Properties

Label 29.3.c.a
Level $29$
Weight $3$
Character orbit 29.c
Analytic conductor $0.790$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 29.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.790192766645\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \(x^{8} + 18 x^{6} + 91 x^{4} + 126 x^{2} + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q -\beta_{6} q^{2} + ( \beta_{2} + \beta_{6} ) q^{3} + ( \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{4} + ( -\beta_{3} - \beta_{4} ) q^{5} + ( -\beta_{1} - \beta_{2} - 2 \beta_{3} - 5 \beta_{4} + \beta_{5} + \beta_{6} ) q^{6} + ( \beta_{1} - \beta_{2} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{7} + ( -5 - \beta_{1} + 2 \beta_{3} + 5 \beta_{4} - \beta_{5} - 2 \beta_{7} ) q^{8} + ( \beta_{1} + \beta_{2} + 3 \beta_{3} + 6 \beta_{4} + \beta_{5} + \beta_{6} ) q^{9} +O(q^{10})\) \( q -\beta_{6} q^{2} + ( \beta_{2} + \beta_{6} ) q^{3} + ( \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{4} + ( -\beta_{3} - \beta_{4} ) q^{5} + ( -\beta_{1} - \beta_{2} - 2 \beta_{3} - 5 \beta_{4} + \beta_{5} + \beta_{6} ) q^{6} + ( \beta_{1} - \beta_{2} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{7} + ( -5 - \beta_{1} + 2 \beta_{3} + 5 \beta_{4} - \beta_{5} - 2 \beta_{7} ) q^{8} + ( \beta_{1} + \beta_{2} + 3 \beta_{3} + 6 \beta_{4} + \beta_{5} + \beta_{6} ) q^{9} + ( \beta_{1} - \beta_{3} + 3 \beta_{5} + \beta_{7} ) q^{10} + ( -1 - 4 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{11} + ( 5 - 2 \beta_{3} - 5 \beta_{4} + 7 \beta_{5} + 2 \beta_{7} ) q^{12} + ( 2 \beta_{1} + 2 \beta_{2} + 5 \beta_{4} - \beta_{5} - \beta_{6} ) q^{13} + ( -5 + \beta_{2} + \beta_{3} - 5 \beta_{4} + \beta_{7} ) q^{14} + ( -2 \beta_{1} - 5 \beta_{5} ) q^{15} + ( -1 - \beta_{1} + \beta_{2} - 6 \beta_{5} + 6 \beta_{6} - 2 \beta_{7} ) q^{16} + ( \beta_{2} - 6 \beta_{6} ) q^{17} + ( 5 - 5 \beta_{1} + \beta_{3} - 5 \beta_{4} - 10 \beta_{5} - \beta_{7} ) q^{18} + ( -1 - \beta_{2} - 4 \beta_{3} - \beta_{4} + 4 \beta_{6} - 4 \beta_{7} ) q^{19} + ( 11 + 5 \beta_{5} - 5 \beta_{6} + 2 \beta_{7} ) q^{20} + ( -5 + 3 \beta_{2} + \beta_{3} - 5 \beta_{4} - 2 \beta_{6} + \beta_{7} ) q^{21} + ( 5 \beta_{1} + 5 \beta_{2} + 3 \beta_{3} + 5 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{22} + ( -2 \beta_{1} + 2 \beta_{2} + 3 \beta_{7} ) q^{23} + ( 15 + 6 \beta_{1} - 6 \beta_{2} + 12 \beta_{5} - 12 \beta_{6} + 3 \beta_{7} ) q^{24} + ( 14 + \beta_{1} - \beta_{2} - 2 \beta_{5} + 2 \beta_{6} ) q^{25} + ( -5 - 4 \beta_{1} - \beta_{3} + 5 \beta_{4} - 7 \beta_{5} + \beta_{7} ) q^{26} + ( -15 + 2 \beta_{1} + 3 \beta_{3} + 15 \beta_{4} + 11 \beta_{5} - 3 \beta_{7} ) q^{27} + ( 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - \beta_{5} - \beta_{6} ) q^{28} + ( 15 - 5 \beta_{1} + 2 \beta_{2} + 6 \beta_{4} + 7 \beta_{5} + 3 \beta_{6} ) q^{29} + ( -25 + 2 \beta_{1} - 2 \beta_{2} - 5 \beta_{5} + 5 \beta_{6} - 7 \beta_{7} ) q^{30} + ( 1 - 7 \beta_{2} + 4 \beta_{3} + \beta_{4} + 9 \beta_{6} + 4 \beta_{7} ) q^{31} + ( -10 - 4 \beta_{2} - \beta_{3} - 10 \beta_{4} + 13 \beta_{6} - \beta_{7} ) q^{32} + ( -3 \beta_{1} - 3 \beta_{2} - 6 \beta_{3} - 45 \beta_{4} - 12 \beta_{5} - 12 \beta_{6} ) q^{33} + ( -\beta_{1} - \beta_{2} + 5 \beta_{3} + 30 \beta_{4} - 6 \beta_{5} - 6 \beta_{6} ) q^{34} + ( -\beta_{1} - \beta_{2} + 5 \beta_{3} - 10 \beta_{4} + \beta_{5} + \beta_{6} ) q^{35} + ( -26 - 11 \beta_{5} + 11 \beta_{6} - 5 \beta_{7} ) q^{36} + ( -10 - 6 \beta_{1} - 8 \beta_{3} + 10 \beta_{4} - 2 \beta_{5} + 8 \beta_{7} ) q^{37} + ( 5 \beta_{1} + 5 \beta_{2} - 11 \beta_{3} - 20 \beta_{4} + 13 \beta_{5} + 13 \beta_{6} ) q^{38} + ( -15 + 3 \beta_{1} + 15 \beta_{4} + 15 \beta_{5} ) q^{39} + ( 25 + 2 \beta_{2} + 3 \beta_{3} + 25 \beta_{4} - 13 \beta_{6} + 3 \beta_{7} ) q^{40} + ( 1 + 7 \beta_{1} + 5 \beta_{3} - \beta_{4} + 6 \beta_{5} - 5 \beta_{7} ) q^{41} + ( -4 \beta_{1} - 4 \beta_{2} + \beta_{3} + 10 \beta_{4} + \beta_{5} + \beta_{6} ) q^{42} + ( -25 + 6 \beta_{2} + \beta_{3} - 25 \beta_{4} - 5 \beta_{6} + \beta_{7} ) q^{43} + ( 6 + 3 \beta_{1} - 6 \beta_{4} - 3 \beta_{5} ) q^{44} + ( 36 - 5 \beta_{1} + 5 \beta_{2} + \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{45} + ( -7 \beta_{2} + \beta_{3} - 6 \beta_{6} + \beta_{7} ) q^{46} + ( 10 + 6 \beta_{1} + 2 \beta_{3} - 10 \beta_{4} - 11 \beta_{5} - 2 \beta_{7} ) q^{47} + ( 40 + 9 \beta_{2} + 13 \beta_{3} + 40 \beta_{4} - 17 \beta_{6} + 13 \beta_{7} ) q^{48} + ( -9 + 3 \beta_{1} - 3 \beta_{2} - 12 \beta_{7} ) q^{49} + ( -10 + 2 \beta_{2} - \beta_{3} - 10 \beta_{4} - 10 \beta_{6} - \beta_{7} ) q^{50} + ( -6 \beta_{1} - 6 \beta_{2} - 11 \beta_{3} - 20 \beta_{4} + 8 \beta_{5} + 8 \beta_{6} ) q^{51} + ( -15 - 3 \beta_{1} + 3 \beta_{2} - 6 \beta_{5} + 6 \beta_{6} - 9 \beta_{7} ) q^{52} + ( 35 - 6 \beta_{1} + 6 \beta_{2} - 7 \beta_{5} + 7 \beta_{6} - 10 \beta_{7} ) q^{53} + ( 55 - 5 \beta_{1} + 5 \beta_{2} - 10 \beta_{5} + 10 \beta_{6} + 7 \beta_{7} ) q^{54} + ( -11 + 7 \beta_{1} + 3 \beta_{3} + 11 \beta_{4} + 7 \beta_{5} - 3 \beta_{7} ) q^{55} + ( -25 - 5 \beta_{1} - 8 \beta_{3} + 25 \beta_{4} + 4 \beta_{5} + 8 \beta_{7} ) q^{56} + ( -\beta_{1} - \beta_{2} + 7 \beta_{3} + 10 \beta_{4} - 23 \beta_{5} - 23 \beta_{6} ) q^{57} + ( 35 + 3 \beta_{1} - 7 \beta_{2} - 5 \beta_{3} - 15 \beta_{4} + 4 \beta_{5} - 19 \beta_{6} + 2 \beta_{7} ) q^{58} + ( -10 + 4 \beta_{1} - 4 \beta_{2} + 10 \beta_{5} - 10 \beta_{6} + 17 \beta_{7} ) q^{59} + ( -25 + 3 \beta_{2} - 10 \beta_{3} - 25 \beta_{4} + 29 \beta_{6} - 10 \beta_{7} ) q^{60} + ( -24 + 9 \beta_{2} + 2 \beta_{3} - 24 \beta_{4} + 8 \beta_{6} + 2 \beta_{7} ) q^{61} + ( 3 \beta_{1} + 3 \beta_{2} + 6 \beta_{3} - 45 \beta_{4} ) q^{62} + ( 3 \beta_{1} + 3 \beta_{2} - 10 \beta_{3} + 20 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{63} + ( \beta_{1} + \beta_{2} - 3 \beta_{3} - 61 \beta_{4} + \beta_{5} + \beta_{6} ) q^{64} + ( 5 - \beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} + 11 \beta_{7} ) q^{65} + ( -60 + 12 \beta_{1} + 9 \beta_{3} + 60 \beta_{4} + 33 \beta_{5} - 9 \beta_{7} ) q^{66} + ( -15 \beta_{1} - 15 \beta_{2} + 6 \beta_{3} + 40 \beta_{4} ) q^{67} + ( -30 - 7 \beta_{1} + 12 \beta_{3} + 30 \beta_{4} - 28 \beta_{5} - 12 \beta_{7} ) q^{68} + ( 20 + 3 \beta_{2} + 2 \beta_{3} + 20 \beta_{4} + 20 \beta_{6} + 2 \beta_{7} ) q^{69} + ( 5 - 3 \beta_{1} + 5 \beta_{3} - 5 \beta_{4} + 2 \beta_{5} - 5 \beta_{7} ) q^{70} + ( 3 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} + 62 \beta_{4} - 5 \beta_{5} - 5 \beta_{6} ) q^{71} + ( -75 - 15 \beta_{2} - 12 \beta_{3} - 75 \beta_{4} + 18 \beta_{6} - 12 \beta_{7} ) q^{72} + ( -20 - 6 \beta_{1} + 4 \beta_{3} + 20 \beta_{4} - 14 \beta_{5} - 4 \beta_{7} ) q^{73} + ( -10 + 14 \beta_{1} - 14 \beta_{2} + 4 \beta_{5} - 4 \beta_{6} + 8 \beta_{7} ) q^{74} + ( 18 \beta_{2} + 3 \beta_{3} + 6 \beta_{6} + 3 \beta_{7} ) q^{75} + ( 61 + 5 \beta_{1} - 13 \beta_{3} - 61 \beta_{4} + 52 \beta_{5} + 13 \beta_{7} ) q^{76} + ( 25 - 17 \beta_{2} - 2 \beta_{3} + 25 \beta_{4} + 10 \beta_{6} - 2 \beta_{7} ) q^{77} + ( 75 - 3 \beta_{1} + 3 \beta_{2} + 18 \beta_{7} ) q^{78} + ( -46 - 14 \beta_{2} - 6 \beta_{3} - 46 \beta_{4} + 7 \beta_{6} - 6 \beta_{7} ) q^{79} + ( -5 \beta_{1} - 5 \beta_{2} + 9 \beta_{3} + 21 \beta_{4} - 24 \beta_{5} - 24 \beta_{6} ) q^{80} + ( -21 + 20 \beta_{1} - 20 \beta_{2} + 11 \beta_{5} - 11 \beta_{6} + 3 \beta_{7} ) q^{81} + ( 30 - 12 \beta_{1} + 12 \beta_{2} - 3 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} ) q^{82} + ( 30 + 7 \beta_{1} - 7 \beta_{2} + 10 \beta_{5} - 10 \beta_{6} - 21 \beta_{7} ) q^{83} + ( -15 - 5 \beta_{1} + 15 \beta_{4} - 2 \beta_{5} ) q^{84} + ( 5 \beta_{1} - 7 \beta_{3} + 16 \beta_{5} + 7 \beta_{7} ) q^{85} + ( -7 \beta_{1} - 7 \beta_{2} + \beta_{3} + 25 \beta_{4} + 18 \beta_{5} + 18 \beta_{6} ) q^{86} + ( 15 + 16 \beta_{1} + 11 \beta_{2} + 8 \beta_{3} + 35 \beta_{4} - 10 \beta_{5} + 33 \beta_{6} - 9 \beta_{7} ) q^{87} + ( 5 - 23 \beta_{1} + 23 \beta_{2} - 5 \beta_{5} + 5 \beta_{6} + 12 \beta_{7} ) q^{88} + ( -11 + 31 \beta_{2} - 8 \beta_{3} - 11 \beta_{4} - 12 \beta_{6} - 8 \beta_{7} ) q^{89} + ( 5 - 13 \beta_{2} - \beta_{3} + 5 \beta_{4} - 44 \beta_{6} - \beta_{7} ) q^{90} + ( 4 \beta_{1} + 4 \beta_{2} + 15 \beta_{3} - 50 \beta_{4} + \beta_{5} + \beta_{6} ) q^{91} + ( -2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 30 \beta_{4} - 8 \beta_{5} - 8 \beta_{6} ) q^{92} + ( 14 \beta_{1} + 14 \beta_{2} + 11 \beta_{3} - 25 \beta_{4} - 6 \beta_{5} - 6 \beta_{6} ) q^{93} + ( -55 - 8 \beta_{1} + 8 \beta_{2} - 5 \beta_{5} + 5 \beta_{6} - 9 \beta_{7} ) q^{94} + ( -41 + 5 \beta_{1} + 2 \beta_{3} + 41 \beta_{4} - 26 \beta_{5} - 2 \beta_{7} ) q^{95} + ( 2 \beta_{1} + 2 \beta_{2} + 22 \beta_{3} + 25 \beta_{4} - 35 \beta_{5} - 35 \beta_{6} ) q^{96} + ( -5 - 7 \beta_{1} - 16 \beta_{3} + 5 \beta_{4} - 2 \beta_{5} + 16 \beta_{7} ) q^{97} + ( 18 \beta_{2} - 9 \beta_{3} + 33 \beta_{6} - 9 \beta_{7} ) q^{98} + ( 81 - 39 \beta_{1} - 18 \beta_{3} - 81 \beta_{4} - 48 \beta_{5} + 18 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 2q^{2} - 2q^{3} - 4q^{7} - 42q^{8} + O(q^{10}) \) \( 8q + 2q^{2} - 2q^{3} - 4q^{7} - 42q^{8} + 6q^{10} - 6q^{11} + 54q^{12} - 40q^{14} - 10q^{15} - 32q^{16} + 12q^{17} + 20q^{18} - 16q^{19} + 108q^{20} - 36q^{21} + 168q^{24} + 104q^{25} - 54q^{26} - 98q^{27} + 128q^{29} - 220q^{30} - 10q^{31} - 106q^{32} - 252q^{36} - 84q^{37} - 90q^{39} + 226q^{40} + 20q^{41} - 190q^{43} + 42q^{44} + 292q^{45} + 12q^{46} + 58q^{47} + 354q^{48} - 72q^{49} - 60q^{50} - 144q^{52} + 252q^{53} + 400q^{54} - 74q^{55} - 192q^{56} + 326q^{58} - 40q^{59} - 258q^{60} - 208q^{61} + 36q^{65} - 414q^{66} - 296q^{68} + 120q^{69} + 44q^{70} - 636q^{72} - 188q^{73} - 64q^{74} - 12q^{75} + 592q^{76} + 180q^{77} + 600q^{78} - 382q^{79} - 124q^{81} + 228q^{82} + 280q^{83} - 124q^{84} + 32q^{85} + 34q^{87} + 20q^{88} - 64q^{89} + 128q^{90} - 460q^{94} - 380q^{95} - 44q^{97} - 66q^{98} + 552q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 18 x^{6} + 91 x^{4} + 126 x^{2} + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{4} + 9 \nu^{2} + 6 \nu + 5 \)\()/6\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{4} - 9 \nu^{2} + 6 \nu - 5 \)\()/6\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{5} - 15 \nu^{3} - 47 \nu \)\()/6\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} - 18 \nu^{5} - 86 \nu^{3} - 81 \nu \)\()/30\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} + 17 \nu^{5} + \nu^{4} + 77 \nu^{3} + 15 \nu^{2} + 82 \nu + 35 \)\()/12\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} + 17 \nu^{5} - \nu^{4} + 77 \nu^{3} - 15 \nu^{2} + 82 \nu - 35 \)\()/12\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{6} + 16 \nu^{4} + 62 \nu^{2} + 35 \)\()/6\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{6} + 2 \beta_{5} + \beta_{2} - \beta_{1} - 10\)\()/2\)
