Properties

Label 4018.2.a.bs
Level 4018
Weight 2
Character orbit 4018.a
Self dual yes
Analytic conductor 32.084
Analytic rank 0
Dimension 10
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 4018 = 2 \cdot 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4018.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.0838915322\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 2 x^{9} - 17 x^{8} + 36 x^{7} + 75 x^{6} - 174 x^{5} - 69 x^{4} + 260 x^{3} - 104 x^{2} - 24 x + 14\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 13 \nu^{9} - 18 \nu^{8} - 234 \nu^{7} + 330 \nu^{6} + 1201 \nu^{5} - 1610 \nu^{4} - 1896 \nu^{3} + 2442 \nu^{2} - 18 \nu - 330 \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( -114 \nu^{9} + 167 \nu^{8} + 2023 \nu^{7} - 3005 \nu^{6} - 10109 \nu^{5} + 14188 \nu^{4} + 15490 \nu^{3} - 20714 \nu^{2} + 230 \nu + 2842 \)\()/14\)
\(\beta_{3}\)\(=\)\((\)\( 117 \nu^{9} - 169 \nu^{8} - 2086 \nu^{7} + 3064 \nu^{6} + 10513 \nu^{5} - 14673 \nu^{4} - 16220 \nu^{3} + 21804 \nu^{2} - 358 \nu - 2996 \)\()/14\)
\(\beta_{4}\)\(=\)\((\)\( -24 \nu^{9} + 35 \nu^{8} + 426 \nu^{7} - 630 \nu^{6} - 2130 \nu^{5} + 2975 \nu^{4} + 3266 \nu^{3} - 4344 \nu^{2} + 52 \nu + 594 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( -249 \nu^{9} + 362 \nu^{8} + 4424 \nu^{7} - 6528 \nu^{6} - 22157 \nu^{5} + 30938 \nu^{4} + 34004 \nu^{3} - 45404 \nu^{2} + 614 \nu + 6230 \)\()/14\)
\(\beta_{6}\)\(=\)\((\)\( -261 \nu^{9} + 377 \nu^{8} + 4641 \nu^{7} - 6799 \nu^{6} - 23290 \nu^{5} + 32206 \nu^{4} + 35916 \nu^{3} - 47216 \nu^{2} + 356 \nu + 6496 \)\()/14\)
\(\beta_{7}\)\(=\)\((\)\( -270 \nu^{9} + 390 \nu^{8} + 4809 \nu^{7} - 7053 \nu^{6} - 24201 \nu^{5} + 33605 \nu^{4} + 37378 \nu^{3} - 49632 \nu^{2} + 558 \nu + 6832 \)\()/14\)
\(\beta_{8}\)\(=\)\((\)\( 41 \nu^{9} - 60 \nu^{8} - 727 \nu^{7} + 1079 \nu^{6} + 3628 \nu^{5} - 5087 \nu^{4} - 5552 \nu^{3} + 7416 \nu^{2} - 84 \nu - 1018 \)\()/2\)
\(\beta_{9}\)\(=\)\((\)\( 319 \nu^{9} - 453 \nu^{8} - 5698 \nu^{7} + 8222 \nu^{6} + 28835 \nu^{5} - 39401 \nu^{4} - 44840 \nu^{3} + 58592 \nu^{2} - 474 \nu - 8050 \)\()/14\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{9} + \beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} - \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\(-\beta_{8} - \beta_{6} + \beta_{3} - \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(4 \beta_{9} - \beta_{7} + 5 \beta_{6} + 4 \beta_{5} - 6 \beta_{4} - \beta_{3} - 2 \beta_{2} - 4 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(-2 \beta_{9} - 9 \beta_{8} + 3 \beta_{7} - 13 \beta_{6} + \beta_{5} + 14 \beta_{3} + \beta_{2} - 7 \beta_{1} + 30\)
