Properties

Label 403.2.a.d
Level 403
Weight 2
Character orbit 403.a
Self dual yes
Analytic conductor 3.218
Analytic rank 1
Dimension 8
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 403 = 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 403.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.21797120146\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{2} -\beta_{6} q^{3} + ( 2 - \beta_{1} + \beta_{2} ) q^{4} + ( -2 - \beta_{2} - \beta_{5} ) q^{5} + ( -1 - \beta_{1} - \beta_{3} + \beta_{5} + 2 \beta_{6} ) q^{6} + ( \beta_{2} + \beta_{3} + \beta_{4} ) q^{7} + ( -3 + \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{8} + ( 1 - \beta_{1} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{2} -\beta_{6} q^{3} + ( 2 - \beta_{1} + \beta_{2} ) q^{4} + ( -2 - \beta_{2} - \beta_{5} ) q^{5} + ( -1 - \beta_{1} - \beta_{3} + \beta_{5} + 2 \beta_{6} ) q^{6} + ( \beta_{2} + \beta_{3} + \beta_{4} ) q^{7} + ( -3 + \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{8} + ( 1 - \beta_{1} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{9} + ( 2 - 3 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} ) q^{10} + ( -2 - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{11} + ( \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{12} - q^{13} + ( \beta_{1} + 2 \beta_{3} + \beta_{5} ) q^{14} + ( -\beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{15} + ( 2 - 3 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} ) q^{16} + ( -2 + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{17} + ( -3 + 2 \beta_{1} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{18} + ( -1 - \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{19} + ( -5 + 3 \beta_{1} - 2 \beta_{2} - \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{20} + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{5} + \beta_{6} ) q^{21} + ( 2 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - 3 \beta_{6} ) q^{22} + ( 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{23} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + 4 \beta_{6} - 3 \beta_{7} ) q^{24} + ( 2 + 3 \beta_{2} + 4 \beta_{5} + \beta_{6} ) q^{25} + ( 1 - \beta_{1} ) q^{26} + ( -1 + 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{27} + ( 3 + 2 \beta_{2} + \beta_{4} - \beta_{6} + \beta_{7} ) q^{28} + ( 1 + 2 \beta_{2} + 2 \beta_{3} - \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{29} + ( 1 + 5 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} - 3 \beta_{5} - 7 \beta_{6} + 2 \beta_{7} ) q^{30} - q^{31} + ( -5 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{32} + ( -2 + 3 \beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{33} + ( 4 - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - 3 \beta_{6} + 2 \beta_{7} ) q^{34} + ( -\beta_{1} - 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} ) q^{35} + ( 3 - 4 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} + 4 \beta_{5} + 6 \beta_{6} - 3 \beta_{7} ) q^{36} + ( -3 + 3 \beta_{1} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{37} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - 