Properties

Label 930.2.j.c
Level $930$
Weight $2$
Character orbit 930.j
Analytic conductor $7.426$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.1698758656.6
Defining polynomial: \(x^{8} + 18 x^{6} + 97 x^{4} + 176 x^{2} + 64\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( -1 - \beta_{1} - \beta_{7} ) q^{3} -\beta_{7} q^{4} + ( \beta_{3} - \beta_{4} + \beta_{5} ) q^{5} + ( \beta_{1} - \beta_{2} - \beta_{7} ) q^{6} + ( \beta_{2} + \beta_{4} + \beta_{6} + \beta_{7} ) q^{7} -\beta_{2} q^{8} + ( 2 \beta_{1} - 2 \beta_{2} + \beta_{7} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( -1 - \beta_{1} - \beta_{7} ) q^{3} -\beta_{7} q^{4} + ( \beta_{3} - \beta_{4} + \beta_{5} ) q^{5} + ( \beta_{1} - \beta_{2} - \beta_{7} ) q^{6} + ( \beta_{2} + \beta_{4} + \beta_{6} + \beta_{7} ) q^{7} -\beta_{2} q^{8} + ( 2 \beta_{1} - 2 \beta_{2} + \beta_{7} ) q^{9} + ( \beta_{2} - \beta_{4} - \beta_{7} ) q^{10} + ( \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} ) q^{11} + ( -1 - \beta_{2} + \beta_{7} ) q^{12} + ( -1 + 2 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{6} ) q^{13} + ( 1 + \beta_{3} + \beta_{5} + \beta_{6} ) q^{14} + ( 1 + \beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{15} - q^{16} + ( 2 \beta_{1} + \beta_{3} - \beta_{4} + 4 \beta_{5} + \beta_{6} - \beta_{7} ) q^{17} + ( -2 + \beta_{2} + 2 \beta_{7} ) q^{18} + ( -3 \beta_{1} - 3 \beta_{2} - \beta_{4} + \beta_{5} ) q^{19} + ( 1 - \beta_{3} - \beta_{6} ) q^{20} + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{21} + ( 1 + \beta_{2} - \beta_{4} - \beta_{6} ) q^{22} + ( -2 + 3 \beta_{2} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{23} + ( -1 + \beta_{1} + \beta_{2} ) q^{24} + ( -1 - 4 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} ) q^{25} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} + 2 \beta_{7} ) q^{26} + ( -1 + 5 \beta_{2} + \beta_{7} ) q^{27} + ( -\beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{28} + ( 3 \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{5} - 2 \beta_{6} ) q^{29} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{30} + q^{31} + \beta_{1} q^{32} + ( 2 - 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{5} - 2 \beta_{6} ) q^{33} + ( 3 \beta_{3} - 4 \beta_{4} + \beta_{5} - 2 \beta_{7} ) q^{34} + ( 2 - \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{35} + ( 1 + 2 \beta_{1} + 2 \beta_{2} ) q^{36} + ( -2 - 2 \beta_{7} ) q^{37} + ( -3 + \beta_{2} - \beta_{4} - \beta_{6} - 4 \beta_{7} ) q^{38} + ( 2 - \beta_{1} + 3 \beta_{2} - \beta_{3} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{39} + ( -\beta_{1} - \beta_{5} - \beta_{6} ) q^{40} + ( 3 \beta_{1} + 3 \beta_{2} + \beta_{3} - \beta_{5} + 4 \beta_{7} ) q^{41} + ( -1 - \beta_{1} - 2 \beta_{4} + \beta_{5} - 3 \beta_{7} ) q^{42} + ( \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{43} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} ) q^{44} + ( -1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{45} + ( 3 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} ) q^{46} + ( 3 + 4 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{6} - 4 \beta_{7} ) q^{47} + ( 1 + \beta_{1} + \beta_{7} ) q^{48} + ( 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + 5 \beta_{4} - 4 \beta_{5} + 3 \beta_{7} ) q^{49} + ( -4 + \beta_{1} - \beta_{3} + \beta_{5} ) q^{50} + ( -2 \beta_{1} + 2 \beta_{2} - \beta_{3} - 4 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{51} + ( 