\(\nu^{3}\)\(=\)\(\beta_{6} + \beta_{5} + 5 \beta_{4} - \beta_{3} - 4 \beta_{2} - 4 \beta_{1}\)
\(\nu^{4}\)\(=\)\((\)\(18 \beta_{6} - 18 \beta_{5} - 15 \beta_{2} + 15 \beta_{1} + 80\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-30 \beta_{6} - 30 \beta_{5} - 150 \beta_{4} + 18 \beta_{3} + 73 \beta_{2} + 73 \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\(6 \beta_{7} - 82 \beta_{6} + 82 \beta_{5} + 89 \beta_{2} - 89 \beta_{1} - 365\)
\(\nu^{7}\)\(=\)\((\)\(368 \beta_{6} + 368 \beta_{5} + 1780 \beta_{4} - 152 \beta_{3} - 707 \beta_{2} - 707 \beta_{1}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/29\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
12.1
2.35663i
3.22189i
1.35225i
0.486981i
2.35663i
3.22189i
1.35225i
0.486981i
−1.45515 1.45515i 3.81178 + 3.81178i 0.234947i 3.14526i 11.0935i 0.342313 −5.47873 + 5.47873i 20.0593i −4.57683 + 4.57683i
12.2 −1.07935 1.07935i −2.14254 2.14254i 1.67001i 0.488689i 4.62511i 8.09117 −6.11992 + 6.11992i 0.180982i −0.527467 + 0.527467i
12.3 0.909588 + 0.909588i 0.442660 + 0.442660i 2.34530i 4.16447i 0.805276i −9.68815 5.77161 5.77161i 8.60810i −3.78796 + 3.78796i
12.4 2.62492 + 2.62492i −3.11190 3.11190i 9.78036i 4.53053i 16.3369i −0.745339 −15.1730 + 15.1730i 10.3678i 11.8923 11.8923i
17.1 −1.45515 + 1.45515i 3.81178 3.81178i 0.234947i 3.14526i 11.0935i 0.342313 −5.47873 5.47873i 20.0593i −4.57683 4.57683i
17.2 −1.07935 + 1.07935i −2.14254 + 2.14254i 1.67001i 0.488689i 4.62511i 8.09117 −6.11992 6.11992i 0.180982i −0.527467 0.527467i
17.3 0.909588 0.909588i 0.442660 0.442660i 2.34530i 4.16447i 0.805276i −9.68815 5.77161 + 5.77161i 8.60810i −3.78796 3.78796i
17.4 2.62492 2.62492i −3.11190 + 3.11190i 9.78036i 4.53053i 16.3369i −0.745339 −15.1730 15.1730i 10.3678i 11.8923 + 11.8923i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 29.3.c.a 8
3.b odd 2 1 261.3.f.a 8
4.b odd 2 1 464.3.l.c 8
29.c odd 4 1 inner 29.3.c.a 8
87.f even 4 1 261.3.f.a 8
116.e even 4 1 464.3.l.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.3.c.a 8 1.a even 1 1 trivial
29.3.c.a 8 29.c odd 4 1 inner
261.3.f.a 8 3.b odd 2 1
261.3.f.a 8 87.f even 4 1
464.3.l.c 8 4.b odd 2 1
464.3.l.c 8 116.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(29, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 225 + 30 T + 2 T^{2} + 10 T^{3} + 70 T^{4} + 18 T^{5} + 2 T^{6} - 2 T^{7} + T^{8} \)
$3$ \( 2025 - 3510 T + 3042 T^{2} + 2094 T^{3} + 694 T^{4} + 22 T^{5} + 2 T^{6} + 2 T^{7} + T^{8} \)
