Properties

Label 241.2.a.a
Level 241
Weight 2
Character orbit 241.a
Self dual yes
Analytic conductor 1.924
Analytic rank 1
Dimension 7
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 241.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.92439468871\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: 7.7.31056073.1
Defining polynomial: \(x^{7} - 3 x^{6} - 3 x^{5} + 11 x^{4} + x^{3} - 9 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - 3 x^{6} - 3 x^{5} + 11 x^{4} + x^{3} - 9 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{5} - 2 \nu^{4} - 4 \nu^{3} + 5 \nu^{2} + 2 \nu - 1 \)
\(\beta_{3}\)\(=\)\( \nu^{5} - 2 \nu^{4} - 4 \nu^{3} + 4 \nu^{2} + 3 \nu + 1 \)
\(\beta_{4}\)\(=\)\( \nu^{6} - 3 \nu^{5} - 2 \nu^{4} + 9 \nu^{3} - 3 \nu^{2} - 4 \nu + 2 \)
\(\beta_{5}\)\(=\)\( \nu^{6} - 2 \nu^{5} - 5 \nu^{4} + 6 \nu^{3} + 7 \nu^{2} - 3 \nu - 2 \)
\(\beta_{6}\)\(=\)\( -\nu^{6} + 3 \nu^{5} + 3 \nu^{4} - 11 \nu^{3} - \nu^{2} + 8 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{3} + \beta_{2} + \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(-\beta_{6} - \beta_{5} - \beta_{3} + 2 \beta_{2} + 4 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-\beta_{6} - 2 \beta_{5} + \beta_{4} - 6 \beta_{3} + 8 \beta_{2} + 8 \beta_{1} + 8\)
\(\nu^{5}\)\(=\)\(-6 \beta_{6} - 8 \beta_{5} + 2 \beta_{4} - 11 \beta_{3} + 20 \beta_{2} + 25 \beta_{1} + 11\)
\(\nu^{6}\)\(=\)\(-11 \beta_{6} - 19 \beta_{5} + 9 \beta_{4} - 39 \beta_{3} + 61 \beta_{2} + 62 \beta_{1} + 44\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.60363
−0.911223
−0.356270
0.369356
1.27758
1.48734
2.73684
−2.60363 0.980039 4.77887 −1.69135 −2.55166 −1.30586 −7.23513 −2.03952 4.40364
1.2 −1.91122 −0.186202 1.65278 −2.25110 0.355874 3.52970 0.663624 −2.96533 4.30235
1.3 −1.35627 −2.45059 −0.160532 2.74184 3.32366 −0.283608 2.93026 3.00540 −3.71867
1.4 −0.630644 2.33806 −1.60229 −3.89634 −1.47449 −3.68231 2.27176 2.46654 2.45721
1.5 0.277577 −0.494846 −1.92295 −1.23324 −0.137358 1.36627 −1.08892 −2.75513 −0.342320
1.6 0.487343 −0.815004 −1.76250 0.961999 −0.397187 −4.61392 −1.83363 −2.33577 0.468824
1.7 1.73684 −2.37146 1.01662 −2.63180 −4.11885 −2.01025 −1.70797 2.62382 −4.57103
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(241\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 241.2.a.a 7
3.b odd 2 1 2169.2.a.e 7
4.b odd 2 1 3856.2.a.j 7
5.b even 2 1 6025.2.a.f 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
241.2.a.a 7 1.a even 1 1 trivial
2169.2.a.e 7 3.b odd 2 1
3856.2.a.j 7 4.b odd 2 1
6025.2.a.f 7 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{7} + 4 T_{2}^{6} - 14 T_{2}^{4} - 10 T_{2}^{3} + 6 T_{2}^{2} + 3 T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(241))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T + 14 T^{2} + 34 T^{3} + 74 T^{4} + 134 T^{5} + 223 T^{6} + 327 T^{7} + 446 T^{8} + 536 T^{9} + 592 T^{10} + 544 T^{11} + 448 T^{12} + 256 T^{13} + 128 T^{14} \)
