Properties

Label 722.2.c.n
Level $722$
Weight $2$
Character orbit 722.c
Analytic conductor $5.765$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 722 = 2 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 722.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.76519902594\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.324000000.2
Defining polynomial: \(x^{8} + 5 x^{6} + 20 x^{4} + 25 x^{2} + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 5 x^{6} + 20 x^{4} + 25 x^{2} + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} - 15 \)\()/20\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} - 35 \nu \)\()/20\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{6} + 4 \nu^{4} + 20 \nu^{2} + 5 \)\()/20\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} + 4 \nu^{5} + 20 \nu^{3} + 5 \nu \)\()/20\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{6} + 8 \nu^{4} + 20 \nu^{2} + 25 \)\()/20\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{7} + 6 \nu^{5} + 20 \nu^{3} + 25 \nu \)\()/10\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{6} + 2 \beta_{4} - \beta_{2}\)
\(\nu^{3}\)\(=\)\(-\beta_{7} + 3 \beta_{5} - \beta_{3}\)
\(\nu^{4}\)\(=\)\(5 \beta_{6} - 5 \beta_{4} - 5\)
\(\nu^{5}\)\(=\)\(5 \beta_{7} - 10 \beta_{5} - 10 \beta_{1}\)
\(\nu^{6}\)\(=\)\(20 \beta_{2} + 15\)
\(\nu^{7}\)\(=\)\(20 \beta_{3} + 35 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/722\mathbb{Z}\right)^\times\).

\(n\) \(363\)
\(\chi(n)\) \(\beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
429.1
−0.587785 + 1.01807i
0.951057 1.64728i
0.587785 1.01807i
−0.951057 + 1.64728i
−0.587785 1.01807i
0.951057 + 1.64728i
0.587785 + 1.01807i
−0.951057 1.64728i
0.500000 0.866025i −1.26007 + 2.18251i −0.500000 0.866025i 1.22982 2.13012i 1.26007 + 2.18251i −2.79360 −1.00000 −1.67557 2.90217i −1.22982 2.13012i
429.2 0.500000 0.866025i 0.221232 0.383185i −0.500000 0.866025i 0.445746 0.772054i −0.221232 0.383185i 2.52015 −1.00000 1.40211 + 2.42853i −0.445746 0.772054i
429.3 0.500000 0.866025i 0.642040 1.11205i −0.500000 0.866025i −1.84786 + 3.20059i −0.642040 1.11205i −0.442463 −1.00000 0.675571 + 1.17012i 1.84786 + 3.20059i
429.4 0.500000 0.866025i 1.39680 2.41933i −0.500000 0.866025i 1.17229 2.03046i −1.39680 2.41933i −1.28408 −1.00000 −2.40211 4.16058i −1.17229 2.03046i
653.1 0.500000 + 0.866025i −1.26007 2.18251i −0.500000 + 0.866025i 1.22982 + 2.13012i 1.26007 2.18251i −2.79360 −1.00000 −1.67557 + 2.90217i −1.22982 + 2.13012i
653.2 0.500000 + 0.866025i 0.221232 + 0.383185i −0.500000 + 0.866025i 0.445746 + 0.772054i −0.221232 + 0.383185i 2.52015 −1.00000 1.40211 2.42853i −0.445746 + 0.772054i
653.3 0.500000 + 0.866025i 0.642040 + 1.11205i −0.500000 + 0.866025i −1.84786 3.20059i −0.642040 + 1.11205i −0.442463 −1.00000 0.675571 1.17012i 1.84786 3.20059i
653.4 0.500000 + 0.866025i 1.39680 + 2.41933i −0.500000 + 0.866025i 1.17229 + 2.03046i −1.39680 + 2.41933i −1.28408 −1.00000 −2.40211 + 4.16058i −1.17229 + 2.03046i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 653.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 722.2.c.n 8
19.b odd 2 1 722.2.c.m 8
19.c even 3 1 722.2.a.m 4
19.c even 3 1 inner 722.2.c.n 8
19.d odd 6 1 722.2.a.n yes 4
19.d odd 6 1 722.2.c.m 8
19.e even 9 6 722.2.e.s 24
19.f odd 18 6 722.2.e.r 24
57.f even 6 1 6498.2.a.bx 4
57.h odd 6 1 6498.2.a.ca 4
76.f even 6 1 5776.2.a.bt 4
76.g odd 6 1 5776.2.a.bv 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
722.2.a.m 4 19.c even 3 1
722.2.a.n yes 4 19.d odd 6 1
722.2.c.m 8 19.b odd 2 1
722.2.c.m 8 19.d odd 6 1
722.2.c.n 8 1.a even 1 1 trivial
722.2.c.n 8 19.c even 3 1 inner
722.2.e.r 24 19.f odd 18 6
722.2.e.s 24 19.e even 9 6
5776.2.a.bt 4 76.f even 6 1
5776.2.a.bv 4 76.g odd 6 1
6498.2.a.bx 4 57.f even 6 1
6498.2.a.ca 4 57.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(722, [\chi])\):

