Theta Functions - 20.4 Values at = 0

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
20.4.E1 θ 1 ⁡ ( 0 , q ) = θ 2 ′ ⁡ ( 0 , q ) Jacobi-theta 1 0 𝑞 diffop Jacobi-theta 2 1 0 𝑞 {\displaystyle{\displaystyle\theta_{1}\left(0,q\right)=\theta_{2}'\left(0,q% \right)}}
\Jacobithetaq{1}@{0}{q} = \Jacobithetaq{2}'@{0}{q}		 

JacobiTheta1(0, q) = diff( JacobiTheta2(0, q), 0$(1) )

EllipticTheta[1, 0, q] == D[EllipticTheta[2, 0, q], {0, 1}]
Error Failure -
Failed [10 / 10]
Result: Times[-1.0, D[EllipticTheta[2, 0.0, Complex[0.8660254037844387, 0.49999999999999994]]
Test Values: {0.0, 1.0}]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Times[-1.0, D[EllipticTheta[2, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]]
Test Values: {0.0, 1.0}]], {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
20.4.E1 θ 2 ′ ⁡ ( 0 , q ) = θ 3 ′ ⁡ ( 0 , q ) diffop Jacobi-theta 2 1 0 𝑞 diffop Jacobi-theta 3 1 0 𝑞 {\displaystyle{\displaystyle\theta_{2}'\left(0,q\right)=\theta_{3}'\left(0,q% \right)}}
\Jacobithetaq{2}'@{0}{q} = \Jacobithetaq{3}'@{0}{q}		 

diff( JacobiTheta2(0, q), 0$(1) ) = diff( JacobiTheta3(0, q), 0$(1) )

D[EllipticTheta[2, 0, q], {0, 1}] == D[EllipticTheta[3, 0, q], {0, 1}]
Error Failure -
Failed [10 / 10]
Result: Plus[D[EllipticTheta[2, 0.0, Complex[0.8660254037844387, 0.49999999999999994]]
Test Values: {0.0, 1.0}], Times[-1.0, D[EllipticTheta[3, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], {0.0, 1.0}]]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[D[EllipticTheta[2, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]]
Test Values: {0.0, 1.0}], Times[-1.0, D[EllipticTheta[3, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]], {0.0, 1.0}]]], {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
20.4.E1 θ 3 ′ ⁡ ( 0 , q ) = θ 4 ′ ⁡ ( 0 , q ) diffop Jacobi-theta 3 1 0 𝑞 diffop Jacobi-theta 4 1 0 𝑞 {\displaystyle{\displaystyle\theta_{3}'\left(0,q\right)=\theta_{4}'\left(0,q% \right)}}
\Jacobithetaq{3}'@{0}{q} = \Jacobithetaq{4}'@{0}{q}		 

diff( JacobiTheta3(0, q), 0$(1) ) = diff( JacobiTheta4(0, q), 0$(1) )

D[EllipticTheta[3, 0, q], {0, 1}] == D[EllipticTheta[4, 0, q], {0, 1}]
Error Failure -
Failed [10 / 10]
Result: Plus[D[EllipticTheta[3, 0.0, Complex[0.8660254037844387, 0.49999999999999994]]
Test Values: {0.0, 1.0}], Times[-1.0, D[EllipticTheta[4, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], {0.0, 1.0}]]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[D[EllipticTheta[3, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]]
Test Values: {0.0, 1.0}], Times[-1.0, D[EllipticTheta[4, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]], {0.0, 1.0}]]], {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
20.4.E1 θ 4 ′ ⁡ ( 0 , q ) = 0 diffop Jacobi-theta 4 1 0 𝑞 0 {\displaystyle{\displaystyle\theta_{4}'\left(0,q\right)=0}}
\Jacobithetaq{4}'@{0}{q} = 0		 

