Properties

Label 125.2.e.b
Level $125$
Weight $2$
Character orbit 125.e
Analytic conductor $0.998$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 125 = 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 125.e (of order \(10\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.998130025266\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: 8.0.58140625.2
Defining polynomial: \(x^{8} - 3 x^{7} + 4 x^{6} - 7 x^{5} + 11 x^{4} + 5 x^{3} - 10 x^{2} - 25 x + 25\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 25)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{7} + 4 x^{6} - 7 x^{5} + 11 x^{4} + 5 x^{3} - 10 x^{2} - 25 x + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 406 \nu^{7} - 714 \nu^{6} + 747 \nu^{5} - 1896 \nu^{4} + 2103 \nu^{3} + 4949 \nu^{2} + 1065 \nu - 7800 \)\()/1355\)
\(\beta_{3}\)\(=\)\((\)\( 420 \nu^{7} - 776 \nu^{6} + 698 \nu^{5} - 1924 \nu^{4} + 2297 \nu^{3} + 5129 \nu^{2} + 1055 \nu - 10265 \)\()/1355\)
\(\beta_{4}\)\(=\)\((\)\( 728 \nu^{7} - 1327 \nu^{6} + 1246 \nu^{5} - 3353 \nu^{4} + 3584 \nu^{3} + 8547 \nu^{2} + 2190 \nu - 15715 \)\()/1355\)
\(\beta_{5}\)\(=\)\((\)\( -857 \nu^{7} + 1666 \nu^{6} - 1743 \nu^{5} + 4424 \nu^{4} - 4907 \nu^{3} - 9470 \nu^{2} - 2485 \nu + 18200 \)\()/1355\)
\(\beta_{6}\)\(=\)\((\)\( 891 \nu^{7} - 1623 \nu^{6} + 1624 \nu^{5} - 4492 \nu^{4} + 4991 \nu^{3} + 9520 \nu^{2} + 3235 \nu - 18960 \)\()/1355\)
\(\beta_{7}\)\(=\)\((\)\( 955 \nu^{7} - 1829 \nu^{6} + 1942 \nu^{5} - 4891 \nu^{4} + 5723 \nu^{3} + 9646 \nu^{2} + 2415 \nu - 20550 \)\()/1355\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{7} + \beta_{5} - 3 \beta_{4} + 4 \beta_{3} + \beta_{2} + \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(5 \beta_{7} - 5 \beta_{6} + 4 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 4 \beta_{1} + 2\)
\(\nu^{5}\)\(=\)\(4 \beta_{7} - 6 \beta_{6} - 7 \beta_{3} + 11 \beta_{2} + 4 \beta_{1} - 13\)
\(\nu^{6}\)\(=\)\(7 \beta_{6} + 7 \beta_{5} - 8 \beta_{4} - 7 \beta_{3} + 21 \beta_{2} - 2 \beta_{1} - 21\)
\(\nu^{7}\)\(=\)\(23 \beta_{7} + 38 \beta_{5} + 23 \beta_{4} - 23 \beta_{3} + 12 \beta_{2}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/125\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\beta_{2}\)

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
1.17421 + 0.0566033i
−0.983224 0.644389i
1.66637 0.917186i
−0.357358 + 1.86824i
1.66637 + 0.917186i
−0.357358 1.86824i
1.17421 0.0566033i
−0.983224 + 0.644389i
−0.174207 0.0566033i 0.865190 1.19083i −1.59089 1.15585i 0 −0.218127 + 0.158479i 3.26086i 0.427051 + 0.587785i 0.257524 + 0.792578i 0
24.2 1.98322 + 0.644389i −1.29224 + 1.77862i 1.89991 + 1.38036i 0 −3.70892 + 2.69469i 0.992398i 0.427051 + 0.587785i −0.566541 1.74363i 0
49.1 −0.666375 + 0.917186i 2.47539 0.804303i 0.220859 + 0.679734i 0 −0.911842 + 2.80636i 0.407162i −2.92705 0.951057i 3.05361 2.21858i 0
49.2 1.35736 1.86824i 0.451659 0.146753i −1.02988 3.16963i 0 0.338893 1.04301i 3.03582i −2.92705 0.951057i −2.24459 + 1.63079i 0
74.1 −0.666375 0.917186i 2.47539 + 0.804303i 0.220859 0.679734i 0 −0.911842 2.80636i 0.407162i −2.92705 + 0.951057i 3.05361 + 2.21858i 0
74.2 1.35736 + 1.86824i 0.451659 + 0.146753i −1.02988 + 3.16963i 0 0.338893 + 1.04301i 3.03582i −2.92705 + 0.951057i −2.24459 1.63079i 0
99.1 −0.174207 + 0.0566033i 0.865190 + 1.19083i −1.59089 + 1.15585i 0 −0.218127 0.158479i 3.26086i 0.427051 0.587785i 0.257524 0.792578i 0
99.2 1.98322 0.644389i −1.29224 1.77862i 1.89991 1.38036i 0 −3.70892 2.69469i 0.992398i 0.427051 0.587785i −0.566541 + 1.74363i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 125.2.e.b 8
5.b even 2 1 25.2.e.a 8
5.c odd 4 2 125.2.d.b 16
15.d odd 2 1 225.2.m.a 8