1 + 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} ) q^{52} + ( -1 - \beta_{2} + 4 \beta_{3} - \beta_{4} - \beta_{6} - 2 \beta_{7} ) q^{53} + ( 5 + \beta_{1} + \beta_{2} ) q^{54} + ( -3 + 3 \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{55} + ( -\beta_{4} + \beta_{5} - 2 \beta_{7} ) q^{56} + ( -2 + 6 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{5} - 2 \beta_{6} - 4 \beta_{7} ) q^{57} + ( -3 - \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} + 4 \beta_{7} ) q^{58} + ( 4 + \beta_{3} + \beta_{5} - 2 \beta_{6} ) q^{59} + ( -1 - \beta_{1} + \beta_{4} - 2 \beta_{5} ) q^{60} + ( 8 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{5} - 2 \beta_{6} ) q^{61} -\beta_{1} q^{62} + ( -2 + \beta_{1} - 3 \beta_{3} - \beta_{4} - \beta_{5} - 3 \beta_{6} - 3 \beta_{7} ) q^{63} + \beta_{7} q^{64} + ( -5 - 4 \beta_{1} - 2 \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{6} + 2 \beta_{7} ) q^{65} + ( 1 - 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{66} + ( -2 - 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{7} ) q^{67} + ( 2 \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{68} + ( 3 - \beta_{1} - 5 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{69} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - 3 \beta_{7} ) q^{70} + ( -3 \beta_{1} - 3 \beta_{2} - \beta_{4} + \beta_{5} + 6 \beta_{7} ) q^{71} + ( 2 - \beta_{1} + 2 \beta_{7} ) q^{72} + ( -5 + \beta_{1} + \beta_{3} - \beta_{4} + 3 \beta_{5} + \beta_{6} + 4 \beta_{7} ) q^{73} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{74} + ( -3 + 5 \beta_{1} + 4 \beta_{2} - \beta_{3} + \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{75} + ( 1 + 3 \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} ) q^{76} + ( 1 - 3 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{77} + ( 3 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{78} + ( 5 \beta_{1} + 5 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + \beta_{5} - 4 \beta_{7} ) q^{79} + ( -\beta_{3} + \beta_{4} - \beta_{5} ) q^{80} + ( 7 - 4 \beta_{1} - 4 \beta_{2} ) q^{81} + ( 3 + 4 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} + 4 \beta_{7} ) q^{82} + ( -1 - 10 \beta_{2} + 3 \beta_{3} - \beta_{4} - \beta_{6} - 2 \beta_{7} ) q^{83} + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{84} + ( 4 + 4 \beta_{1} - 10 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + \beta_{5} - 10 \beta_{7} ) q^{85} + ( \beta_{3} + \beta_{4} - 2 \beta_{5} + 2 \beta_{7} ) q^{86} + ( -3 + 6 \beta_{2} - \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + \beta_{6} + 6 \beta_{7} ) q^{87} + ( 1 - \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{88} + ( 5 - \beta_{6} ) q^{89} + ( -2 + \beta_{1} + 2 \beta_{4} - \beta_{5} + 3 \beta_{6} ) q^{90} + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - 2 \beta_{6} ) q^{91} + ( -2 - 3 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{92} + ( -1 - \beta_{1} - \beta_{7} ) q^{93} + ( -3 \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{5} + 4 \beta_{7} ) q^{94} + ( -3 - \beta_{1} + \beta_{2} + \beta_{3} - 4 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} - 6 \beta_{7} ) q^{95} + ( -\beta_{1} + \beta_{2} + \beta_{7} ) q^{96} + ( 2 + 8 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{97} + ( 2 - 2 \beta_{2} + \beta_{3} + 4 \beta_{4} + 4 \beta_{6} + 6 \beta_{7} ) q^{98} + ( -1 - \beta_{1} - 3 \beta_{2} - \beta_{3} + 