diff( JacobiTheta4(0, q), 0$(1) ) = 0

D[EllipticTheta[4, 0, q], {0, 1}] == 0
Error Failure -
Failed [10 / 10]
Result: D[EllipticTheta[4, 0.0, Complex[0.8660254037844387, 0.49999999999999994]]
Test Values: {0.0, 1.0}], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: D[EllipticTheta[4, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]]
Test Values: {0.0, 1.0}], {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
20.4.E6 θ 1 ′ ⁡ ( 0 , q ) = θ 2 ⁡ ( 0 , q ) ⁢ θ 3 ⁡ ( 0 , q ) ⁢ θ 4 ⁡ ( 0 , q ) diffop Jacobi-theta 1 1 0 𝑞 Jacobi-theta 2 0 𝑞 Jacobi-theta 3 0 𝑞 Jacobi-theta 4 0 𝑞 {\displaystyle{\displaystyle\theta_{1}'\left(0,q\right)=\theta_{2}\left(0,q% \right)\theta_{3}\left(0,q\right)\theta_{4}\left(0,q\right)}}
\Jacobithetaq{1}'@{0}{q} = \Jacobithetaq{2}@{0}{q}\Jacobithetaq{3}@{0}{q}\Jacobithetaq{4}@{0}{q}		 

diff( JacobiTheta1(0, q), 0$(1) ) = JacobiTheta2(0, q)*JacobiTheta3(0, q)*JacobiTheta4(0, q)

D[EllipticTheta[1, 0, q], {0, 1}] == EllipticTheta[2, 0, q]*EllipticTheta[3, 0, q]*EllipticTheta[4, 0, q]
Error Failure -
Failed [10 / 10]
Result: Plus[D[0.0
Test Values: {0.0, 1.0}], Times[-1.0, EllipticTheta[2, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], EllipticTheta[3, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], EllipticTheta[4, 0.0, Complex[0.8660254037844387, 0.49999999999999994]]]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[D[0.0
Test Values: {0.0, 1.0}], Times[-1.0, EllipticTheta[2, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]], EllipticTheta[3, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]], EllipticTheta[4, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]]]], {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
20.4.E7 θ 1 ′′ ⁢ ( 0 , q ) = θ 2 ′′′ ⁡ ( 0 , q ) diffop Jacobi-theta 1 2 0 𝑞 diffop Jacobi-theta 2 3 0 𝑞 {\displaystyle{\displaystyle\theta_{1}''(0,q)=\theta_{2}'''\left(0,q\right)}}
\Jacobithetaq{1}''(0,q) = \Jacobithetaq{2}'''@{0}{q}		 

subs( temp=(0 , q) , diff( JacobiTheta1(temp, =), temp$(2) ) )*diff( JacobiTheta2(0, q), 0$(3) )

(D[EllipticTheta[1, temp, ==], {temp, 2}]/.temp-> (0 , q) )*D[EllipticTheta[2, 0, q], {0, 3}]
Translation Error Translation Error - Skip - symbolical successful subtest
20.4.E7 θ 2 ′′′ ⁡ ( 0 , q ) = θ 3 ′′′ ⁡ ( 0 , q ) diffop Jacobi-theta 2 3 0 𝑞 diffop Jacobi-theta 3 3 0 𝑞 {\displaystyle{\displaystyle\theta_{2}'''\left(0,q\right)=\theta_{3}'''\left(0% ,q\right)}}
\Jacobithetaq{2}'''@{0}{q} = \Jacobithetaq{3}'''@{0}{q}		 

diff( JacobiTheta2(0, q), 0$(3) ) = diff( JacobiTheta3(0, q), 0$(3) )

D[EllipticTheta[2, 0, q], {0, 3}] == D[EllipticTheta[3, 0, q], {0, 3}]
Error Failure -
Failed [10 / 10]
Result: Plus[D[EllipticTheta[2, 0.0, Complex[0.8660254037844387, 0.49999999999999994]]
Test Values: {0.0, 3.0}], Times[-1.0, D[EllipticTheta[3, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], {0.0, 3.0}]]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[D[EllipticTheta[2, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]]
Test Values: {0.0, 3.0}], Times[-1.0, D[EllipticTheta[3, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]], {0.0, 3.0}]]], {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
20.4.E7 θ 3 ′′′ ⁡ ( 0 , q ) = θ 4 ′′′ ⁡ ( 0 , q ) diffop Jacobi-theta 3 3 0 𝑞 diffop Jacobi-theta 4 3 0 𝑞 {\displaystyle{\displaystyle\theta_{3}'''\left(0,q\right)=\theta_{4}'''\left(0% ,q\right)}}
\Jacobithetaq{3}'''@{0}{q} = \Jacobithetaq{4}'''@{0}{q}		 

diff( JacobiTheta3(0, q), 0$(3) ) = diff( JacobiTheta4(0, q), 0$(3) )