\(T_{3}^{8} - \cdots\)
\(T_{5}^{8} - \cdots\)
\( T_{7}^{4} + 2 T_{7}^{3} - 6 T_{7}^{2} - 12 T_{7} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{4} \)
$3$ \( 16 - 48 T + 120 T^{2} - 88 T^{3} + 64 T^{4} - 12 T^{5} + 10 T^{6} - 2 T^{7} + T^{8} \)
$5$ \( 361 - 608 T + 815 T^{2} - 428 T^{3} + 204 T^{4} - 42 T^{5} + 15 T^{6} - 2 T^{7} + T^{8} \)
$7$ \( ( -4 - 12 T - 6 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$11$ \( ( 76 + 12 T - 26 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$13$ \( 3721 - 14152 T + 47175 T^{2} - 23092 T^{3} + 7644 T^{4} - 1498 T^{5} + 215 T^{6} - 18 T^{7} + T^{8} \)
$17$ \( 361 + 1406 T + 5305 T^{2} + 894 T^{3} + 544 T^{4} + 94 T^{5} + 45 T^{6} + 6 T^{7} + T^{8} \)
$19$ \( T^{8} \)
$23$ \( 1488400 - 805200 T + 374600 T^{2} - 57400 T^{3} + 10320 T^{4} - 820 T^{5} + 150 T^{6} - 10 T^{7} + T^{8} \)
$29$ \( 361 + 1748 T + 7875 T^{2} + 2928 T^{3} + 1164 T^{4} + 122 T^{5} + 35 T^{6} + 2 T^{7} + T^{8} \)
$31$ \( ( 1436 + 996 T + 246 T^{2} + 26 T^{3} + T^{4} )^{2} \)
$37$ \( ( -19 - 46 T - 19 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$41$ \( 128881 + 54568 T + 29925 T^{2} + 5728 T^{3} + 2544 T^{4} + 532 T^{5} + 125 T^{6} + 12 T^{7} + T^{8} \)
$43$ \( 10000 - 10000 T + 11000 T^{2} - 1000 T^{3} + 1200 T^{4} - 300 T^{5} + 90 T^{6} - 10 T^{7} + T^{8} \)
$47$ \( 5776 + 8512 T + 12240 T^{2} + 2272 T^{3} + 1284 T^{4} - 272 T^{5} + 140 T^{6} - 12 T^{7} + T^{8} \)
$53$ \( 32761 + 17738 T + 15215 T^{2} - 142 T^{3} + 1564 T^{4} + 52 T^{5} + 95 T^{6} - 8 T^{7} + T^{8} \)
$59$ \( 26896 + 14432 T + 8400 T^{2} + 2272 T^{3} + 884 T^{4} + 208 T^{5} + 60 T^{6} + 8 T^{7} + T^{8} \)
$61$ \( 1050625 - 153750 T + 140375 T^{2} + 17250 T^{3} + 12200 T^{4} + 300 T^{5} + 115 T^{6} + T^{8} \)
$67$ \( 400 - 2800 T + 19000 T^{2} - 4600 T^{3} + 2320 T^{4} + 20 T^{5} + 130 T^{6} - 10 T^{7} + T^{8} \)
$71$ \( 400 + 7200 T + 127600 T^{2} + 36000 T^{3} + 10020 T^{4} + 720 T^{5} + 100 T^{6} + T^{8} \)
$73$ \( 19321 - 834 T + 7125 T^{2} - 3586 T^{3} + 2824 T^{4} - 726 T^{5} + 145 T^{6} - 14 T^{7} + T^{8} \)
$79$ \( 22316176 - 3457968 T + 979880 T^{2} - 139048 T^{3} + 29664 T^{4} - 3532 T^{5} + 390 T^{6} - 22 T^{7} + T^{8} \)
$83$ \( ( -6884 - 2832 T - 196 T^{2} + 12 T^{3} + T^{4} )^{2} \)
$89$ \( 1343281 + 839116 T + 490565 T^{2} + 58084 T^{3} + 13584 T^{4} + 984 T^{5} + 285 T^{6} + 16 T^{7} + T^{8} \)
$97$ \( 63664441 - 15543092 T + 4584625 T^{2} - 253972 T^{3} + 72324 T^{4} - 6668 T^{5} + 685 T^{6} - 28 T^{7} + T^{8} \)
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