Properties

Label 8034.2.a.r
Level 8034
Weight 2
Character orbit 8034.a
Self dual yes
Analytic conductor 64.152
Analytic rank 1
Dimension 9
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - 4 x^{8} - 9 x^{7} + 45 x^{6} + 7 x^{5} - 123 x^{4} + 37 x^{3} + 87 x^{2} - 54 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - 4 x^{8} - 9 x^{7} + 45 x^{6} + 7 x^{5} - 123 x^{4} + 37 x^{3} + 87 x^{2} - 54 x + 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 55 \nu^{8} - 272 \nu^{7} - 307 \nu^{6} + 2869 \nu^{5} - 1267 \nu^{4} - 6789 \nu^{3} + 3935 \nu^{2} + 4811 \nu - 2412 \)\()/634\)
\(\beta_{3}\)\(=\)\((\)\( 129 \nu^{8} - 442 \nu^{7} - 1331 \nu^{6} + 4781 \nu^{5} + 2815 \nu^{4} - 11785 \nu^{3} - 223 \nu^{2} + 7065 \nu - 3444 \)\()/634\)
\(\beta_{4}\)\(=\)\((\)\( -131 \nu^{8} + 498 \nu^{7} + 1273 \nu^{6} - 5381 \nu^{5} - 2377 \nu^{4} + 12931 \nu^{3} + 1175 \nu^{2} - 5995 \nu + 1964 \)\()/634\)
\(\beta_{5}\)\(=\)\((\)\( 92 \nu^{8} - 357 \nu^{7} - 819 \nu^{6} + 3825 \nu^{5} + 774 \nu^{4} - 9287 \nu^{3} + 1539 \nu^{2} + 5938 \nu - 1977 \)\()/317\)
\(\beta_{6}\)\(=\)\((\)\( -359 \nu^{8} + 1176 \nu^{7} + 4171 \nu^{6} - 13551 \nu^{5} - 12675 \nu^{4} + 38965 \nu^{3} + 11433 \nu^{2} - 29201 \nu + 5692 \)\()/634\)
\(\beta_{7}\)\(=\)\((\)\( -415 \nu^{8} + 1476 \nu^{7} + 4449 \nu^{6} - 17037 \nu^{5} - 10555 \nu^{4} + 49497 \nu^{3} + 3219 \nu^{2} - 39183 \nu + 10534 \)\()/634\)
\(\beta_{8}\)\(=\)\((\)\( 261 \nu^{8} - 968 \nu^{7} - 2575 \nu^{6} + 10779 \nu^{5} + 4656 \nu^{4} - 28776 \nu^{3} + 28 \nu^{2} + 20133 \nu - 5619 \)\()/317\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{5} + \beta_{3} + \beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{8} + \beta_{6} - 2 \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 7 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(2 \beta_{8} + 2 \beta_{7} - \beta_{6} - 11 \beta_{5} + \beta_{4} + 8 \beta_{3} + 10 \beta_{2} + \beta_{1} + 21\)
\(\nu^{5}\)\(=\)\(14 \beta_{8} + 2 \beta_{7} + 10 \beta_{6} - 27 \beta_{5} - 7 \beta_{4} + 12 \beta_{3} - 7 \beta_{2} + 54 \beta_{1} + 17\)
\(\nu^{6}\)\(=\)\(34 \beta_{8} + 27 \beta_{7} - 7 \beta_{6} - 114 \beta_{5} + 17 \beta_{4} + 73 \beta_{3} + 86 \beta_{2} + 17 \beta_{1} + 177\)
\(\nu^{7}\)\(=\)\(165 \beta_{8} + 45 \beta_{7} + 86 \beta_{6} - 305 \beta_{5} - 25 \beta_{4} + 141 \beta_{3} - 35 \beta_{2} + 434 \beta_{1} + 213\)
\(\nu^{8}\)\(=\)\(445 \beta_{8} + 315 \beta_{7} - 35 \beta_{6} - 1165 \beta_{5} + 236 \beta_{4} + 715 \beta_{3} + 719 \beta_{2} + 224 \beta_{1} + 1591\)

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
3.20969
2.80812
1.90297
0.840786
0.435443
0.254984
−1.25914
−1.44298
−2.74988
1.00000 −1.00000 1.00000 −3.20969 −1.00000 −2.02210 1.00000 1.00000 −3.20969
1.2 1.00000 −1.00000 1.00000 −2.80812 −1.00000 −1.72587 1.00000 1.00000 −2.80812
1.3 1.00000 −1.00000 1.00000 −1.90297 −1.00000 −5.10863 1.00000 1.00000 −1.90297
