Properties

Label 6042.2.a.bc
Level 6042
Weight 2
Character orbit 6042.a
Self dual yes
Analytic conductor 48.246
Analytic rank 1
Dimension 9
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6042.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.2456129013\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - 3 x^{8} - 20 x^{7} + 69 x^{6} + 27 x^{5} - 185 x^{4} + 8 x^{3} + 109 x^{2} - 8 x - 14\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - 3 x^{8} - 20 x^{7} + 69 x^{6} + 27 x^{5} - 185 x^{4} + 8 x^{3} + 109 x^{2} - 8 x - 14\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 18 \nu^{8} + 96 \nu^{7} - 599 \nu^{6} - 2018 \nu^{5} + 6181 \nu^{4} + 7311 \nu^{3} - 11661 \nu^{2} - 11054 \nu + 4020 \)\()/2078\)
\(\beta_{3}\)\(=\)\((\)\( -133 \nu^{8} + 676 \nu^{7} + 2752 \nu^{6} - 16721 \nu^{5} - 3014 \nu^{4} + 77067 \nu^{3} - 19643 \nu^{2} - 79830 \nu + 774 \)\()/8312\)
\(\beta_{4}\)\(=\)\((\)\( 895 \nu^{8} - 2846 \nu^{7} - 18066 \nu^{6} + 64977 \nu^{5} + 27438 \nu^{4} - 173817 \nu^{3} - 22041 \nu^{2} + 94666 \nu + 23946 \)\()/8312\)
\(\beta_{5}\)\(=\)\((\)\( -586 \nu^{8} + 1377 \nu^{7} + 12805 \nu^{6} - 32777 \nu^{5} - 39950 \nu^{4} + 96198 \nu^{3} + 43687 \nu^{2} - 43032 \nu - 2730 \)\()/4156\)
\(\beta_{6}\)\(=\)\((\)\( -695 \nu^{8} + 2181 \nu^{7} + 13373 \nu^{6} - 49418 \nu^{5} - 7016 \nu^{4} + 120673 \nu^{3} - 37220 \nu^{2} - 51710 \nu + 20028 \)\()/4156\)
\(\beta_{7}\)\(=\)\((\)\( -1449 \nu^{8} + 4740 \nu^{7} + 26920 \nu^{6} - 105613 \nu^{5} + 4786 \nu^{4} + 227599 \nu^{3} - 94575 \nu^{2} - 52526 \nu + 25494 \)\()/8312\)
\(\beta_{8}\)\(=\)\((\)\( 1541 \nu^{8} - 2864 \nu^{7} - 35292 \nu^{6} + 69901 \nu^{5} + 144790 \nu^{4} - 206163 \nu^{3} - 242785 \nu^{2} + 95310 \nu + 83714 \)\()/8312\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_{2} + 6\)
\(\nu^{3}\)\(=\)\(-\beta_{8} - 3 \beta_{7} + 4 \beta_{6} - 2 \beta_{5} - 2 \beta_{3} + 2 \beta_{2} + 13 \beta_{1} - 5\)
\(\nu^{4}\)\(=\)\(4 \beta_{8} - 2 \beta_{7} + \beta_{6} - 13 \beta_{5} - 20 \beta_{4} + 16 \beta_{3} - 40 \beta_{2} - 12 \beta_{1} + 86\)
\(\nu^{5}\)\(=\)\(-14 \beta_{8} - 61 \beta_{7} + 77 \beta_{6} - 25 \beta_{5} - 55 \beta_{3} + 50 \beta_{2} + 212 \beta_{1} - 151\)
\(\nu^{6}\)\(=\)\(97 \beta_{8} - 27 \beta_{7} - 9 \beta_{6} - 184 \beta_{5} - 367 \beta_{4} + 273 \beta_{3} - 722 \beta_{2} - 339 \beta_{1} + 1457\)
\(\nu^{7}\)\(=\)\(-231 \beta_{8} - 1080 \beta_{7} + 1338 \beta_{6} - 280 \beta_{5} + 96 \beta_{4} - 1147 \beta_{3} + 1170 \beta_{2} + 3687 \beta_{1} - 3426\)
\(\nu^{8}\)\(=\)\(1923 \beta_{8} - 72 \beta_{7} - 771 \beta_{6} - 2804 \beta_{5} - 6505 \beta_{4} + 5002 \beta_{3} - 12918 \beta_{2} - 7723 \beta_{1} + 25992\)

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
3.89305
2.61835
2.08688
0.801114
0.487125
−0.369644
−0.744377
−1.40257