$3$ \( 1 + 3 T + 16 T^{2} + 35 T^{3} + 110 T^{4} + 191 T^{5} + 467 T^{6} + 679 T^{7} + 1401 T^{8} + 1719 T^{9} + 2970 T^{10} + 2835 T^{11} + 3888 T^{12} + 2187 T^{13} + 2187 T^{14} \)
$5$ \( 1 + 8 T + 47 T^{2} + 190 T^{3} + 660 T^{4} + 1907 T^{5} + 5037 T^{6} + 11697 T^{7} + 25185 T^{8} + 47675 T^{9} + 82500 T^{10} + 118750 T^{11} + 146875 T^{12} + 125000 T^{13} + 78125 T^{14} \)
$7$ \( 1 + 7 T + 46 T^{2} + 196 T^{3} + 786 T^{4} + 2528 T^{5} + 7897 T^{6} + 21047 T^{7} + 55279 T^{8} + 123872 T^{9} + 269598 T^{10} + 470596 T^{11} + 773122 T^{12} + 823543 T^{13} + 823543 T^{14} \)
$11$ \( 1 + 18 T + 194 T^{2} + 1471 T^{3} + 8839 T^{4} + 43563 T^{5} + 182353 T^{6} + 651389 T^{7} + 2005883 T^{8} + 5271123 T^{9} + 11764709 T^{10} + 21536911 T^{11} + 31243894 T^{12} + 31888098 T^{13} + 19487171 T^{14} \)
$13$ \( 1 + T + 43 T^{2} + 16 T^{3} + 962 T^{4} + 171 T^{5} + 16575 T^{6} + 3431 T^{7} + 215475 T^{8} + 28899 T^{9} + 2113514 T^{10} + 456976 T^{11} + 15965599 T^{12} + 4826809 T^{13} + 62748517 T^{14} \)
$17$ \( 1 + 2 T + 54 T^{2} + 118 T^{3} + 1511 T^{4} + 4204 T^{5} + 32789 T^{6} + 93345 T^{7} + 557413 T^{8} + 1214956 T^{9} + 7423543 T^{10} + 9855478 T^{11} + 76672278 T^{12} + 48275138 T^{13} + 410338673 T^{14} \)
$19$ \( 1 + 6 T + 77 T^{2} + 408 T^{3} + 3328 T^{4} + 14851 T^{5} + 91875 T^{6} + 346087 T^{7} + 1745625 T^{8} + 5361211 T^{9} + 22826752 T^{10} + 53170968 T^{11} + 190659623 T^{12} + 282275286 T^{13} + 893871739 T^{14} \)
$23$ \( 1 + 22 T + 329 T^{2} + 3499 T^{3} + 30354 T^{4} + 215473 T^{5} + 1308858 T^{6} + 6746533 T^{7} + 30103734 T^{8} + 113985217 T^{9} + 369317118 T^{10} + 979163659 T^{11} + 2117556847 T^{12} + 3256789558 T^{13} + 3404825447 T^{14} \)
$29$ \( 1 + 16 T + 228 T^{2} + 1945 T^{3} + 15113 T^{4} + 86708 T^{5} + 503802 T^{6} + 2527253 T^{7} + 14610258 T^{8} + 72921428 T^{9} + 368590957 T^{10} + 1375661545 T^{11} + 4676541972 T^{12} + 9517173136 T^{13} + 17249876309 T^{14} \)
$31$ \( 1 + 18 T + 321 T^{2} + 3457 T^{3} + 35295 T^{4} + 269386 T^{5} + 1944797 T^{6} + 11129437 T^{7} + 60288707 T^{8} + 258879946 T^{9} + 1051473345 T^{10} + 3192612097 T^{11} + 9189957471 T^{12} + 15975066258 T^{13} + 27512614111 T^{14} \)
$37$ \( 1 - 8 T + 140 T^{2} - 446 T^{3} + 5906 T^{4} + 5317 T^{5} + 140628 T^{6} + 725991 T^{7} + 5203236 T^{8} + 7278973 T^{9} + 299156618 T^{10} - 835875806 T^{11} + 9708153980 T^{12} - 20525811272 T^{13} + 94931877133 T^{14} \)
$41$ \( 1 + 15 T + 165 T^{2} + 1716 T^{3} + 15349 T^{4} + 112837 T^{5} + 825907 T^{6} + 5652081 T^{7} + 33862187 T^{8} + 189678997 T^{9} + 1057868429 T^{10} + 4849005876 T^{11} + 19116273165 T^{12} + 71251563615 T^{13} + 194754273881 T^{14} \)
$43$ \( 1 - 14 T + 160 T^{2} - 1152 T^{3} + 8465 T^{4} - 39218 T^{5} + 236797 T^{6} - 1042279 T^{7} + 10182271 T^{8} - 72514082 T^{9} + 673026755 T^{10} - 3938458752 T^{11} + 23521350880 T^{12} - 88499082686 T^{13} + 271818611107 T^{14} \)