D[EllipticTheta[3, 0, q], {0, 3}] == D[EllipticTheta[4, 0, q], {0, 3}]
Error Failure -
Failed [10 / 10]
Result: Plus[D[EllipticTheta[3, 0.0, Complex[0.8660254037844387, 0.49999999999999994]]
Test Values: {0.0, 3.0}], Times[-1.0, D[EllipticTheta[4, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], {0.0, 3.0}]]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[D[EllipticTheta[3, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]]
Test Values: {0.0, 3.0}], Times[-1.0, D[EllipticTheta[4, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]], {0.0, 3.0}]]], {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
20.4.E7 θ 4 ′′′ ⁡ ( 0 , q ) = 0 diffop Jacobi-theta 4 3 0 𝑞 0 {\displaystyle{\displaystyle\theta_{4}'''\left(0,q\right)=0}}
\Jacobithetaq{4}'''@{0}{q} = 0		 

diff( JacobiTheta4(0, q), 0$(3) ) = 0

D[EllipticTheta[4, 0, q], {0, 3}] == 0
Error Failure -
Failed [10 / 10]
Result: D[EllipticTheta[4, 0.0, Complex[0.8660254037844387, 0.49999999999999994]]
Test Values: {0.0, 3.0}], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: D[EllipticTheta[4, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]]
Test Values: {0.0, 3.0}], {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
20.4.E8 θ 1 ′′′ ⁡ ( 0 , q ) θ 1 ′ ⁡ ( 0 , q ) = - 1 + 24 ⁢ ∑ n = 1 ∞ q 2 ⁢ n ( 1 - q 2 ⁢ n ) 2 diffop Jacobi-theta 1 3 0 𝑞 diffop Jacobi-theta 1 1 0 𝑞 1 24 superscript subscript 𝑛 1 superscript 𝑞 2 𝑛 superscript 1 superscript 𝑞 2 𝑛 2 {\displaystyle{\displaystyle\frac{\theta_{1}'''\left(0,q\right)}{\theta_{1}'% \left(0,q\right)}=-1+24\sum_{n=1}^{\infty}\frac{q^{2n}}{(1-q^{2n})^{2}}}}
\frac{\Jacobithetaq{1}'''@{0}{q}}{\Jacobithetaq{1}'@{0}{q}} = -1+24\sum_{n=1}^{\infty}\frac{q^{2n}}{(1-q^{2n})^{2}}		 

(diff( JacobiTheta1(0, q), 0$(3) ))/(diff( JacobiTheta1(0, q), 0$(1) )) = - 1 + 24*sum(((q)^(2*n))/((1 - (q)^(2*n))^(2)), n = 1..infinity)

Divide[D[EllipticTheta[1, 0, q], {0, 3}],D[EllipticTheta[1, 0, q], {0, 1}]] == - 1 + 24*Sum[Divide[(q)^(2*n),(1 - (q)^(2*n))^(2)], {n, 1, Infinity}, GenerateConditions->None]
Error Failure -
Failed [10 / 10]
Result: Plus[1.0, Times[Power[D[0.0
Test Values: {0.0, 1.0}], -1], D[0.0, {0.0, 3.0}]], Times[-24.0, NSum[Times[Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Times[2, n]], Power[Plus[1, Times[-1, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Times[2, n]]]], -2]], {n, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[1.0, Times[Power[D[0.0
Test Values: {0.0, 1.0}], -1], D[0.0, {0.0, 3.0}]], Times[-24.0, NSum[Times[Power[Power[E, Times[Complex[0, Rational[2, 3]], Pi]], Times[2, n]], Power[Plus[1, Times[-1, Power[Power[E, Times[Complex[0, Rational[2, 3]], Pi]], Times[2, n]]]], -2]], {n, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
20.4.E9 θ 2 ′′ ⁡ ( 0 , q ) θ 2 ⁡ ( 0 , q ) = - 1 - 8 ⁢ ∑ n = 1 ∞ q 2 ⁢ n ( 1 + q 2 ⁢ n ) 2 diffop Jacobi-theta 2 2 0 𝑞 Jacobi-theta 2 0 𝑞 1 8 superscript subscript 𝑛 1 superscript 𝑞 2 𝑛 superscript 1 superscript 𝑞 2 𝑛 2 {\displaystyle{\displaystyle\frac{\theta_{2}''\left(0,q\right)}{\theta_{2}% \left(0,q\right)}=-1-8\sum_{n=1}^{\infty}\frac{q^{2n}}{(1+q^{2n})^{2}}}}
\frac{\Jacobithetaq{2}''@{0}{q}}{\Jacobithetaq{2}@{0}{q}} = -1-8\sum_{n=1}^{\infty}\frac{q^{2n}}{(1+q^{2n})^{2}}		 