−4.36993
1.00000 −1.00000 1.00000 −3.89305 −1.00000 −4.93000 1.00000 1.00000 −3.89305
1.2 1.00000 −1.00000 1.00000 −2.61835 −1.00000 3.95780 1.00000 1.00000 −2.61835
1.3 1.00000 −1.00000 1.00000 −2.08688 −1.00000 −3.74967 1.00000 1.00000 −2.08688
1.4 1.00000 −1.00000 1.00000 −0.801114 −1.00000 1.86180 1.00000 1.00000 −0.801114
1.5 1.00000 −1.00000 1.00000 −0.487125 −1.00000 4.75176 1.00000 1.00000 −0.487125
1.6 1.00000 −1.00000 1.00000 0.369644 −1.00000 −1.63265 1.00000 1.00000 0.369644
1.7 1.00000 −1.00000 1.00000 0.744377 −1.00000 −0.530367 1.00000 1.00000 0.744377
1.8 1.00000 −1.00000 1.00000 1.40257 −1.00000 −2.54548 1.00000 1.00000 1.40257
1.9 1.00000 −1.00000 1.00000 4.36993 −1.00000 −4.18318 1.00000 1.00000 4.36993
\(n\): 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 6042.2.a.bc 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6042.2.a.bc 9 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(19\) \(-1\)
\(53\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6042))\):

\(T_{5}^{9} + \cdots\)
\(T_{7}^{9} + \cdots\)
\(T_{11}^{9} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{9} \)
$3$ \( ( 1 + T )^{9} \)
$5$ \( 1 + 3 T + 25 T^{2} + 51 T^{3} + 227 T^{4} + 215 T^{5} + 683 T^{6} - 1284 T^{7} - 1888 T^{8} - 14576 T^{9} - 9440 T^{10} - 32100 T^{11} + 85375 T^{12} + 134375 T^{13} + 709375 T^{14} + 796875 T^{15} + 1953125 T^{16} + 1171875 T^{17} + 1953125 T^{18} \)
$7$ \( 1 + 7 T + 34 T^{2} + 105 T^{3} + 298 T^{4} + 753 T^{5} + 2365 T^{6} + 6913 T^{7} + 21168 T^{8} + 55436 T^{9} + 148176 T^{10} + 338737 T^{11} + 811195 T^{12} + 1807953 T^{13} + 5008486 T^{14} + 12353145 T^{15} + 28000462 T^{16} + 40353607 T^{17} + 40353607 T^{18} \)
$11$ \( 1 - 4 T + 50 T^{2} - 203 T^{3} + 1176 T^{4} - 4492 T^{5} + 17628 T^{6} - 61773 T^{7} + 205021 T^{8} - 695056 T^{9} + 2255231 T^{10} - 7474533 T^{11} + 23462868 T^{12} - 65767372 T^{13} + 189395976 T^{14} - 359626883 T^{15} + 974358550 T^{16} - 857435524 T^{17} + 2357947691 T^{18} \)
$13$ \( 1 + 5 T + 56 T^{2} + 273 T^{3} + 1902 T^{4} + 7937 T^{5} + 43145 T^{6} + 161365 T^{7} + 736960 T^{8} + 2391904 T^{9} + 9580480 T^{10} + 27270685 T^{11} + 94789565 T^{12} + 226688657 T^{13} + 706199286 T^{14} + 1317718857 T^{15} + 3513916952 T^{16} + 4078653605 T^{17} + 10604499373 T^{18} \)
$17$ \( 1 + 28 T + 463 T^{2} + 5504 T^{3} + 51856 T^{4} + 404158 T^{5} + 2680492 T^{6} + 15365899 T^{7} + 76944328 T^{8} + 338242374 T^{9} + 1308053576 T^{10} + 4440744811 T^{11} + 13169257196 T^{12} + 33755680318 T^{13} + 73628104592 T^{14} + 132853179776 T^{15} + 189986805599 T^{16} + 195321208348 T^{17} + 118587876497 T^{18} \)
$19$ \( ( 1 - T )^{9} \)