\(\nu^{5}\)\(=\)\(38 \beta_{9} + 6 \beta_{8} - 15 \beta_{7} + 54 \beta_{6} + 40 \beta_{5} - 60 \beta_{4} - 17 \beta_{3} - 11 \beta_{2} - 35 \beta_{1} + 8\)
\(\nu^{6}\)\(=\)\(-33 \beta_{9} - 82 \beta_{8} + 44 \beta_{7} - 148 \beta_{6} + 10 \beta_{5} + 12 \beta_{4} + 159 \beta_{3} + 18 \beta_{2} - 52 \beta_{1} + 265\)
\(\nu^{7}\)\(=\)\(382 \beta_{9} + 107 \beta_{8} - 174 \beta_{7} + 586 \beta_{6} + 406 \beta_{5} - 603 \beta_{4} - 220 \beta_{3} - 73 \beta_{2} - 323 \beta_{1} - 14\)
\(\nu^{8}\)\(=\)\(-438 \beta_{9} - 789 \beta_{8} + 529 \beta_{7} - 1643 \beta_{6} + 45 \beta_{5} + 266 \beta_{4} + 1712 \beta_{3} + 245 \beta_{2} - 399 \beta_{1} + 2520\)
\(\nu^{9}\)\(=\)\(3933 \beta_{9} + 1434 \beta_{8} - 1905 \beta_{7} + 6349 \beta_{6} + 4129 \beta_{5} - 6123 \beta_{4} - 2655 \beta_{3} - 584 \beta_{2} - 3076 \beta_{1} - 947\)

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
−3.30199
−0.339114
0.835626
3.06975
1.92458
0.560789
−1.84058
2.39018
0.553933
−1.85317
−1.00000 −2.26016 1.00000 0.772879 2.26016 0 −1.00000 2.10833 −0.772879
1.2 −1.00000 −2.13421 1.00000 0.374303 2.13421 0 −1.00000 1.55487 −0.374303
1.3 −1.00000 −1.96764 1.00000 −1.63790 1.96764 0 −1.00000 0.871591 1.63790
1.4 −1.00000 −0.539617 1.00000 3.38661 0.539617 0 −1.00000 −2.70881 −3.38661
1.5 −1.00000 0.280978 1.00000 −0.181985 −0.280978 0 −1.00000 −2.92105 0.181985
1.6 −1.00000 0.748903 1.00000 −1.86270 −0.748903 0 −1.00000 −2.43914 1.86270
1.7 −1.00000 1.68800 1.00000 4.36119 −1.68800 0 −1.00000 −0.150660 −4.36119
1.8 −1.00000 2.25734 1.00000 −3.63252 −2.25734 0 −1.00000 2.09559 3.63252
1.9 −1.00000 2.84461 1.00000 0.100034 −2.84461 0 −1.00000 5.09179 −0.100034
1.10 −1.00000 3.08180 1.00000 2.32009 −3.08180 0 −1.00000 6.49750 −2.32009
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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 4018.2.a.bs yes 10
7.b odd 2 1 4018.2.a.br 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4018.2.a.br 10 7.b odd 2 1
4018.2.a.bs yes 10 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(1\)
\(41\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4018))\):

\(T_{3}^{10} - \cdots\)
\(T_{5}^{10} - \cdots\)
\(T_{11}^{10} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{10} \)
$3$ \( 1 - 4 T + 18 T^{2} - 52 T^{3} + 158 T^{4} - 384 T^{5} + 952 T^{6} - 2004 T^{7} + 4229 T^{8} - 7788 T^{9} + 14376 T^{10} - 23364 T^{11} + 38061 T^{12} - 54108 T^{13} + 77112 T^{14} - 93312 T^{15} + 115182 T^{16} - 113724 T^{17} + 118098 T^{18} - 78732 T^{19} + 59049 T^{20} \)
$5$ \( 1 - 4 T + 30 T^{2} - 104 T^{3} + 426 T^{4} - 1280 T^{5} + 3870 T^{6} - 10220 T^{7} + 25849 T^{8} - 61816 T^{9} + 140242 T^{10} - 309080 T^{11} + 646225 T^{12} - 1277500 T^{13} + 2418750 T^{14} - 4000000 T^{15} + 6656250 T^{16} - 8125000 T^{17} + 11718750 T^{18} - 7812500 T^{19} + 9765625 T^{20} \)