4 \beta_{4} - \beta_{5} + \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{3} + 4q^{7} + O(q^{10}) \) \( 8q - 8q^{3} + 4q^{7} - 8q^{12} - 4q^{13} + 12q^{14} + 4q^{15} - 8q^{16} + 4q^{17} - 16q^{18} + 4q^{20} + 12q^{21} + 4q^{22} - 12q^{23} - 8q^{24} - 4q^{25} - 8q^{27} + 4q^{28} - 8q^{29} + 8q^{31} + 8q^{33} + 24q^{35} + 8q^{36} - 16q^{37} - 28q^{38} + 8q^{39} - 4q^{40} - 8q^{42} - 8q^{43} + 4q^{44} - 12q^{45} + 28q^{46} + 28q^{47} + 8q^{48} - 32q^{50} - 8q^{51} + 4q^{52} - 12q^{53} + 40q^{54} - 20q^{55} - 24q^{57} - 28q^{58} + 24q^{59} - 8q^{60} + 56q^{61} - 28q^{63} - 36q^{65} + 4q^{66} - 16q^{67} - 4q^{68} + 28q^{69} + 20q^{70} + 16q^{72} - 36q^{73} - 32q^{75} + 4q^{76} + 12q^{77} + 20q^{78} + 56q^{81} + 28q^{82} - 12q^{83} - 8q^{84} + 32q^{85} - 20q^{87} + 4q^{88} + 36q^{89} - 4q^{90} + 8q^{91} - 12q^{92} - 8q^{93} - 36q^{95} + 12q^{97} + 32q^{98} - 4q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 18 x^{6} + 97 x^{4} + 176 x^{2} + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{7} + 2 \nu^{6} + 18 \nu^{5} + 28 \nu^{4} + 89 \nu^{3} + 74 \nu^{2} + 104 \nu - 16 \)\()/64\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} - 2 \nu^{6} + 18 \nu^{5} - 28 \nu^{4} + 89 \nu^{3} - 74 \nu^{2} + 104 \nu + 16 \)\()/64\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} - 2 \nu^{6} - 10 \nu^{5} - 20 \nu^{4} + 15 \nu^{3} - 2 \nu^{2} + 120 \nu + 80 \)\()/64\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} - 2 \nu^{6} - 10 \nu^{5} - 20 \nu^{4} + 15 \nu^{3} - 2 \nu^{2} + 184 \nu + 80 \)\()/64\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} - 2 \nu^{6} + 10 \nu^{5} - 20 \nu^{4} - 15 \nu^{3} - 2 \nu^{2} - 120 \nu + 80 \)\()/64\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{6} + 14 \nu^{4} + 45 \nu^{2} + 32 \)\()/8\)
\(\beta_{7}\)\(=\)\((\)\( 3 \nu^{7} + 46 \nu^{5} + 179 \nu^{3} + 168 \nu \)\()/64\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{4} - \beta_{3}\)
\(\nu^{2}\)\(=\)\(\beta_{6} + 2 \beta_{2} - 2 \beta_{1} - 5\)
\(\nu^{3}\)\(=\)\(4 \beta_{7} - 2 \beta_{5} - 5 \beta_{4} + 7 \beta_{3} - 4 \beta_{2} - 4 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-9 \beta_{6} + 4 \beta_{5} + 4 \beta_{3} - 22 \beta_{2} + 22 \beta_{1} + 37\)
\(\nu^{5}\)\(=\)\(-52 \beta_{7} + 22 \beta_{5} + 37 \beta_{4} - 59 \beta_{3} + 56 \beta_{2} + 56 \beta_{1}\)
\(\nu^{6}\)\(=\)\(89 \beta_{6} - 56 \beta_{5} - 56 \beta_{3} + 218 \beta_{2} - 218 \beta_{1} - 325\)
\(\nu^{7}\)\(=\)\(580 \beta_{7} - 218 \beta_{5} - 325 \beta_{4} + 543 \beta_{3} - 620 \beta_{2} - 620 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/930\mathbb{Z}\right)^\times\).

\(n\) \(187\) \(311\) \(871\)
\(\chi(n)\) \(\beta_{7}\) \(-1\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
497.1
2.16053i
3.16053i
1.69230i
0.692297i
2.16053i
3.16053i
1.69230i
0.692297i
−0.707107 + 0.707107i −1.70711 0.292893i 1.00000i −2.23483 + 0.0743018i 1.41421 1.00000i 0.218591 + 0.218591i 0.707107 + 0.707107i 2.82843 + 1.00000i 1.52773 1.63280i
497.2 −0.707107 + 0.707107i −1.70711 0.292893i 1.00000i 1.52773 + 1.63280i 1.41421 1.00000i −1.33991 1.33991i 0.707107 + 0.707107i 2.82843 + 1.00000i −2.23483 0.0743018i
497.3 0.707107 0.707107i −0.292893 1.70711i 1.00000i −0.489528 + 2.18183i −1.41421 1.00000i −0.474719 0.474719i −0.707107 0.707107i −2.82843 + 1.00000i 1.19663 + 1.88893i
497.4 0.707107 0.707107i −0.292893 1.70711i 1.00000i 1.19663 1.88893i −1.41421 1.00000i 3.59604 + 3.59604i −0.707107 0.707107i −2.82843 + 1.00000i −0.489528 2.18183i