$23$ \( 1 + 10 T + 140 T^{2} + 1061 T^{3} + 9154 T^{4} + 59854 T^{5} + 395560 T^{6} + 2254083 T^{7} + 12218033 T^{8} + 60528768 T^{9} + 281014759 T^{10} + 1192409907 T^{11} + 4812778520 T^{12} + 16749603214 T^{13} + 58918283822 T^{14} + 157066078229 T^{15} + 476675562580 T^{16} + 783109852810 T^{17} + 1801152661463 T^{18} \)
$29$ \( 1 + 91 T^{2} + 306 T^{3} + 4427 T^{4} + 23594 T^{5} + 204156 T^{6} + 910810 T^{7} + 7917781 T^{8} + 28101124 T^{9} + 229615649 T^{10} + 765991210 T^{11} + 4979160684 T^{12} + 16687587914 T^{13} + 90802856623 T^{14} + 182015936226 T^{15} + 1569738744119 T^{16} + 14507145975869 T^{18} \)
$31$ \( 1 - 5 T + 98 T^{2} - 296 T^{3} + 4910 T^{4} - 2740 T^{5} + 140640 T^{6} + 312096 T^{7} + 3503411 T^{8} + 17203362 T^{9} + 108605741 T^{10} + 299924256 T^{11} + 4189806240 T^{12} - 2530447540 T^{13} + 140569131410 T^{14} - 262701089576 T^{15} + 2696236182878 T^{16} - 4264455187205 T^{17} + 26439622160671 T^{18} \)
$37$ \( 1 + 25 T + 334 T^{2} + 3231 T^{3} + 26434 T^{4} + 198031 T^{5} + 1451457 T^{6} + 10527915 T^{7} + 72728030 T^{8} + 462404356 T^{9} + 2690937110 T^{10} + 14412715635 T^{11} + 73520651421 T^{12} + 371141976991 T^{13} + 1833038159338 T^{14} + 8289862027479 T^{15} + 31707246962422 T^{16} + 87811986348025 T^{17} + 129961739795077 T^{18} \)
$41$ \( 1 + 7 T + 229 T^{2} + 779 T^{3} + 20930 T^{4} + 18604 T^{5} + 1241203 T^{6} - 747759 T^{7} + 61086109 T^{8} - 57378350 T^{9} + 2504530469 T^{10} - 1256982879 T^{11} + 85544951963 T^{12} + 52570457644 T^{13} + 2424870286930 T^{14} + 3700331203739 T^{15} + 44598728718749 T^{16} + 55894476603847 T^{17} + 327381934393961 T^{18} \)
$43$ \( 1 + 16 T + 370 T^{2} + 4261 T^{3} + 57936 T^{4} + 527928 T^{5} + 5341518 T^{6} + 40173363 T^{7} + 328239423 T^{8} + 2070822992 T^{9} + 14114295189 T^{10} + 74280548187 T^{11} + 424688071626 T^{12} + 1804880774328 T^{13} + 8517081153648 T^{14} + 26935327951789 T^{15} + 100572886109590 T^{16} + 187011204441616 T^{17} + 502592611936843 T^{18} \)
$47$ \( 1 + 9 T + 217 T^{2} + 1377 T^{3} + 17386 T^{4} + 55778 T^{5} + 463249 T^{6} - 2301431 T^{7} - 12983767 T^{8} - 264529406 T^{9} - 610237049 T^{10} - 5083861079 T^{11} + 48095900927 T^{12} + 272178846818 T^{13} + 3987392291702 T^{14} + 14842979508033 T^{15} + 109937217140471 T^{16} + 214301579955849 T^{17} + 1119130473102767 T^{18} \)
$53$ \( ( 1 + T )^{9} \)
$59$ \( 1 + 3 T + 95 T^{2} + 36 T^{3} + 10809 T^{4} + 21035 T^{5} + 734662 T^{6} + 259951 T^{7} + 52896539 T^{8} + 109591438 T^{9} + 3120895801 T^{10} + 904889431 T^{11} + 150884146898 T^{12} + 254888688635 T^{13} + 7727616747891 T^{14} + 1518499211076 T^{15} + 236421891057805 T^{16} + 440491312812963 T^{17} + 8662995818654939 T^{18} \)
$61$ \( 1 + 16 T + 479 T^{2} + 6584 T^{3} + 107417 T^{4} + 1230672 T^{5} + 14591754 T^{6} + 137872698 T^{7} + 1308299009 T^{8} + 10207642764 T^{9} + 79806239549 T^{10} + 513024309258 T^{11} + 3312050914674 T^{12} + 17039688835152 T^{13} + 90724000864517 T^{14} + 339210144792824 T^{15} + 1505373818454059 T^{16} + 3067317007956496 T^{17} + 11694146092834141 T^{18} \)
