystyle{\displaystyle\theta_{3}\left(z\middle|\tau+1\right)=\theta_{4}\left(z\middle|\tau\right)}}$ \Jacobithetatau{3}@{z}{\tau+1} = \Jacobithetatau{4}@{z}{\tau}		JacobiTheta3(z,exp(I*Pi*tau + 1)) = JacobiTheta4(z,exp(I*Pi*tau))	EllipticTheta[3, z, Exp[I*Pi*(\[Tau]+ 1)]] == EllipticTheta[4, z, Exp[I*Pi*(\[Tau])]]	Failure	Failure	Failed [70 / 70] Result: 1.500564535+2.208881092*I Test Values: {tau = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I} Result: -1.492914692-.5532090072*I Test Values: {tau = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Successful [Tested: 70]
20.7.E29	${\displaystyle{\displaystyle\theta_{4}\left(z\middle|\tau+1\right)=\theta_{3}\left(z\middle|\tau\right)}}$ \Jacobithetatau{4}@{z}{\tau+1} = \Jacobithetatau{3}@{z}{\tau}		JacobiTheta4(z,exp(I*Pi*tau + 1)) = JacobiTheta3(z,exp(I*Pi*tau))	EllipticTheta[4, z, Exp[I*Pi*(\[Tau]+ 1)]] == EllipticTheta[3, z, Exp[I*Pi*(\[Tau])]]	Failure	Failure	Failed [70 / 70] Result: -.8770870366-.8516489897*I Test Values: {tau = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I} Result: 7.362801863+2.459098613*I Test Values: {tau = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Successful [Tested: 70]
20.7.E34	${\displaystyle{\displaystyle\frac{\theta_{1}\left(z,q^{2}\right)\theta_{3}\left(z,q^{2}\right)}{\theta_{1}\left(z,iq\right)}=\frac{\theta_{2}\left(z,q^{2}\right)\theta_{4}\left(z,q^{2}\right)}{\theta_{2}\left(z,iq\right)}}}$ \frac{\Jacobithetaq{1}@{z}{q^{2}}\Jacobithetaq{3}@{z}{q^{2}}}{\Jacobithetaq{1}@{z}{iq}} = \frac{\Jacobithetaq{2}@{z}{q^{2}}\Jacobithetaq{4}@{z}{q^{2}}}{\Jacobithetaq{2}@{z}{iq}}		(JacobiTheta1(z, (q)^(2))*JacobiTheta3(z, (q)^(2)))/(JacobiTheta1(z, I*q)) = (JacobiTheta2(z, (q)^(2))*JacobiTheta4(z, (q)^(2)))/(JacobiTheta2(z, I*q))	Divide[EllipticTheta[1, z, (q)^(2)]*EllipticTheta[3, z, (q)^(2)],EllipticTheta[1, z, I*q]] == Divide[EllipticTheta[2, z, (q)^(2)]*EllipticTheta[4, z, (q)^(2)],EllipticTheta[2, z, I*q]]	Failure	Failure	Error	Successful [Tested: 70]
20.7.E34	${\displaystyle{\displaystyle\frac{\theta_{2}\left(z,q^{2}\right)\theta_{4}\left(z,q^{2}\right)}{\theta_{2}\left(z,iq\right)}=i^{-1/4}\sqrt{\frac{\theta_{2}\left(0,q^{2}\right)\theta_{4}\left(0,q^{2}\right)}{2}}}}$ \frac{\Jacobithetaq{2}@{z}{q^{2}}\Jacobithetaq{4}@{z}{q^{2}}}{\Jacobithetaq{2}@{z}{iq}} = i^{-1/4}\sqrt{\frac{\Jacobithetaq{2}@{0}{q^{2}}\Jacobithetaq{4}@{0}{q^{2}}}{2}}		(JacobiTheta2(z, (q)^(2))*JacobiTheta4(z, (q)^(2)))/(JacobiTheta2(z, I*q)) = (I)^(- 1/4)*sqrt((JacobiTheta2(0, (q)^(2))*JacobiTheta4(0, (q)^(2)))/(2))	Divide[EllipticTheta[2, z, (q)^(2)]*EllipticTheta[4, z, (q)^(2)],EllipticTheta[2, z, I*q]] == (I)^(- 1/4)*Sqrt[Divide[EllipticTheta[2, 0, (q)^(2)]*EllipticTheta[4, 0, (q)^(2)],2]]	Failure	Failure	Error	Failed [7 / 70] Result: Complex[-1.1102230246251565*^-16, 0.47279727016045703] Test Values: {Rule[q, -0.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[4.440892098500626*^-16, 0.4727972701604571] Test Values: {Rule[q, -0.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data