$7$ 1
$11$ \( 1 - 4 T + 56 T^{2} - 212 T^{3} + 1703 T^{4} - 5724 T^{5} + 35078 T^{6} - 105956 T^{7} + 541704 T^{8} - 1474696 T^{9} + 6642204 T^{10} - 16221656 T^{11} + 65546184 T^{12} - 141027436 T^{13} + 513576998 T^{14} - 921855924 T^{15} + 3016968383 T^{16} - 4131280252 T^{17} + 12004097336 T^{18} - 9431790764 T^{19} + 25937424601 T^{20} \)
$13$ \( 1 - 4 T + 84 T^{2} - 376 T^{3} + 3589 T^{4} - 16004 T^{5} + 102524 T^{6} - 415772 T^{7} + 2114898 T^{8} - 7444340 T^{9} + 32048656 T^{10} - 96776420 T^{11} + 357417762 T^{12} - 913451084 T^{13} + 2928187964 T^{14} - 5942173172 T^{15} + 17323417501 T^{16} - 23593442392 T^{17} + 68521380564 T^{18} - 42417997492 T^{19} + 137858491849 T^{20} \)
$17$ \( 1 - 20 T + 270 T^{2} - 2732 T^{3} + 23170 T^{4} - 168248 T^{5} + 1082200 T^{6} - 6215012 T^{7} + 32344925 T^{8} - 152774628 T^{9} + 659622840 T^{10} - 2597168676 T^{11} + 9347683325 T^{12} - 30534353956 T^{13} + 90386426200 T^{14} - 238888100536 T^{15} + 559267473730 T^{16} - 1121045254636 T^{17} + 1883454509070 T^{18} - 2371757529940 T^{19} + 2015993900449 T^{20} \)
$19$ \( 1 + 56 T^{2} + 220 T^{3} + 1745 T^{4} + 11364 T^{5} + 65212 T^{6} + 305044 T^{7} + 1873190 T^{8} + 7472140 T^{9} + 38409720 T^{10} + 141970660 T^{11} + 676221590 T^{12} + 2092296796 T^{13} + 8498493052 T^{14} + 28138389036 T^{15} + 82095062345 T^{16} + 196651782580 T^{17} + 951079530296 T^{18} + 6131066257801 T^{20} \)
$23$ \( 1 - 4 T + 154 T^{2} - 704 T^{3} + 11405 T^{4} - 54132 T^{5} + 550428 T^{6} - 2477604 T^{7} + 19320594 T^{8} - 77660236 T^{9} + 510605492 T^{10} - 1786185428 T^{11} + 10220594226 T^{12} - 30145007868 T^{13} + 154032321948 T^{14} - 348412119276 T^{15} + 1688349314045 T^{16} - 2396997114688 T^{17} + 12059891733274 T^{18} - 7204610645852 T^{19} + 41426511213649 T^{20} \)
$29$ \( 1 + 4 T + 154 T^{2} + 676 T^{3} + 12796 T^{4} + 54880 T^{5} + 736946 T^{6} + 2928168 T^{7} + 31527057 T^{8} + 113549636 T^{9} + 1036422246 T^{10} + 3292939444 T^{11} + 26514254937 T^{12} + 71415089352 T^{13} + 521227903826 T^{14} + 1125651857120 T^{15} + 7611359215516 T^{16} + 11660916384884 T^{17} + 77037947595994 T^{18} + 58028583903476 T^{19} + 420707233300201 T^{20} \)
$31$ \( 1 + 4 T + 198 T^{2} + 508 T^{3} + 17854 T^{4} + 24492 T^{5} + 1008216 T^{6} + 396612 T^{7} + 41612613 T^{8} - 8285408 T^{9} + 1397476804 T^{10} - 256847648 T^{11} + 39989721093 T^{12} + 11815468092 T^{13} + 931108648536 T^{14} + 701185166292 T^{15} + 15845490720574 T^{16} + 13976407968388 T^{17} + 168872425413318 T^{18} + 105758488642684 T^{19} + 819628286980801 T^{20} \)