4 \beta_{5} - \beta_{7} ) q^{38} + \beta_{6} q^{39} + ( 7 - 3 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + \beta_{4} + 3 \beta_{5} + 3 \beta_{6} ) q^{40} + ( -2 - \beta_{1} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{41} + ( -2 \beta_{1} - 3 \beta_{2} - \beta_{3} - 3 \beta_{4} - \beta_{5} - 2 \beta_{7} ) q^{42} + ( -\beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - 3 \beta_{7} ) q^{43} + ( -7 - 3 \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + 5 \beta_{6} - 3 \beta_{7} ) q^{44} + ( -6 + 5 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - 4 \beta_{6} + 3 \beta_{7} ) q^{45} + ( -1 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{46} + ( -3 + 3 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{47} + ( 4 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} + 3 \beta_{7} ) q^{48} + ( 3 \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{49} + ( -1 + 6 \beta_{1} + \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - \beta_{5} - 6 \beta_{6} + 4 \beta_{7} ) q^{50} + ( -4 + 3 \beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} + 3 \beta_{7} ) q^{51} + ( -2 + \beta_{1} - \beta_{2} ) q^{52} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{53} + ( 8 - 5 \beta_{1} + 5 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} + 6 \beta_{6} - 2 \beta_{7} ) q^{54} + ( 3 + 2 \beta_{1} + \beta_{2} + 5 \beta_{3} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{55} + ( -5 + 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{56} + ( \beta_{1} + \beta_{2} + 4 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{57} + ( -4 + \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + 3 \beta_{5} + 5 \beta_{6} - 2 \beta_{7} ) q^{58} + ( -\beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{59} + ( 5 - 6 \beta_{1} + 5 \beta_{2} - 4 \beta_{3} + 7 \beta_{5} + 11 \beta_{6} - 3 \beta_{7} ) q^{60} + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{61} + ( 1 - \beta_{1} ) q^{62} + ( -4 + \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{6} ) q^{63} + ( 7 - \beta_{1} - \beta_{3} - 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{64} + ( 2 + \beta_{2} + \beta_{5} ) q^{65} + ( 11 - \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{66} + ( 2 - \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{67} + ( -5 - \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + 5 \beta_{6} - \beta_{7} ) q^{68} + ( -3 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{7} ) q^{69} + ( -4 - 3 \beta_{1} - \beta_{2} - 6 \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{70} + ( -1 - \beta_{2} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{71} + ( 6 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - \beta_{4} - 6 \beta_{5} - 10 \beta_{6} + 5 \beta_{7} ) q^{72} + ( 3 - 2 \beta_{1} - 3 \beta_{2} - \beta_{5} - \beta_{6} ) q^{73} + ( 10 - 3 \beta_{1} + 7 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} ) q^{74} + ( -3 + 5 \beta_{1} - 7 \beta_{2} - 3 \beta_{4} - 4 \beta_{5} - 7 \beta_{6} + 3 \beta_{7} ) q^{75} + ( -3 - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{76} + ( -6 - 3 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} ) q^{77} + ( 1 + \beta_{1} + \beta_{3} - \beta_{5} - 2 \beta_{6} ) q^{78} + ( 4 - 2 \beta_{2} - 2 \beta_{3} - 4 \beta_{5} + \beta_{7} ) q^{79} + ( -3 + 8 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} - 5 \beta_{6} + \beta_{7} ) q^{80} + ( 3 - 6 \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{5} + 4 \beta_{6} - 4 \beta_{7} ) q^{81} + ( -3 - 3 \beta_{1} - 5 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + \beta_{5} + 4 \beta_{6} - 3 \beta_{7} ) q^{82} + ( -1 + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{83} + ( -\beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{84} + ( 3 - \beta_{2} + 3 \beta_{3} - \beta_{4} - 2 \beta_{5} - 4 \beta_{6} - \beta_{7} ) q^{85} + ( 5 + \beta_{1} - \beta_{2} - \beta_{3} - 7 \beta_{5} - 4 \beta_{6} + 3 \beta_{7} ) q^{86} + ( 4 - 5 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{87} + ( 11 - \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - 7 \beta_{5} - 6 \beta_{6} + 5 \beta_{7} ) q^{88} + ( -3 - 3 \beta_{1} + \beta_{2} - 4 \beta_{3} + \beta_{4} + 2 \beta_{6} - 2 \beta_{7} ) q^{89} + ( 14 - 11 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} + 9 \beta_{5} + 11 \beta_{6} - 6 \beta_{7} ) q^{90} + ( -\beta_{2} - \beta_{3} - \beta_{4} ) q^{91} + ( 6 - 4 \beta_{1} + 3 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{92} + \beta_{6} q^{93} + ( 6 - 2 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - 4 \beta_{6} + 4 \beta_{7} ) q^{94} + ( 4 + 6 \beta_{1} + 5 \beta_{2} + \beta_{3} + 3 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} ) q^{95} + ( 3 - 2 \beta_{4} + 5 \beta_{5} + 4 \beta_{6} - \beta_{7} ) q^{96} + ( -3 - 2 \beta_{2} - 2 \beta_{3} - \beta_{6} - 2 \beta_{7} ) q^{97} + ( 6 + 4 \beta_{2} + \beta_{3} + 3 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{98} + ( 6 - 3 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + 3 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 5q^{2} - 3q^{3} + 9q^{4} - 15q^{5} - 4q^{7} - 15q^{8} + 9q^{9} + O(q^{10}) \) \( 8q - 5q^{2} - 3q^{3} + 9q^{4} - 15q^{5} - 4q^{7} - 15q^{8} + 9q^{9} + 9q^{10} - 5q^{11} - 9q^{12} - 8q^{13} + 2q^{14} + 2q^{15} + 3q^{16} - 11q^{17} - 30q^{18} - 9q^{19} - 31q^{20} - 16q^{21} - 2q^{22} + 13q^{24} + 19q^{25} + 5q^{26} - 9q^{27} + 16q^{28} - 12q^{29} - 7q^{30} - 8q^{31} - 25q^{32} - 14q^{33} + 22q^{34} + 7q^{35} + 37q^{36} - 9q^{37} - 9q^{38} + 3q^{39} + 55q^{40} - 25q^{41} - 3q^{42} + 7q^{43} - 26q^{44} - 45q^{45} + 5q^{46} - 17q^{47} - 9q^{48} - 11q^{50} - 10q^{51} - 9q^{52} - 15q^{53} + 54q^{54} + 7q^{55} - 14q^{56} - 7q^{57} - 5q^{58} - 15q^{59} + 61q^{60} + 11q^{61} + 5q^{62} - 21q^{63} + 47q^{64} + 15q^{65} + 83q^{66} + 18q^{67} - 16q^{68} - 15q^{69} - 24q^{70} - 7q^{71} - 21q^{72} + 24q^{73} + 48q^{74} - 17q^{75} - 3q^{76} - 49q^{77} + 33q^{79} - 16q^{80} + 20q^{81} - q^{82} - 13q^{83} - 6q^{84} + q^{85} + 19q^{86} + 18q^{87} + 37q^{88} - 23q^{89} + 117q^{90} + 4q^{91} + 22q^{92} + 3q^{93} + 10q^{94} + 43q^{95} + 46q^{96} - 17q^{97} + 52q^{98} + 51q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{7} - 7 x^{6} + 19 x^{5} + 21 x^{4} - 31 x^{3} - 29 x^{2} - x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 4 \nu \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 3 \nu^{3} - 3 \nu^{2} + 9 \nu + 3 \)