$5$ \( 841 + 3696 T^{2} + 742 T^{4} + 48 T^{6} + T^{8} \)
$7$ \( ( 20 - 32 T - 78 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$11$ \( 81 - 1674 T + 17298 T^{2} + 58722 T^{3} + 99838 T^{4} - 1710 T^{5} + 18 T^{6} + 6 T^{7} + T^{8} \)
$13$ \( 137945025 + 5330916 T^{2} + 74710 T^{4} + 452 T^{6} + T^{8} \)
$17$ \( 660490000 + 31456800 T + 749088 T^{2} - 154272 T^{3} + 91484 T^{4} + 3312 T^{5} + 72 T^{6} - 12 T^{7} + T^{8} \)
$19$ \( 5996334096 + 111507840 T + 1036800 T^{2} - 2526336 T^{3} + 644364 T^{4} - 15744 T^{5} + 128 T^{6} + 16 T^{7} + T^{8} \)
$23$ \( ( 19940 - 204 T - 318 T^{2} + T^{4} )^{2} \)
$29$ \( 500246412961 - 76137385088 T + 5233879400 T^{2} - 226329920 T^{3} + 8004638 T^{4} - 269120 T^{5} + 7400 T^{6} - 128 T^{7} + T^{8} \)
$31$ \( 21518249481 + 631064682 T + 9253602 T^{2} - 11817522 T^{3} + 5495454 T^{4} - 28362 T^{5} + 50 T^{6} + 10 T^{7} + T^{8} \)
$37$ \( 1885238841600 - 162853528320 T + 7033928832 T^{2} + 113340864 T^{3} + 971104 T^{4} - 43344 T^{5} + 3528 T^{6} + 84 T^{7} + T^{8} \)
$41$ \( 24625141776 - 11456707392 T + 2665084032 T^{2} - 151636752 T^{3} + 4451004 T^{4} - 32328 T^{5} + 200 T^{6} - 20 T^{7} + T^{8} \)
$43$ \( 401582027025 + 91693311270 T + 10468176818 T^{2} + 720557578 T^{3} + 32511934 T^{4} + 959586 T^{5} + 18050 T^{6} + 190 T^{7} + T^{8} \)
$47$ \( 4236057225 - 2392134090 T + 675428258 T^{2} - 60155566 T^{3} + 2483326 T^{4} + 52218 T^{5} + 1682 T^{6} - 58 T^{7} + T^{8} \)
$53$ \( ( -3983085 + 255882 T + 884 T^{2} - 126 T^{3} + T^{4} )^{2} \)
$59$ \( ( -2998700 - 351292 T - 9886 T^{2} + 20 T^{3} + T^{4} )^{2} \)
$61$ \( 8832379812624 + 31264724640 T + 55335200 T^{2} + 555820336 T^{3} + 41061340 T^{4} + 1243128 T^{5} + 21632 T^{6} + 208 T^{7} + T^{8} \)
$67$ \( 32439859360000 + 142118817664 T^{2} + 133818864 T^{4} + 24904 T^{6} + T^{8} \)
$71$ \( 163478420513424 + 248898665520 T^{2} + 113378296 T^{4} + 19004 T^{6} + T^{8} \)
$73$ \( 23666745600 + 3369711360 T + 239892608 T^{2} - 33460672 T^{3} + 8418784 T^{4} + 557328 T^{5} + 17672 T^{6} + 188 T^{7} + T^{8} \)
$79$ \( 101810622515625 - 5398721381250 T + 143139251250 T^{2} + 15566672250 T^{3} + 499352350 T^{4} + 7826930 T^{5} + 72962 T^{6} + 382 T^{7} + T^{8} \)
$83$ \( ( -10359540 + 784836 T - 4838 T^{2} - 140 T^{3} + T^{4} )^{2} \)
$89$ \( 622709901672336 + 2273024161728 T + 4148511872 T^{2} - 3555640160 T^{3} + 412427692 T^{4} - 1467216 T^{5} + 2048 T^{6} + 64 T^{7} + T^{8} \)
$97$ \( 502668058467600 - 795829548960 T + 629983008 T^{2} - 587161440 T^{3} + 81721980 T^{4} - 459504 T^{5} + 968 T^{6} + 44 T^{7} + T^{8} \)
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