$37$ \( 1 + 16 T + 358 T^{2} + 4432 T^{3} + 57353 T^{4} + 569536 T^{5} + 5474048 T^{6} + 44713024 T^{7} + 346781998 T^{8} + 2366254304 T^{9} + 15280917108 T^{10} + 87551409248 T^{11} + 474744555262 T^{12} + 2264848804672 T^{13} + 10259247273728 T^{14} + 39493879893952 T^{15} + 147152106735377 T^{16} + 420738079453456 T^{17} + 1257467644503718 T^{18} + 2079387836721232 T^{19} + 4808584372417849 T^{20} \)
$41$ \( ( 1 - T )^{10} \)
$43$ \( 1 + 8 T + 156 T^{2} + 1708 T^{3} + 17262 T^{4} + 158080 T^{5} + 1448222 T^{6} + 10630400 T^{7} + 85213937 T^{8} + 589574964 T^{9} + 3956312860 T^{10} + 25351723452 T^{11} + 157560569513 T^{12} + 845191212800 T^{13} + 4951182821822 T^{14} + 23239094669440 T^{15} + 109119368951838 T^{16} + 464266187770756 T^{17} + 1823359243305756 T^{18} + 4020740895494744 T^{19} + 21611482313284249 T^{20} \)
$47$ \( 1 - 24 T + 416 T^{2} - 4844 T^{3} + 48025 T^{4} - 393252 T^{5} + 3157396 T^{6} - 24584516 T^{7} + 202236454 T^{8} - 1564612916 T^{9} + 11446650088 T^{10} - 73536807052 T^{11} + 446740326886 T^{12} - 2552438204668 T^{13} + 15407085270676 T^{14} - 90190382692764 T^{15} + 517671816175225 T^{16} - 2454082395522772 T^{17} + 9905495251292576 T^{18} - 26859131354466408 T^{19} + 52599132235830049 T^{20} \)
$53$ \( 1 + 4 T + 268 T^{2} + 1160 T^{3} + 36052 T^{4} + 190492 T^{5} + 3281812 T^{6} + 20651104 T^{7} + 228036701 T^{8} + 1544728772 T^{9} + 13081613782 T^{10} + 81870624916 T^{11} + 640555093109 T^{12} + 3074474410208 T^{13} + 25895075231572 T^{14} + 79662895852556 T^{15} + 799069547422708 T^{16} + 1362664922210920 T^{17} + 16685597030244748 T^{18} + 13199054367208532 T^{19} + 174887470365513049 T^{20} \)
$59$ \( 1 + 392 T^{2} - 124 T^{3} + 72653 T^{4} - 53436 T^{5} + 8565058 T^{6} - 9694268 T^{7} + 730980722 T^{8} - 964042332 T^{9} + 48398429980 T^{10} - 56878497588 T^{11} + 2544543893282 T^{12} - 1990999067572 T^{13} + 103785899771938 T^{14} - 38202694841364 T^{15} + 3064542310619573 T^{16} - 308592784117556 T^{17} + 57557531540893832 T^{18} + 511116753300641401 T^{20} \)
$61$ \( 1 + 380 T^{2} - 36 T^{3} + 72834 T^{4} - 9300 T^{5} + 9196908 T^{6} - 1146808 T^{7} + 844205981 T^{8} - 93522728 T^{9} + 58775394162 T^{10} - 5704886408 T^{11} + 3141290455301 T^{12} - 260303626648 T^{13} + 127338925859628 T^{14} - 7854745599300 T^{15} + 3752434946209074 T^{16} - 113138742096756 T^{17} + 72848778938966780 T^{18} + 713342911662882601 T^{20} \)
$67$ \( 1 - 8 T + 470 T^{2} - 2780 T^{3} + 100899 T^{4} - 430000 T^{5} + 13375686 T^{6} - 39953160 T^{7} + 1265966028 T^{8} - 2789089532 T^{9} + 94002716856 T^{10} - 186868998644 T^{11} + 5682921499692 T^{12} - 12016432261080 T^{13} + 269535067044006 T^{14} - 580553796010000 T^{15} + 9127160302469931 T^{16} - 16848778262797940 T^{17} + 190851808451621270 T^{18} - 217652275170359576 T^{19} + 1822837804551761449 T^{20} \)