\(\beta_{5}\)\(=\)\( \nu^{5} - 3 \nu^{4} - 3 \nu^{3} + 9 \nu^{2} + 3 \nu \)
\(\beta_{6}\)\(=\)\( \nu^{7} - 4 \nu^{6} - 3 \nu^{5} + 21 \nu^{4} + 3 \nu^{3} - 30 \nu^{2} - 9 \nu + 2 \)
\(\beta_{7}\)\(=\)\( \nu^{7} - 3 \nu^{6} - 7 \nu^{5} + 20 \nu^{4} + 18 \nu^{3} - 34 \nu^{2} - 20 \nu + 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 5 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(\beta_{4} + 3 \beta_{3} + 6 \beta_{2} + 9 \beta_{1} + 15\)
\(\nu^{5}\)\(=\)\(\beta_{5} + 3 \beta_{4} + 12 \beta_{3} + 12 \beta_{2} + 30 \beta_{1} + 27\)
\(\nu^{6}\)\(=\)\(\beta_{7} - \beta_{6} + 4 \beta_{5} + 13 \beta_{4} + 36 \beta_{3} + 43 \beta_{2} + 69 \beta_{1} + 90\)
\(\nu^{7}\)\(=\)\(4 \beta_{7} - 3 \beta_{6} + 19 \beta_{5} + 40 \beta_{4} + 114 \beta_{3} + 109 \beta_{2} + 201 \beta_{1} + 205\)

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
−1.66697
−1.57233
−0.759212
−0.247616
0.158875
1.93310
2.39479
2.75938
−2.66697 1.08287 5.11275 −1.31345 −2.88800 1.98184 −8.30164 −1.82738 3.50294
1.2 −2.57233 −3.30456 4.61688 −4.29347 8.50042 0.180190 −6.73148 7.92012 11.0442
1.3 −1.75921 2.94965 1.09483 −3.30948 −5.18907 −3.55859 1.59239 5.70046 5.82207
1.4 −1.24762 −2.42435 −0.443454 0.848757 3.02465 −1.14028 3.04849 2.87746 −1.05892
1.5 −0.841125 0.162061 −1.29251 0.443680 −0.136313 0.552392 2.76941 −2.97374 −0.373191
1.6 0.933096 1.42306 −1.12933 −3.66529 1.32785 −3.96149 −2.91997 −0.974910 −3.42006
1.7 1.39479 −1.21460 −0.0545724 −0.0316546 −1.69410 −2.20383 −2.86569 −1.52475 −0.0441514
1.8 1.75938 −1.67414 1.09540 −3.67909 −2.94544 4.14977 −1.59152 −0.197253 −6.47291
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

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 403.2.a.d 8
3.b odd 2 1 3627.2.a.q 8
4.b odd 2 1 6448.2.a.bf 8
13.b even 2 1 5239.2.a.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.a.d 8 1.a even 1 1 trivial
3627.2.a.q 8 3.b odd 2 1
5239.2.a.j 8 13.b even 2 1
6448.2.a.bf 8 4.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(13\) \(1\)
\(31\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 5 T_{2}^{7} - 30 T_{2}^{5} - 24 T_{2}^{4} + 54 T_{2}^{3} + 54 T_{2}^{2} - 28 T_{2} - 29 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(403))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 5 T + 16 T^{2} + 40 T^{3} + 88 T^{4} + 174 T^{5} + 310 T^{6} + 496 T^{7} + 731 T^{8} + 992 T^{9} + 1240 T^{10} + 1392 T^{11} + 1408 T^{12} + 1280 T^{13} + 1024 T^{14} + 640 T^{15} + 256 T^{16} \)
$3$ \( 1 + 3 T + 12 T^{2} + 27 T^{3} + 67 T^{4} + 124 T^{5} + 235 T^{6} + 396 T^{7} + 702 T^{8} + 1188 T^{9} + 2115 T^{10} + 3348 T^{11} + 5427 T^{12} + 6561 T^{13} + 8748 T^{14} + 6561 T^{15} + 6561 T^{16} \)