1.4 1.00000 −1.00000 1.00000 −0.840786 −1.00000 2.87036 1.00000 1.00000 −0.840786
1.5 1.00000 −1.00000 1.00000 −0.435443 −1.00000 1.91967 1.00000 1.00000 −0.435443
1.6 1.00000 −1.00000 1.00000 −0.254984 −1.00000 2.89551 1.00000 1.00000 −0.254984
1.7 1.00000 −1.00000 1.00000 1.25914 −1.00000 −0.797445 1.00000 1.00000 1.25914
1.8 1.00000 −1.00000 1.00000 1.44298 −1.00000 0.367530 1.00000 1.00000 1.44298
1.9 1.00000 −1.00000 1.00000 2.74988 −1.00000 −2.39902 1.00000 1.00000 2.74988
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
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 8034.2.a.r 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.r 9 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(13\) \(-1\)
\(103\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

\(T_{5}^{9} + \cdots\)
\(T_{7}^{9} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{9} \)
$3$ \( ( 1 + T )^{9} \)
$5$ \( 1 + 4 T + 36 T^{2} + 115 T^{3} + 592 T^{4} + 1573 T^{5} + 5987 T^{6} + 13498 T^{7} + 41626 T^{8} + 80072 T^{9} + 208130 T^{10} + 337450 T^{11} + 748375 T^{12} + 983125 T^{13} + 1850000 T^{14} + 1796875 T^{15} + 2812500 T^{16} + 1562500 T^{17} + 1953125 T^{18} \)
$7$ \( 1 + 4 T + 41 T^{2} + 150 T^{3} + 825 T^{4} + 2836 T^{5} + 10829 T^{6} + 34281 T^{7} + 101917 T^{8} + 285898 T^{9} + 713419 T^{10} + 1679769 T^{11} + 3714347 T^{12} + 6809236 T^{13} + 13865775 T^{14} + 17647350 T^{15} + 33765263 T^{16} + 23059204 T^{17} + 40353607 T^{18} \)
$11$ \( 1 + 5 T + 92 T^{2} + 380 T^{3} + 3797 T^{4} + 13159 T^{5} + 93013 T^{6} + 271467 T^{7} + 1497576 T^{8} + 3653008 T^{9} + 16473336 T^{10} + 32847507 T^{11} + 123800303 T^{12} + 192660919 T^{13} + 611510647 T^{14} + 673193180 T^{15} + 1792819732 T^{16} + 1071794405 T^{17} + 2357947691 T^{18} \)
$13$ \( ( 1 - T )^{9} \)
$17$ \( 1 + 6 T + 105 T^{2} + 535 T^{3} + 5230 T^{4} + 23865 T^{5} + 168151 T^{6} + 689302 T^{7} + 3862362 T^{8} + 13892394 T^{9} + 65660154 T^{10} + 199208278 T^{11} + 826125863 T^{12} + 1993228665 T^{13} + 7425852110 T^{14} + 12913599415 T^{15} + 43085560665 T^{16} + 41854544646 T^{17} + 118587876497 T^{18} \)
$19$ \( 1 + 4 T + 89 T^{2} + 116 T^{3} + 3173 T^{4} - 2110 T^{5} + 87057 T^{6} - 93359 T^{7} + 2295634 T^{8} - 1501406 T^{9} + 43617046 T^{10} - 33702599 T^{11} + 597123963 T^{12} - 274977310 T^{13} + 7856662127 T^{14} + 5457322196 T^{15} + 79554584771 T^{16} + 67934252164 T^{17} + 322687697779 T^{18} \)
$23$ \( 1 + 6 T + 122 T^{2} + 663 T^{3} + 7508 T^{4} + 35794 T^{5} + 303460 T^{6} + 1273572 T^{7} + 9005105 T^{8} + 33522378 T^{9} + 207117415 T^{10} + 673719588 T^{11} + 3692197820 T^{12} + 10016628754 T^{13} + 48324063244 T^{14} + 98147794407 T^{15} + 415388704534 T^{16} + 469865911686 T^{17} + 1801152661463 T^{18} \)