(diff( JacobiTheta2(0, q), 0$(2) ))/(JacobiTheta2(0, q)) = - 1 - 8*sum(((q)^(2*n))/((1 + (q)^(2*n))^(2)), n = 1..infinity)

Divide[D[EllipticTheta[2, 0, q], {0, 2}],EllipticTheta[2, 0, q]] == - 1 - 8*Sum[Divide[(q)^(2*n),(1 + (q)^(2*n))^(2)], {n, 1, Infinity}, GenerateConditions->None]
Error Failure - Skipped - Because timed out
20.4.E10 θ 3 ′′ ⁡ ( 0 , q ) θ 3 ⁡ ( 0 , q ) = - 8 ⁢ ∑ n = 1 ∞ q 2 ⁢ n - 1 ( 1 + q 2 ⁢ n - 1 ) 2 diffop Jacobi-theta 3 2 0 𝑞 Jacobi-theta 3 0 𝑞 8 superscript subscript 𝑛 1 superscript 𝑞 2 𝑛 1 superscript 1 superscript 𝑞 2 𝑛 1 2 {\displaystyle{\displaystyle\frac{\theta_{3}''\left(0,q\right)}{\theta_{3}% \left(0,q\right)}=-8\sum_{n=1}^{\infty}\frac{q^{2n-1}}{(1+q^{2n-1})^{2}}}}
\frac{\Jacobithetaq{3}''@{0}{q}}{\Jacobithetaq{3}@{0}{q}} = -8\sum_{n=1}^{\infty}\frac{q^{2n-1}}{(1+q^{2n-1})^{2}}		 

(diff( JacobiTheta3(0, q), 0$(2) ))/(JacobiTheta3(0, q)) = - 8*sum(((q)^(2*n - 1))/((1 + (q)^(2*n - 1))^(2)), n = 1..infinity)

Divide[D[EllipticTheta[3, 0, q], {0, 2}],EllipticTheta[3, 0, q]] == - 8*Sum[Divide[(q)^(2*n - 1),(1 + (q)^(2*n - 1))^(2)], {n, 1, Infinity}, GenerateConditions->None]
Error Aborted - Skipped - Because timed out
20.4.E11 θ 4 ′′ ⁡ ( 0 , q ) θ 4 ⁡ ( 0 , q ) = 8 ⁢ ∑ n = 1 ∞ q 2 ⁢ n - 1 ( 1 - q 2 ⁢ n - 1 ) 2 diffop Jacobi-theta 4 2 0 𝑞 Jacobi-theta 4 0 𝑞 8 superscript subscript 𝑛 1 superscript 𝑞 2 𝑛 1 superscript 1 superscript 𝑞 2 𝑛 1 2 {\displaystyle{\displaystyle\frac{\theta_{4}''\left(0,q\right)}{\theta_{4}% \left(0,q\right)}=8\sum_{n=1}^{\infty}\frac{q^{2n-1}}{(1-q^{2n-1})^{2}}}}
\frac{\Jacobithetaq{4}''@{0}{q}}{\Jacobithetaq{4}@{0}{q}} = 8\sum_{n=1}^{\infty}\frac{q^{2n-1}}{(1-q^{2n-1})^{2}}		 

(diff( JacobiTheta4(0, q), 0$(2) ))/(JacobiTheta4(0, q)) = 8*sum(((q)^(2*n - 1))/((1 - (q)^(2*n - 1))^(2)), n = 1..infinity)

