Properties

Label 115.2.b.b
Level $115$
Weight $2$
Character orbit 115.b
Analytic conductor $0.918$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 115.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.918279623245\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.527896576.2
Defining polynomial: \(x^{8} - 2 x^{7} + 2 x^{6} + 2 x^{5} + 7 x^{4} - 10 x^{3} + 8 x^{2} + 4 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} + 2 x^{6} + 2 x^{5} + 7 x^{4} - 10 x^{3} + 8 x^{2} + 4 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -27 \nu^{7} + 31 \nu^{6} + 7 \nu^{5} - 221 \nu^{4} - 187 \nu^{3} + 128 \nu^{2} + 66 \nu - 882 \)\()/467\)
\(\beta_{3}\)\(=\)\((\)\( 52 \nu^{7} - 77 \nu^{6} + 73 \nu^{5} + 97 \nu^{4} + 585 \nu^{3} - 333 \nu^{2} + 288 \nu + 142 \)\()/467\)
\(\beta_{4}\)\(=\)\((\)\( 138 \nu^{7} - 366 \nu^{6} + 535 \nu^{5} - 12 \nu^{4} + 852 \nu^{3} - 1692 \nu^{2} + 2309 \nu - 162 \)\()/467\)
\(\beta_{5}\)\(=\)\((\)\( -142 \nu^{7} + 336 \nu^{6} - 361 \nu^{5} - 211 \nu^{4} - 897 \nu^{3} + 2005 \nu^{2} - 1469 \nu - 280 \)\()/467\)
\(\beta_{6}\)\(=\)\((\)\( 273 \nu^{7} - 521 \nu^{6} + 500 \nu^{5} + 626 \nu^{4} + 1787 \nu^{3} - 2332 \nu^{2} + 1979 \nu + 979 \)\()/467\)
\(\beta_{7}\)\(=\)\((\)\( -284 \nu^{7} + 672 \nu^{6} - 722 \nu^{5} - 422 \nu^{4} - 1794 \nu^{3} + 3543 \nu^{2} - 2938 \nu - 560 \)\()/467\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{7} + 2 \beta_{5}\)
\(\nu^{3}\)\(=\)\(-\beta_{7} - \beta_{6} + \beta_{5} + 2 \beta_{3} - \beta_{2} - 1\)
\(\nu^{4}\)\(=\)\(-\beta_{6} + \beta_{4} - 5 \beta_{2} - 7\)
\(\nu^{5}\)\(=\)\(6 \beta_{7} - 6 \beta_{5} + 5 \beta_{4} - 6 \beta_{2} - 5 \beta_{1} - 6\)
\(\nu^{6}\)\(=\)\(22 \beta_{7} + 6 \beta_{6} - 27 \beta_{5} + 6 \beta_{4} - \beta_{3} - \beta_{1}\)
\(\nu^{7}\)\(=\)\(29 \beta_{7} + 22 \beta_{6} - 30 \beta_{5} - 15 \beta_{3} + 29 \beta_{2} + 30\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/115\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1
1.47984 + 1.47984i
−0.199724 + 0.199724i
0.790245 0.790245i
−1.07037 1.07037i
−1.07037 + 1.07037i
0.790245 + 0.790245i
−0.199724 0.199724i
1.47984 1.47984i
2.37988i 1.95969i −3.66382 0.479844 + 2.18398i −4.66382 2.28394i 3.95969i −0.840379 5.19760 1.14197i
24.2 1.92022i 1.39945i −1.68725 −1.19972 1.88697i −2.68725 4.60747i 0.600553i 1.04155 −3.62340 + 2.30373i
24.3 0.751024i 0.580491i 1.43596 −0.209755 + 2.22621i 0.435963 0.315061i 2.58049i 2.66303 1.67194 + 0.157531i
24.4 0.291367i 3.14073i 1.91511 −2.07037 0.844739i 0.915105 1.20647i 1.14073i −6.86420 −0.246129 + 0.603236i
24.5 0.291367i 3.14073i 1.91511 −2.07037 + 0.844739i 0.915105 1.20647i 1.14073i −6.86420 −0.246129 0.603236i
24.6 0.751024i 0.580491i 1.43596 −0.209755 2.22621i 0.435963 0.315061i 2.58049i 2.66303 1.67194 0.157531i