20.d odd 2 1 400.2.y.c 8
25.d even 5 1 25.2.e.a 8
25.d even 5 1 625.2.b.c 8
25.d even 5 1 625.2.e.a 8
25.d even 5 1 625.2.e.i 8
25.e even 10 1 inner 125.2.e.b 8
25.e even 10 1 625.2.b.c 8
25.e even 10 1 625.2.e.a 8
25.e even 10 1 625.2.e.i 8
25.f odd 20 2 125.2.d.b 16
25.f odd 20 2 625.2.a.f 8
25.f odd 20 4 625.2.d.o 16
75.j odd 10 1 225.2.m.a 8
75.l even 20 2 5625.2.a.x 8
100.j odd 10 1 400.2.y.c 8
100.l even 20 2 10000.2.a.bj 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.2.e.a 8 5.b even 2 1
25.2.e.a 8 25.d even 5 1
125.2.d.b 16 5.c odd 4 2
125.2.d.b 16 25.f odd 20 2
125.2.e.b 8 1.a even 1 1 trivial
125.2.e.b 8 25.e even 10 1 inner
225.2.m.a 8 15.d odd 2 1
225.2.m.a 8 75.j odd 10 1
400.2.y.c 8 20.d odd 2 1
400.2.y.c 8 100.j odd 10 1
625.2.a.f 8 25.f odd 20 2
625.2.b.c 8 25.d even 5 1
625.2.b.c 8 25.e even 10 1
625.2.d.o 16 25.f odd 20 4
625.2.e.a 8 25.d even 5 1
625.2.e.a 8 25.e even 10 1
625.2.e.i 8 25.d even 5 1
625.2.e.i 8 25.e even 10 1
5625.2.a.x 8 75.l even 20 2
10000.2.a.bj 8 100.l even 20 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{8} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(125, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 10 T + 26 T^{2} - 10 T^{3} + T^{4} - 10 T^{5} + 11 T^{6} - 5 T^{7} + T^{8} \)
$3$ \( 16 - 80 T + 144 T^{2} - 110 T^{3} + 51 T^{4} - 15 T^{5} + 9 T^{6} - 5 T^{7} + T^{8} \)
$5$ \( T^{8} \)
$7$ \( 16 + 116 T^{2} + 121 T^{4} + 21 T^{6} + T^{8} \)
$11$ \( ( 16 + 8 T + 4 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$13$ \( 1 + 5 T + 4 T^{2} - 5 T^{3} + 21 T^{4} + 5 T^{5} + 4 T^{6} - 5 T^{7} + T^{8} \)
$17$ \( 1936 - 880 T - 784 T^{2} + 550 T^{3} + 31 T^{4} - 125 T^{5} + 56 T^{6} - 10 T^{7} + T^{8} \)
$19$ \( 400 + 200 T + 400 T^{2} - 100 T^{3} - 15 T^{4} + 40 T^{5} + 30 T^{6} + 5 T^{7} + T^{8} \)
$23$ \( 256 + 320 T - 16 T^{2} + 60 T^{3} + 241 T^{4} + 15 T^{5} - T^{6} + 5 T^{7} + T^{8} \)
$29$ \( 483025 + 142475 T + 49350 T^{2} + 4525 T^{3} + 485 T^{4} - 5 T^{5} + 30 T^{6} + 5 T^{7} + T^{8} \)
$31$ \( 1936 + 23496 T + 110532 T^{2} + 31178 T^{3} + 6855 T^{4} + 917 T^{5} + 117 T^{6} + 9 T^{7} + T^{8} \)
$37$ \( 116281 + 192665 T + 141631 T^{2} + 61170 T^{3} + 17321 T^{4} + 3270 T^{5} + 406 T^{6} + 30 T^{7} + T^{8} \)
$41$ \( 13456 - 1624 T + 9492 T^{2} - 7622 T^{3} + 2655 T^{4} + 457 T^{5} + 52 T^{6} + 4 T^{7} + T^{8} \)
$43$ \( 246016 + 56784 T^{2} + 4421 T^{4} + 129 T^{6} + T^{8} \)
$47$ \( 65536 + 61440 T - 4864 T^{2} - 9840 T^{3} + 4101 T^{4} - 615 T^{5} + 16 T^{6} + T^{8} \)
$53$ \( 8755681 - 4157395 T + 722619 T^{2} - 29590 T^{3} - 8079 T^{4} + 1290 T^{5} - 6 T^{6} - 10 T^{7} + T^{8} \)
$59$ \( 4080400 + 1333200 T + 407000 T^{2} + 54150 T^{3} + 5635 T^{4} + 15 T^{5} + T^{8} \)
$61$ \( 116281 - 59334 T + 139032 T^{2} + 12978 T^{3} + 16405 T^{4} - 1068 T^{5} - 43 T^{6} + 9 T^{7} + T^{8} \)
$67$ \( 246016 - 198400 T + 74816 T^{2} - 6080 T^{3} - 2384 T^{4} - 80 T^{5} + 116 T^{6} + 20 T^{7} + T^{8} \)
$71$ \( 24245776 - 5889104 T + 966712 T^{2} - 53922 T^{3} + 3455 T^{4} + 297 T^{5} + 142 T^{6} - 6 T^{7} + T^{8} \)
$73$ \( 1 + 4 T^{2} - 30 T^{3} + 91 T^{4} - 120 T^{5} + 49 T^{6} + 15 T^{7} + T^{8} \)
$79$ \( 33408400 + 4913000 T + 1416100 T^{2} + 79050 T^{3} + 12185 T^{4} + 600 T^{5} + 100 T^{6} - 15 T^{7} + T^{8} \)
$83$ \( 99856 - 284400 T + 329344 T^{2} - 199950 T^{3} + 70651 T^{4} - 11175 T^{5} + 949 T^{6} - 45 T^{7} + T^{8} \)
$89$ \( 1392400 + 1640200 T + 888600 T^{2} + 258800 T^{3} + 47985 T^{4} + 5890 T^{5} + 520 T^{6} + 25 T^{7} + T^{8} \)
$97$ \( 301334881 - 63360350 T + 5529361 T^{2} - 971040 T^{3} + 213086 T^{4} - 24990 T^{5} + 1636 T^{6} - 60 T^{7} + T^{8} \)
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