Divide[D[EllipticTheta[4, 0, q], {0, 2}],EllipticTheta[4, 0, q]] == 8*Sum[Divide[(q)^(2*n - 1),(1 - (q)^(2*n - 1))^(2)], {n, 1, Infinity}, GenerateConditions->None]
Error Aborted - Skipped - Because timed out
20.4.E12 θ 1 ′′′ ⁡ ( 0 , q ) θ 1 ′ ⁡ ( 0 , q ) = θ 2 ′′ ⁡ ( 0 , q ) θ 2 ⁡ ( 0 , q ) + θ 3 ′′ ⁡ ( 0 , q ) θ 3 ⁡ ( 0 , q ) + θ 4 ′′ ⁡ ( 0 , q ) θ 4 ⁡ ( 0 , q ) diffop Jacobi-theta 1 3 0 𝑞 diffop Jacobi-theta 1 1 0 𝑞 diffop Jacobi-theta 2 2 0 𝑞 Jacobi-theta 2 0 𝑞 diffop Jacobi-theta 3 2 0 𝑞 Jacobi-theta 3 0 𝑞 diffop Jacobi-theta 4 2 0 𝑞 Jacobi-theta 4 0 𝑞 {\displaystyle{\displaystyle\frac{\theta_{1}'''\left(0,q\right)}{\theta_{1}'% \left(0,q\right)}=\frac{\theta_{2}''\left(0,q\right)}{\theta_{2}\left(0,q% \right)}+\frac{\theta_{3}''\left(0,q\right)}{\theta_{3}\left(0,q\right)}+\frac% {\theta_{4}''\left(0,q\right)}{\theta_{4}\left(0,q\right)}}}
\frac{\Jacobithetaq{1}'''@{0}{q}}{\Jacobithetaq{1}'@{0}{q}} = \frac{\Jacobithetaq{2}''@{0}{q}}{\Jacobithetaq{2}@{0}{q}}+\frac{\Jacobithetaq{3}''@{0}{q}}{\Jacobithetaq{3}@{0}{q}}+\frac{\Jacobithetaq{4}''@{0}{q}}{\Jacobithetaq{4}@{0}{q}}		 

(diff( JacobiTheta1(0, q), 0$(3) ))/(diff( JacobiTheta1(0, q), 0$(1) )) = (diff( JacobiTheta2(0, q), 0$(2) ))/(JacobiTheta2(0, q))+(diff( JacobiTheta3(0, q), 0$(2) ))/(JacobiTheta3(0, q))+(diff( JacobiTheta4(0, q), 0$(2) ))/(JacobiTheta4(0, q))

Divide[D[EllipticTheta[1, 0, q], {0, 3}],D[EllipticTheta[1, 0, q], {0, 1}]] == Divide[D[EllipticTheta[2, 0, q], {0, 2}],EllipticTheta[2, 0, q]]+Divide[D[EllipticTheta[3, 0, q], {0, 2}],EllipticTheta[3, 0, q]]+Divide[D[EllipticTheta[4, 0, q], {0, 2}],EllipticTheta[4, 0, q]]
Error Failure -
Failed [10 / 10]
Result: Plus[Times[Power[D[0.0
Test Values: {0.0, 1.0}], -1], D[0.0, {0.0, 3.0}]], Times[-1.0, D[EllipticTheta[2, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], {0.0, 2.0}], Power[EllipticTheta[2, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], -1]], Times[-1.0, D[EllipticTheta[3, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], {0.0, 2.0}], Power[EllipticTheta[3, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], -1]], Times[-1.0, D[EllipticTheta[4, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], {0.0, 2.0}], Power[EllipticTheta[4, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], -1]]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Times[Power[D[0.0
Test Values: {0.0, 1.0}], -1], D[0.0, {0.0, 3.0}]], Times[-1.0, D[EllipticTheta[2, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]], {0.0, 2.0}], Power[EllipticTheta[2, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]], -1]], Times[-1.0, D[EllipticTheta[3, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]], {0.0, 2.0}], Power[EllipticTheta[3, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]], -1]], Times[-1.0, D[EllipticTheta[4, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]], {0.0, 2.0}], Power[EllipticTheta[4, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]], -1]]], {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
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