$29$ \( 1 + 19 T + 306 T^{2} + 3152 T^{3} + 27740 T^{4} + 185659 T^{5} + 1075537 T^{6} + 5043049 T^{7} + 22812464 T^{8} + 107108802 T^{9} + 661561456 T^{10} + 4241204209 T^{11} + 26231271893 T^{12} + 131313083179 T^{13} + 568979273260 T^{14} + 1874883107792 T^{15} + 5278462150554 T^{16} + 9504681846259 T^{17} + 14507145975869 T^{18} \)
$31$ \( 1 + 6 T + 173 T^{2} + 812 T^{3} + 13665 T^{4} + 51436 T^{5} + 675453 T^{6} + 2100801 T^{7} + 24902436 T^{8} + 68887654 T^{9} + 771975516 T^{10} + 2018869761 T^{11} + 20122420323 T^{12} + 47502226156 T^{13} + 391217348415 T^{14} + 720652988972 T^{15} + 4759682241203 T^{16} + 5117346224646 T^{17} + 26439622160671 T^{18} \)
$37$ \( 1 + 13 T + 292 T^{2} + 2370 T^{3} + 30442 T^{4} + 164673 T^{5} + 1660233 T^{6} + 6131065 T^{7} + 63235570 T^{8} + 194561306 T^{9} + 2339716090 T^{10} + 8393427985 T^{11} + 84095782149 T^{12} + 308623714353 T^{13} + 2110968738994 T^{14} + 6080771589330 T^{15} + 27720108122836 T^{16} + 45662232900973 T^{17} + 129961739795077 T^{18} \)
$41$ \( 1 + 18 T + 470 T^{2} + 5998 T^{3} + 88496 T^{4} + 866722 T^{5} + 9144477 T^{6} + 71249051 T^{7} + 582284380 T^{8} + 3653553638 T^{9} + 23873659580 T^{10} + 119769654731 T^{11} + 630246499317 T^{12} + 2449149225442 T^{13} + 10252810363696 T^{14} + 28491125237518 T^{15} + 91534508724070 T^{16} + 143728654124178 T^{17} + 327381934393961 T^{18} \)
$43$ \( 1 + 20 T + 341 T^{2} + 4024 T^{3} + 42622 T^{4} + 368278 T^{5} + 2977167 T^{6} + 21109515 T^{7} + 147645727 T^{8} + 959590918 T^{9} + 6348766261 T^{10} + 39031493235 T^{11} + 236705616669 T^{12} + 1259069194678 T^{13} + 6265793857546 T^{14} + 25437164909176 T^{15} + 92690146387487 T^{16} + 233764005552020 T^{17} + 502592611936843 T^{18} \)
$47$ \( 1 - 14 T + 269 T^{2} - 1986 T^{3} + 22311 T^{4} - 81366 T^{5} + 835281 T^{6} + 691675 T^{7} + 16401744 T^{8} + 159981022 T^{9} + 770881968 T^{10} + 1527910075 T^{11} + 86721379263 T^{12} - 397040124246 T^{13} + 5116916451177 T^{14} - 21407521643394 T^{15} + 136281619404547 T^{16} - 333358013264654 T^{17} + 1119130473102767 T^{18} \)
$53$ \( 1 + 3 T + 319 T^{2} + 565 T^{3} + 47654 T^{4} + 34022 T^{5} + 4489511 T^{6} - 127050 T^{7} + 306209494 T^{8} - 83666246 T^{9} + 16229103182 T^{10} - 356883450 T^{11} + 668384929147 T^{12} + 268449944582 T^{13} + 19928688023422 T^{14} + 12522864037885 T^{15} + 374732853608003 T^{16} + 186779071234083 T^{17} + 3299763591802133 T^{18} \)
$59$ \( 1 + 9 T + 280 T^{2} + 2715 T^{3} + 43245 T^{4} + 413654 T^{5} + 4593623 T^{6} + 40696966 T^{7} + 359358817 T^{8} + 2826164840 T^{9} + 21202170203 T^{10} + 141666138646 T^{11} + 943433698117 T^{12} + 5012394847094 T^{13} + 30916901310255 T^{14} + 114520148835315 T^{15} + 696822415749320 T^{16} + 1321473938438889 T^{17} + 8662995818654939 T^{18} \)