24.7 1.92022i 1.39945i −1.68725 −1.19972 + 1.88697i −2.68725 4.60747i 0.600553i 1.04155 −3.62340 2.30373i
24.8 2.37988i 1.95969i −3.66382 0.479844 2.18398i −4.66382 2.28394i 3.95969i −0.840379 5.19760 + 1.14197i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 24.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 115.2.b.b 8
3.b odd 2 1 1035.2.b.e 8
4.b odd 2 1 1840.2.e.d 8
5.b even 2 1 inner 115.2.b.b 8
5.c odd 4 1 575.2.a.i 4
5.c odd 4 1 575.2.a.j 4
15.d odd 2 1 1035.2.b.e 8
15.e even 4 1 5175.2.a.bv 4
15.e even 4 1 5175.2.a.bw 4
20.d odd 2 1 1840.2.e.d 8
20.e even 4 1 9200.2.a.ck 4
20.e even 4 1 9200.2.a.cq 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.2.b.b 8 1.a even 1 1 trivial
115.2.b.b 8 5.b even 2 1 inner
575.2.a.i 4 5.c odd 4 1
575.2.a.j 4 5.c odd 4 1
1035.2.b.e 8 3.b odd 2 1
1035.2.b.e 8 15.d odd 2 1
1840.2.e.d 8 4.b odd 2 1
1840.2.e.d 8 20.d odd 2 1
5175.2.a.bv 4 15.e even 4 1
5175.2.a.bw 4 15.e even 4 1
9200.2.a.ck 4 20.e even 4 1
9200.2.a.cq 4 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 10 T_{2}^{6} + 27 T_{2}^{4} + 14 T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(115, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 14 T^{2} + 27 T^{4} + 10 T^{6} + T^{8} \)
$3$ \( 25 + 96 T^{2} + 70 T^{4} + 16 T^{6} + T^{8} \)
$5$ \( 625 + 750 T + 650 T^{2} + 410 T^{3} + 206 T^{4} + 82 T^{5} + 26 T^{6} + 6 T^{7} + T^{8} \)
$7$ \( 16 + 176 T^{2} + 152 T^{4} + 28 T^{6} + T^{8} \)
$11$ \( ( -28 + 44 T - 16 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$13$ \( 3721 + 5872 T^{2} + 1174 T^{4} + 64 T^{6} + T^{8} \)
$17$ \( 400 + 1776 T^{2} + 1612 T^{4} + 80 T^{6} + T^{8} \)
$19$ \( ( -20 + 28 T - 6 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$23$ \( ( 1 + T^{2} )^{4} \)
$29$ \( ( 5 - 4 T - 22 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$31$ \( ( -167 - 256 T - 74 T^{2} + T^{4} )^{2} \)
$37$ \( 226576 + 75856 T^{2} + 5752 T^{4} + 148 T^{6} + T^{8} \)
$41$ \( ( 2485 - 368 T - 94 T^{2} + 8 T^{3} + T^{4} )^{2} \)
$43$ \( 3857296 + 653600 T^{2} + 21340 T^{4} + 252 T^{6} + T^{8} \)
$47$ \( 20367169 + 1273144 T^{2} + 28734 T^{4} + 280 T^{6} + T^{8} \)
$53$ \( 4804864 + 468480 T^{2} + 16096 T^{4} + 224 T^{6} + T^{8} \)
$59$ \( ( 16 - 16 T - 20 T^{2} + T^{4} )^{2} \)
$61$ \( ( -2756 - 1148 T - 102 T^{2} + 8 T^{3} + T^{4} )^{2} \)
$67$ \( 16 + 208 T^{2} + 120 T^{4} + 20 T^{6} + T^{8} \)
$71$ \( ( -7435 - 1568 T + 66 T^{2} + 24 T^{3} + T^{4} )^{2} \)
$73$ \( 69538921 + 3070400 T^{2} + 50550 T^{4} + 368 T^{6} + T^{8} \)
$79$ \( ( 28 + 412 T + 178 T^{2} + 24 T^{3} + T^{4} )^{2} \)
$83$ \( 223442704 + 9024496 T^{2} + 114508 T^{4} + 576 T^{6} + T^{8} \)
$89$ \( ( 2380 + 372 T - 86 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$97$ \( 21864976 + 3638896 T^{2} + 69592 T^{4} + 460 T^{6} + T^{8} \)
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