683.1 −0.707107 0.707107i −1.70711 + 0.292893i 1.00000i −2.23483 0.0743018i 1.41421 + 1.00000i 0.218591 0.218591i 0.707107 0.707107i 2.82843 1.00000i 1.52773 + 1.63280i
683.2 −0.707107 0.707107i −1.70711 + 0.292893i 1.00000i 1.52773 1.63280i 1.41421 + 1.00000i −1.33991 + 1.33991i 0.707107 0.707107i 2.82843 1.00000i −2.23483 + 0.0743018i
683.3 0.707107 + 0.707107i −0.292893 + 1.70711i 1.00000i −0.489528 2.18183i −1.41421 + 1.00000i −0.474719 + 0.474719i −0.707107 + 0.707107i −2.82843 1.00000i 1.19663 1.88893i
683.4 0.707107 + 0.707107i −0.292893 + 1.70711i 1.00000i 1.19663 + 1.88893i −1.41421 + 1.00000i 3.59604 3.59604i −0.707107 + 0.707107i −2.82843 1.00000i −0.489528 + 2.18183i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 683.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.j.c 8
3.b odd 2 1 930.2.j.f yes 8
5.c odd 4 1 930.2.j.f yes 8
15.e even 4 1 inner 930.2.j.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.j.c 8 1.a even 1 1 trivial
930.2.j.c 8 15.e even 4 1 inner
930.2.j.f yes 8 3.b odd 2 1
930.2.j.f yes 8 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\):

\(T_{7}^{8} - \cdots\)
\( T_{11}^{8} + 22 T_{11}^{6} + 97 T_{11}^{4} + 144 T_{11}^{2} + 64 \)
\(T_{17}^{8} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{4} )^{2} \)
$3$ \( ( 9 + 12 T + 8 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$5$ \( 625 + 50 T^{2} + 80 T^{3} + 2 T^{4} + 16 T^{5} + 2 T^{6} + T^{8} \)
$7$ \( 4 - 8 T + 8 T^{2} + 52 T^{3} + 117 T^{4} + 48 T^{5} + 8 T^{6} - 4 T^{7} + T^{8} \)
$11$ \( 64 + 144 T^{2} + 97 T^{4} + 22 T^{6} + T^{8} \)
$13$ \( 3136 + 6272 T + 6272 T^{2} + 2688 T^{3} + 596 T^{4} + 24 T^{5} + 8 T^{6} + 4 T^{7} + T^{8} \)
$17$ \( 3717184 + 339328 T + 15488 T^{2} - 7424 T^{3} + 3540 T^{4} + 168 T^{5} + 8 T^{6} - 4 T^{7} + T^{8} \)
$19$ \( 107584 + 53776 T^{2} + 4417 T^{4} + 118 T^{6} + T^{8} \)
$23$ \( 37636 + 42680 T + 24200 T^{2} + 3868 T^{3} + 437 T^{4} + 136 T^{5} + 72 T^{6} + 12 T^{7} + T^{8} \)
$29$ \( ( 248 + 8 T - 62 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$31$ \( ( -1 + T )^{8} \)
$37$ \( ( 8 + 4 T + T^{2} )^{4} \)
$41$ \( 1048576 + 208896 T^{2} + 11140 T^{4} + 196 T^{6} + T^{8} \)
$43$ \( 85264 - 56064 T + 18432 T^{2} + 4768 T^{3} + 785 T^{4} - 104 T^{5} + 32 T^{6} + 8 T^{7} + T^{8} \)
$47$ \( 16384 + 16384 T + 8192 T^{2} - 16128 T^{3} + 9860 T^{4} - 2616 T^{5} + 392 T^{6} - 28 T^{7} + T^{8} \)
$53$ \( 15376 - 75392 T + 184832 T^{2} + 68432 T^{3} + 12977 T^{4} - 772 T^{5} + 72 T^{6} + 12 T^{7} + T^{8} \)
$59$ \( ( -1024 + 512 T - 30 T^{2} - 12 T^{3} + T^{4} )^{2} \)
$61$ \( ( -7792 + 720 T + 186 T^{2} - 28 T^{3} + T^{4} )^{2} \)
$67$ \( 2408704 - 1489920 T + 460800 T^{2} + 59648 T^{3} + 4640 T^{4} - 448 T^{5} + 128 T^{6} + 16 T^{7} + T^{8} \)
$71$ \( 583696 + 369784 T^{2} + 22561 T^{4} + 286 T^{6} + T^{8} \)
$73$ \( 38416 + 120736 T + 189728 T^{2} + 114296 T^{3} + 38417 T^{4} + 6476 T^{5} + 648 T^{6} + 36 T^{7} + T^{8} \)
$79$ \( 602176 + 346288 T^{2} + 20289 T^{4} + 278 T^{6} + T^{8} \)
$83$ \( 18800896 + 15193344 T + 6139008 T^{2} + 815904 T^{3} + 56196 T^{4} + 888 T^{5} + 72 T^{6} + 12 T^{7} + T^{8} \)
$89$ \( ( 248 - 288 T + 113 T^{2} - 18 T^{3} + T^{4} )^{2} \)
$97$ \( 4129024 + 2178304 T + 574592 T^{2} - 102112 T^{3} + 9860 T^{4} + 344 T^{5} + 72 T^{6} - 12 T^{7} + T^{8} \)
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