$47$ \( 1 + 10 T + 213 T^{2} + 1823 T^{3} + 23033 T^{4} + 162514 T^{5} + 1573229 T^{6} + 9290969 T^{7} + 73941763 T^{8} + 358993426 T^{9} + 2391355159 T^{10} + 8895658463 T^{11} + 48850486491 T^{12} + 107792153290 T^{13} + 506623120463 T^{14} \)
$53$ \( 1 - 15 T + 248 T^{2} - 2459 T^{3} + 26359 T^{4} - 226500 T^{5} + 2031262 T^{6} - 14891311 T^{7} + 107656886 T^{8} - 636238500 T^{9} + 3924248843 T^{10} - 19402692779 T^{11} + 103712482264 T^{12} - 332465416935 T^{13} + 1174711139837 T^{14} \)
$59$ \( 1 + 18 T + 283 T^{2} + 3342 T^{3} + 39787 T^{4} + 377400 T^{5} + 3440044 T^{6} + 26583077 T^{7} + 202962596 T^{8} + 1313729400 T^{9} + 8171414273 T^{10} + 40496220462 T^{11} + 202323576617 T^{12} + 759249605538 T^{13} + 2488651484819 T^{14} \)
$61$ \( 1 - 4 T + 173 T^{2} + 229 T^{3} + 13815 T^{4} + 53354 T^{5} + 1189845 T^{6} + 3012271 T^{7} + 72580545 T^{8} + 198530234 T^{9} + 3135742515 T^{10} + 3170697589 T^{11} + 146115160073 T^{12} - 206081497444 T^{13} + 3142742836021 T^{14} \)
$67$ \( 1 - 18 T + 312 T^{2} - 3657 T^{3} + 46177 T^{4} - 442394 T^{5} + 4388546 T^{6} - 34987571 T^{7} + 294032582 T^{8} - 1985906666 T^{9} + 13888333051 T^{10} - 73692649497 T^{11} + 421239033384 T^{12} - 1628250879042 T^{13} + 6060711605323 T^{14} \)
$71$ \( 1 + 50 T + 1452 T^{2} + 29886 T^{3} + 479876 T^{4} + 6256313 T^{5} + 67998420 T^{6} + 622620957 T^{7} + 4827887820 T^{8} + 31538073833 T^{9} + 171752899036 T^{10} + 759453498366 T^{11} + 2619741017652 T^{12} + 6405014196050 T^{13} + 9095120158391 T^{14} \)
$73$ \( 1 + 133 T^{2} + 1068 T^{3} + 10948 T^{4} + 119175 T^{5} + 1571649 T^{6} + 6028685 T^{7} + 114730377 T^{8} + 635083575 T^{9} + 4258958116 T^{10} + 30329321388 T^{11} + 275718521869 T^{12} + 11047398519097 T^{14} \)
$79$ \( 1 + 15 T + 468 T^{2} + 5553 T^{3} + 98197 T^{4} + 946068 T^{5} + 12199714 T^{6} + 95010077 T^{7} + 963777406 T^{8} + 5904410388 T^{9} + 48414950683 T^{10} + 216289799793 T^{11} + 1440062394732 T^{12} + 3646311832815 T^{13} + 19203908986159 T^{14} \)
$83$ \( 1 + 24 T + 705 T^{2} + 11938 T^{3} + 194891 T^{4} + 2473798 T^{5} + 28249166 T^{6} + 273618813 T^{7} + 2344680778 T^{8} + 17041994422 T^{9} + 111436140217 T^{10} + 566557436098 T^{11} + 2777023653315 T^{12} + 7846568960856 T^{13} + 27136050989627 T^{14} \)
$89$ \( 1 + 13 T + 260 T^{2} + 2275 T^{3} + 16962 T^{4} + 2186 T^{5} - 807494 T^{6} - 17411725 T^{7} - 71866966 T^{8} + 17315306 T^{9} + 11957684178 T^{10} + 142738598275 T^{11} + 1451855456740 T^{12} + 6460756782493 T^{13} + 44231334895529 T^{14} \)
$97$ \( 1 - T + 396 T^{2} + 357 T^{3} + 75395 T^{4} + 180194 T^{5} + 9562600 T^{6} + 26454385 T^{7} + 927572200 T^{8} + 1695445346 T^{9} + 68810980835 T^{10} + 31604953317 T^{11} + 3400586741772 T^{12} - 832972004929 T^{13} + 80798284478113 T^{14} \)
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