$67$ \( 1 + 13 T + 293 T^{2} + 1957 T^{3} + 33772 T^{4} + 184886 T^{5} + 3544848 T^{6} + 18535043 T^{7} + 296098764 T^{8} + 1325387994 T^{9} + 19838617188 T^{10} + 83203808027 T^{11} + 1066159119024 T^{12} + 3725660157206 T^{13} + 45596425113604 T^{14} + 177027053904733 T^{15} + 1775788500359639 T^{16} + 5278879808236333 T^{17} + 27206534396294947 T^{18} \)
$71$ \( 1 + 4 T + 130 T^{2} + 583 T^{3} + 15287 T^{4} + 96846 T^{5} + 1598307 T^{6} + 10813961 T^{7} + 124006133 T^{8} + 711461340 T^{9} + 8804435443 T^{10} + 54513177401 T^{11} + 572051656677 T^{12} + 2461019658126 T^{13} + 27581254088737 T^{14} + 74682465525943 T^{15} + 1182365620590830 T^{16} + 2583014124983044 T^{17} + 45848500718449031 T^{18} \)
$73$ \( 1 + 29 T + 635 T^{2} + 9791 T^{3} + 136916 T^{4} + 1627014 T^{5} + 18475288 T^{6} + 186639953 T^{7} + 1810039792 T^{8} + 15782839098 T^{9} + 132132904816 T^{10} + 994604309537 T^{11} + 7187201111896 T^{12} + 46204335682374 T^{13} + 283836670227188 T^{14} + 1481713409595599 T^{15} + 7015098059626595 T^{16} + 23387342664928349 T^{17} + 58871586708267913 T^{18} \)
$79$ \( 1 - 13 T + 517 T^{2} - 6255 T^{3} + 128837 T^{4} - 1416535 T^{5} + 20305237 T^{6} - 198575152 T^{7} + 2225833152 T^{8} - 18848897366 T^{9} + 175840819008 T^{10} - 1239307523632 T^{11} + 10011273745243 T^{12} - 55174152989335 T^{13} + 396438715277963 T^{14} - 1520512034283855 T^{15} + 9928420945844203 T^{16} - 19722414528785293 T^{17} + 119851595982618319 T^{18} \)
$83$ \( 1 + 35 T + 856 T^{2} + 15883 T^{3} + 250682 T^{4} + 3470149 T^{5} + 43270557 T^{6} + 489084637 T^{7} + 5059367602 T^{8} + 48046886080 T^{9} + 419927510966 T^{10} + 3369304064293 T^{11} + 24741541975359 T^{12} + 164687445159829 T^{13} + 987446586468526 T^{14} + 5192793950219827 T^{15} + 23228459647120712 T^{16} + 78830228124866435 T^{17} + 186940255267540403 T^{18} \)
$89$ \( 1 + 19 T + 518 T^{2} + 7530 T^{3} + 123908 T^{4} + 1512318 T^{5} + 19662938 T^{6} + 209238302 T^{7} + 2312049979 T^{8} + 21500675150 T^{9} + 205772448131 T^{10} + 1657376590142 T^{11} + 13861761738922 T^{12} + 94886220424638 T^{13} + 691909638206692 T^{14} + 3742269120936330 T^{15} + 22911831475884022 T^{16} + 74795187308339539 T^{17} + 350356403707485209 T^{18} \)
$97$ \( 1 + 12 T + 502 T^{2} + 3297 T^{3} + 97924 T^{4} + 221390 T^{5} + 11188394 T^{6} - 18864561 T^{7} + 1046986971 T^{8} - 3929804244 T^{9} + 101557736187 T^{10} - 177496654449 T^{11} + 10211345117162 T^{12} + 19599497520590 T^{13} + 840906707326468 T^{14} + 2746308700250913 T^{15} + 40560738808012726 T^{16} + 94049203132523532 T^{17} + 760231058654565217 T^{18} \)
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