$5$ \( 1 + 15 T + 123 T^{2} + 717 T^{3} + 3289 T^{4} + 12450 T^{5} + 39947 T^{6} + 110340 T^{7} + 264523 T^{8} + 551700 T^{9} + 998675 T^{10} + 1556250 T^{11} + 2055625 T^{12} + 2240625 T^{13} + 1921875 T^{14} + 1171875 T^{15} + 390625 T^{16} \)
$7$ \( 1 + 4 T + 36 T^{2} + 108 T^{3} + 578 T^{4} + 1371 T^{5} + 5846 T^{6} + 11754 T^{7} + 45123 T^{8} + 82278 T^{9} + 286454 T^{10} + 470253 T^{11} + 1387778 T^{12} + 1815156 T^{13} + 4235364 T^{14} + 3294172 T^{15} + 5764801 T^{16} \)
$11$ \( 1 + 5 T + 32 T^{2} + 76 T^{3} + 397 T^{4} + 704 T^{5} + 4685 T^{6} + 6125 T^{7} + 48318 T^{8} + 67375 T^{9} + 566885 T^{10} + 937024 T^{11} + 5812477 T^{12} + 12239876 T^{13} + 56689952 T^{14} + 97435855 T^{15} + 214358881 T^{16} \)
$13$ \( ( 1 + T )^{8} \)
$17$ \( 1 + 11 T + 126 T^{2} + 996 T^{3} + 7123 T^{4} + 42792 T^{5} + 234463 T^{6} + 1116927 T^{7} + 4923742 T^{8} + 18987759 T^{9} + 67759807 T^{10} + 210237096 T^{11} + 594920083 T^{12} + 1414177572 T^{13} + 3041333694 T^{14} + 4513725403 T^{15} + 6975757441 T^{16} \)
$19$ \( 1 + 9 T + 106 T^{2} + 708 T^{3} + 4938 T^{4} + 25924 T^{5} + 141398 T^{6} + 623397 T^{7} + 3004911 T^{8} + 11844543 T^{9} + 51044678 T^{10} + 177812716 T^{11} + 643525098 T^{12} + 1753078092 T^{13} + 4986863386 T^{14} + 8044845651 T^{15} + 16983563041 T^{16} \)
$23$ \( 1 + 98 T^{2} + 84 T^{3} + 4275 T^{4} + 9392 T^{5} + 115707 T^{6} + 424422 T^{7} + 2628074 T^{8} + 9761706 T^{9} + 61209003 T^{10} + 114272464 T^{11} + 1196320275 T^{12} + 540652812 T^{13} + 14507517122 T^{14} + 78310985281 T^{16} \)
$29$ \( 1 + 12 T + 148 T^{2} + 1103 T^{3} + 8738 T^{4} + 53277 T^{5} + 358535 T^{6} + 1929446 T^{7} + 11586852 T^{8} + 55953934 T^{9} + 301527935 T^{10} + 1299372753 T^{11} + 6180221378 T^{12} + 22623797347 T^{13} + 88033851508 T^{14} + 206998515708 T^{15} + 500246412961 T^{16} \)
$31$ \( ( 1 + T )^{8} \)
$37$ \( 1 + 9 T + 165 T^{2} + 991 T^{3} + 12935 T^{4} + 67343 T^{5} + 740502 T^{6} + 3375839 T^{7} + 31495474 T^{8} + 124906043 T^{9} + 1013747238 T^{10} + 3411124979 T^{11} + 24242272535 T^{12} + 68719861387 T^{13} + 423344857485 T^{14} + 854386894197 T^{15} + 3512479453921 T^{16} \)
$41$ \( 1 + 25 T + 476 T^{2} + 6372 T^{3} + 72959 T^{4} + 693235 T^{5} + 5892153 T^{6} + 43853249 T^{7} + 297848517 T^{8} + 1797983209 T^{9} + 9904709193 T^{10} + 47778449435 T^{11} + 206164696799 T^{12} + 738235712772 T^{13} + 2261049618716 T^{14} + 4868856847025 T^{15} + 7984925229121 T^{16} \)
$43$ \( 1 - 7 T + 145 T^{2} - 711 T^{3} + 10108 T^{4} - 44326 T^{5} + 602614 T^{6} - 2834324 T^{7} + 30515756 T^{8} - 121875932 T^{9} + 1114233286 T^{10} - 3524227282 T^{11} + 34557240508 T^{12} - 104523002973 T^{13} + 916597642105 T^{14} - 1902730277749 T^{15} + 11688200277601 T^{16} \)
$47$ \( 1 + 17 T + 241 T^{2} + 1981 T^{3} + 14608 T^{4} + 51402 T^{5} + 85212 T^{6} - 2325046 T^{7} - 16365036 T^{8} - 109277162 T^{9} + 188233308 T^{10} + 5336709846 T^{11} + 71282380048 T^{12} + 454332458867 T^{13} + 2597790894289 T^{14} + 8612593047871 T^{15} + 23811286661761 T^{16} \)