$71$ \( 1 - 4 T + 484 T^{2} - 2076 T^{3} + 115006 T^{4} - 485372 T^{5} + 17684686 T^{6} - 69611060 T^{7} + 1941910673 T^{8} - 6877557968 T^{9} + 158787895516 T^{10} - 488306615728 T^{11} + 9789171702593 T^{12} - 24914564095660 T^{13} + 449397599217166 T^{14} - 875722408553572 T^{15} + 14732301252618526 T^{16} - 18881469448819716 T^{17} + 312544709122948324 T^{18} - 183394002873796124 T^{19} + 3255243551009881201 T^{20} \)
$73$ \( 1 + 24 T + 540 T^{2} + 7768 T^{3} + 107529 T^{4} + 1154336 T^{5} + 12519760 T^{6} + 113733984 T^{7} + 1100583206 T^{8} + 9222885584 T^{9} + 84373616616 T^{10} + 673270647632 T^{11} + 5865007904774 T^{12} + 44244453253728 T^{13} + 355539161742160 T^{14} + 2393021170377248 T^{15} + 16272818018629881 T^{16} + 85816191696345496 T^{17} + 435488449622803740 T^{18} + 1412918080998429912 T^{19} + 4297625829703557649 T^{20} \)
$79$ \( 1 - 24 T + 552 T^{2} - 8800 T^{3} + 125510 T^{4} - 1488232 T^{5} + 16674002 T^{6} - 163960520 T^{7} + 1599848841 T^{8} - 14392945192 T^{9} + 131572455532 T^{10} - 1137042670168 T^{11} + 9984656616681 T^{12} - 80838930820280 T^{13} + 649453728494162 T^{14} - 4579373798796568 T^{15} + 30509906542440710 T^{16} - 168994399078199200 T^{17} + 837444063068421672 T^{18} - 2876438303582839656 T^{19} + 9468276082626847201 T^{20} \)
$83$ \( 1 - 48 T + 1548 T^{2} - 36724 T^{3} + 724417 T^{4} - 12098348 T^{5} + 177754882 T^{6} - 2311132348 T^{7} + 27054069462 T^{8} - 285199949076 T^{9} + 2731301924780 T^{10} - 23671595773308 T^{11} + 186375484523718 T^{12} - 1321475431865876 T^{13} + 8435948249273122 T^{14} - 47655884485157764 T^{15} + 236841164454850873 T^{16} - 996544336543061948 T^{17} + 3486548375351235468 T^{18} - 8973132252841939344 T^{19} + 15516041187205853449 T^{20} \)
$89$ \( 1 - 20 T + 464 T^{2} - 6416 T^{3} + 107990 T^{4} - 1259284 T^{5} + 17047382 T^{6} - 177374772 T^{7} + 2141128769 T^{8} - 19963266644 T^{9} + 211631427220 T^{10} - 1776730731316 T^{11} + 16959880979249 T^{12} - 125043715642068 T^{13} + 1069590949863062 T^{14} - 7031916719174516 T^{15} + 53669009610878390 T^{16} - 283788244689714064 T^{17} + 1826577205845765584 T^{18} - 7007128074149704180 T^{19} + 31181719929966183601 T^{20} \)
$97$ \( 1 - 4 T + 456 T^{2} - 1808 T^{3} + 105758 T^{4} - 345612 T^{5} + 17311342 T^{6} - 44385844 T^{7} + 2229385937 T^{8} - 5003762220 T^{9} + 236215189236 T^{10} - 485364935340 T^{11} + 20976292281233 T^{12} - 40509761401012 T^{13} + 1532560660405102 T^{14} - 2967887840902284 T^{15} + 88093453297281182 T^{16} - 146083298336428304 T^{17} + 3573869719035894216 T^{18} - 3040924234618260868 T^{19} + 73742412689492826049 T^{20} \)
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