$61$ \( 1 + 24 T + 703 T^{2} + 11630 T^{3} + 196570 T^{4} + 2464330 T^{5} + 30146861 T^{6} + 298996083 T^{7} + 2852048483 T^{8} + 22752947118 T^{9} + 173974957463 T^{10} + 1112564424843 T^{11} + 6842764656641 T^{12} + 34120721351530 T^{13} + 166022294887570 T^{14} + 599181953818430 T^{15} + 2209348213722763 T^{16} + 4600975511934744 T^{17} + 11694146092834141 T^{18} \)
$67$ \( 1 + 4 T + 252 T^{2} + 1098 T^{3} + 35440 T^{4} + 133165 T^{5} + 3487683 T^{6} + 11366594 T^{7} + 275587775 T^{8} + 787775524 T^{9} + 18464380925 T^{10} + 51024640466 T^{11} + 1048966002129 T^{12} + 2683424027965 T^{13} + 47848433792080 T^{14} + 99323303621562 T^{15} + 1527299324541396 T^{16} + 1624270710226564 T^{17} + 27206534396294947 T^{18} \)
$71$ \( 1 + 9 T + 330 T^{2} + 1637 T^{3} + 43158 T^{4} + 29483 T^{5} + 2830734 T^{6} - 18603230 T^{7} + 102528747 T^{8} - 2231638070 T^{9} + 7279541037 T^{10} - 93778882430 T^{11} + 1013150836674 T^{12} + 749212590923 T^{13} + 77866930330458 T^{14} + 209700164778677 T^{15} + 3001389652269030 T^{16} + 5811781781211849 T^{17} + 45848500718449031 T^{18} \)
$73$ \( 1 + 24 T + 545 T^{2} + 7036 T^{3} + 90751 T^{4} + 845619 T^{5} + 9305813 T^{6} + 83904070 T^{7} + 901932383 T^{8} + 7390785746 T^{9} + 65841063959 T^{10} + 447124789030 T^{11} + 3620119455821 T^{12} + 24014092156179 T^{13} + 188133320136343 T^{14} + 1064787616169404 T^{15} + 6020832192907865 T^{16} + 19355042205457944 T^{17} + 58871586708267913 T^{18} \)
$79$ \( 1 + 15 T + 382 T^{2} + 3782 T^{3} + 58990 T^{4} + 483541 T^{5} + 6600977 T^{6} + 54987597 T^{7} + 678518210 T^{8} + 5226799074 T^{9} + 53602938590 T^{10} + 343177592877 T^{11} + 3254539099103 T^{12} + 18833961116821 T^{13} + 181515556977010 T^{14} + 919356756780422 T^{15} + 7335893232712738 T^{16} + 22756632148598415 T^{17} + 119851595982618319 T^{18} \)
$83$ \( 1 - 20 T + 604 T^{2} - 7955 T^{3} + 133514 T^{4} - 1257238 T^{5} + 15305398 T^{6} - 109977758 T^{7} + 1204553977 T^{8} - 8133004818 T^{9} + 99977980091 T^{10} - 757636774862 T^{11} + 8751427606226 T^{12} - 59666404577398 T^{13} + 525917072409502 T^{14} - 2600810670150395 T^{15} + 16390174797734708 T^{16} - 45045844642780820 T^{17} + 186940255267540403 T^{18} \)
$89$ \( 1 - 3 T + 350 T^{2} - 1263 T^{3} + 58339 T^{4} - 206631 T^{5} + 6828624 T^{6} - 16185020 T^{7} + 686865500 T^{8} - 1029803626 T^{9} + 61131029500 T^{10} - 128201543420 T^{11} + 4813968232656 T^{12} - 12964492000071 T^{13} + 325768444195211 T^{14} - 627687370483743 T^{15} + 15480967213435150 T^{16} - 11809766417106243 T^{17} + 350356403707485209 T^{18} \)
$97$ \( 1 + 19 T + 793 T^{2} + 12520 T^{3} + 287601 T^{4} + 3773718 T^{5} + 62226404 T^{6} + 681492277 T^{7} + 8847368525 T^{8} + 80684784892 T^{9} + 858194746925 T^{10} + 6412160834293 T^{11} + 56792358817892 T^{12} + 334084541236758 T^{13} + 2469727645253457 T^{14} + 10428809501711080 T^{15} + 64073039591143609 T^{16} + 148911238293162259 T^{17} + 760231058654565217 T^{18} \)
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