$53$ \( 1 + 15 T + 424 T^{2} + 4950 T^{3} + 77587 T^{4} + 727866 T^{5} + 8123777 T^{6} + 61843695 T^{7} + 535702502 T^{8} + 3277715835 T^{9} + 22819689593 T^{10} + 108362506482 T^{11} + 612198749347 T^{12} + 2070067690350 T^{13} + 9397689118696 T^{14} + 17620667097555 T^{15} + 62259690411361 T^{16} \)
$59$ \( 1 + 15 T + 379 T^{2} + 4031 T^{3} + 62177 T^{4} + 536492 T^{5} + 6357685 T^{6} + 46119098 T^{7} + 447458939 T^{8} + 2721026782 T^{9} + 22131101485 T^{10} + 110184190468 T^{11} + 753421154897 T^{12} + 2881859849269 T^{13} + 15986422249939 T^{14} + 37329772272285 T^{15} + 146830437604321 T^{16} \)
$61$ \( 1 - 11 T + 294 T^{2} - 3523 T^{3} + 47281 T^{4} - 506350 T^{5} + 5141323 T^{6} - 44680182 T^{7} + 381642698 T^{8} - 2725491102 T^{9} + 19130862883 T^{10} - 114931829350 T^{11} + 654645208321 T^{12} - 2975512768423 T^{13} + 15146990062134 T^{14} - 34570171196231 T^{15} + 191707312997281 T^{16} \)
$67$ \( 1 - 18 T + 598 T^{2} - 7908 T^{3} + 146860 T^{4} - 1515607 T^{5} + 20041239 T^{6} - 165114013 T^{7} + 1686479924 T^{8} - 11062638871 T^{9} + 89965121871 T^{10} - 455838508141 T^{11} + 2959393630060 T^{12} - 10676789346156 T^{13} + 54094112537062 T^{14} - 109092808895814 T^{15} + 406067677556641 T^{16} \)
$71$ \( 1 + 7 T + 390 T^{2} + 2460 T^{3} + 74246 T^{4} + 409796 T^{5} + 8984378 T^{6} + 42737787 T^{7} + 755659863 T^{8} + 3034382877 T^{9} + 45290249498 T^{10} + 146670496156 T^{11} + 1886715667526 T^{12} + 4438404203460 T^{13} + 49959110729190 T^{14} + 63665841108737 T^{15} + 645753531245761 T^{16} \)
$73$ \( 1 - 24 T + 654 T^{2} - 10155 T^{3} + 163680 T^{4} - 1907249 T^{5} + 22776963 T^{6} - 212473704 T^{7} + 2032799812 T^{8} - 15510580392 T^{9} + 121378435827 T^{10} - 741952284233 T^{11} + 4648224086880 T^{12} - 21052042026915 T^{13} + 98972583993006 T^{14} - 265137564458328 T^{15} + 806460091894081 T^{16} \)
$79$ \( 1 - 33 T + 797 T^{2} - 13771 T^{3} + 206359 T^{4} - 2593505 T^{5} + 29672938 T^{6} - 299405543 T^{7} + 2810094306 T^{8} - 23653037897 T^{9} + 185188806058 T^{10} - 1278699111695 T^{11} + 8037699765079 T^{12} - 42374143670629 T^{13} + 193740702050237 T^{14} - 633728996543247 T^{15} + 1517108809906561 T^{16} \)
$83$ \( 1 + 13 T + 473 T^{2} + 5847 T^{3} + 113012 T^{4} + 1228906 T^{5} + 16911750 T^{6} + 156075296 T^{7} + 1700063772 T^{8} + 12954249568 T^{9} + 116505045750 T^{10} + 702672475022 T^{11} + 5363359772852 T^{12} + 23031570639621 T^{13} + 154642796603537 T^{14} + 352768662865151 T^{15} + 2252292232139041 T^{16} \)
$89$ \( 1 + 23 T + 618 T^{2} + 10671 T^{3} + 178919 T^{4} + 2353266 T^{5} + 30161411 T^{6} + 320307670 T^{7} + 3279691286 T^{8} + 28507382630 T^{9} + 238908536531 T^{10} + 1658979578754 T^{11} + 11225779017479 T^{12} + 59587498380279 T^{13} + 307134437813898 T^{14} + 1017320702597167 T^{15} + 3936588805702081 T^{16} \)
$97$ \( 1 + 17 T + 651 T^{2} + 8770 T^{3} + 194932 T^{4} + 2156514 T^{5} + 35045998 T^{6} + 321297201 T^{7} + 4144181257 T^{8} + 31165828497 T^{9} + 329747795182 T^{10} + 1968192101922 T^{11} + 17257189803892 T^{12} + 75310974053890 T^{13} + 542264775208779 T^{14} + 1373570836127921 T^{15} + 7837433594376961 T^{16} \)
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