Properties

Label 6042.2.a.bb
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: \(x^{9} - 4 x^{8} - 16 x^{7} + 76 x^{6} + 30 x^{5} - 366 x^{4} + 300 x^{3} + 101 x^{2} - 106 x + 16\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - 4 x^{8} - 16 x^{7} + 76 x^{6} + 30 x^{5} - 366 x^{4} + 300 x^{3} + 101 x^{2} - 106 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -615 \nu^{8} + 2940 \nu^{7} + 11484 \nu^{6} - 57016 \nu^{5} - 46158 \nu^{4} + 285404 \nu^{3} - 71814 \nu^{2} - 146543 \nu - 19336 \)\()/26914\)
\(\beta_{3}\)\(=\)\((\)\( -779 \nu^{8} + 3724 \nu^{7} + 11855 \nu^{6} - 70426 \nu^{5} - 12713 \nu^{4} + 336392 \nu^{3} - 276671 \nu^{2} - 90525 \nu + 44587 \)\()/13457\)
\(\beta_{4}\)\(=\)\((\)\( 1831 \nu^{8} - 4158 \nu^{7} - 33928 \nu^{6} + 78330 \nu^{5} + 148714 \nu^{4} - 371346 \nu^{3} + 48122 \nu^{2} + 70569 \nu + 17350 \)\()/26914\)
\(\beta_{5}\)\(=\)\((\)\( -2002 \nu^{8} + 7273 \nu^{7} + 34561 \nu^{6} - 138099 \nu^{5} - 110149 \nu^{4} + 678224 \nu^{3} - 350096 \nu^{2} - 320194 \nu + 120443 \)\()/13457\)
\(\beta_{6}\)\(=\)\((\)\( 4037 \nu^{8} - 12078 \nu^{7} - 74202 \nu^{6} + 227530 \nu^{5} + 299972 \nu^{4} - 1097896 \nu^{3} + 371756 \nu^{2} + 451057 \nu - 166196 \)\()/26914\)
\(\beta_{7}\)\(=\)\((\)\( -2164 \nu^{8} + 7391 \nu^{7} + 38702 \nu^{6} - 140186 \nu^{5} - 141607 \nu^{4} + 684937 \nu^{3} - 279212 \nu^{2} - 309497 \nu + 63294 \)\()/13457\)
\(\beta_{8}\)\(=\)\((\)\( -3474 \nu^{8} + 12997 \nu^{7} + 58897 \nu^{6} - 249600 \nu^{5} - 167719 \nu^{4} + 1236964 \nu^{3} - 727251 \nu^{2} - 539153 \nu + 224837 \)\()/13457\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{7} + \beta_{6} - \beta_{5} - \beta_{2} + 5\)
\(\nu^{3}\)\(=\)\(2 \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} - 2 \beta_{2} + 9 \beta_{1} - 2\)
\(\nu^{4}\)\(=\)\(\beta_{8} + 23 \beta_{7} + 9 \beta_{6} - 14 \beta_{5} + 3 \beta_{4} - 14 \beta_{2} + \beta_{1} + 44\)
\(\nu^{5}\)\(=\)\(-3 \beta_{8} + 28 \beta_{7} + 11 \beta_{6} + 14 \beta_{4} - 8 \beta_{3} - 29 \beta_{2} + 91 \beta_{1} - 17\)
\(\nu^{6}\)\(=\)\(15 \beta_{8} + 253 \beta_{7} + 82 \beta_{6} - 169 \beta_{5} + 51 \beta_{4} - \beta_{3} - 157 \beta_{2} + 29 \beta_{1} + 436\)
\(\nu^{7}\)\(=\)\(-63 \beta_{8} + 350 \beta_{7} + 124 \beta_{6} + 8 \beta_{5} + 170 \beta_{4} - 55 \beta_{3} - 344 \beta_{2} + 949 \beta_{1} - 74\)
\(\nu^{8}\)\(=\)\(182 \beta_{8} + 2770 \beta_{7} + 776 \beta_{6} - 1950 \beta_{5} + 706 \beta_{4} - 4 \beta_{3} - 1692 \beta_{2} + 505 \beta_{1} + 4518\)

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
1.91020
1.49919
−3.25110
3.42636
−0.613026
0.217879
−2.88457
0.394407
3.30066
1.00000 −1.00000 1.00000 −4.16096 −1.00000 0.910204 1.00000 1.00000 −4.16096
1.2 1.00000 −1.00000 1.00000 −3.26675 −1.00000 0.499186 1.00000 1.00000 −3.26675
1.3 1.00000 −1.00000 1.00000 −2.51610 −1.00000 −4.25110 1.00000 1.00000 −2.51610
1.4 1.00000 −1.00000 1.00000 −2.03181 −1.00000 2.42636 1.00000 1.00000 −2.03181
1.5 1.00000 −1.00000 1.00000 −0.133530 −1.00000 −1.61303 1.00000 1.00000 −0.133530
1.6 1.00000 −1.00000 1.00000 0.926601 −1.00000 −0.782121 1.00000 1.00000 0.926601
1.7 1.00000 −1.00000 1.00000 1.64105 −1.00000 −3.88457 1.00000 1.00000 1.64105
1.8 1.00000 −1.00000 1.00000 1.69036 −1.00000 −0.605593 1.00000 1.00000 1.69036
1.9 1.00000 −1.00000 1.00000 2.85114 −1.00000 2.30066 1.00000 1.00000 2.85114
\(n\): 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.bb 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6042.2.a.bb 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

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{9} \)
$3$ \( ( 1 + T )^{9} \)
$5$ \( 1 + 5 T + 31 T^{2} + 121 T^{3} + 481 T^{4} + 1519 T^{5} + 4687 T^{6} + 12508 T^{7} + 32108 T^{8} + 73198 T^{9} + 160540 T^{10} + 312700 T^{11} + 585875 T^{12} + 949375 T^{13} + 1503125 T^{14} + 1890625 T^{15} + 2421875 T^{16} + 1953125 T^{17} + 1953125 T^{18} \)
$7$ \( 1 + 5 T + 51 T^{2} + 216 T^{3} + 1228 T^{4} + 4382 T^{5} + 18240 T^{6} + 54713 T^{7} + 183025 T^{8} + 460744 T^{9} + 1281175 T^{10} + 2680937 T^{11} + 6256320 T^{12} + 10521182 T^{13} + 20638996 T^{14} + 25412184 T^{15} + 42000693 T^{16} + 28824005 T^{17} + 40353607 T^{18} \)
$11$ \( 1 - 3 T + 61 T^{2} - 161 T^{3} + 1854 T^{4} - 4418 T^{5} + 37246 T^{6} - 80327 T^{7} + 545424 T^{8} - 1039798 T^{9} + 5999664 T^{10} - 9719567 T^{11} + 49574426 T^{12} - 64683938 T^{13} + 298588554 T^{14} - 285221321 T^{15} + 1188717431 T^{16} - 643076643 T^{17} + 2357947691 T^{18} \)
$13$ \( 1 + 4 T + 53 T^{2} + 114 T^{3} + 1286 T^{4} + 1624 T^{5} + 23272 T^{6} + 20609 T^{7} + 371106 T^{8} + 309494 T^{9} + 4824378 T^{10} + 3482921 T^{11} + 51128584 T^{12} + 46383064 T^{13} + 477482798 T^{14} + 550256226 T^{15} + 3325671401 T^{16} + 3262922884 T^{17} + 10604499373 T^{18} \)
$17$ \( 1 + 16 T + 169 T^{2} + 1248 T^{3} + 7778 T^{4} + 41056 T^{5} + 200696 T^{6} + 905301 T^{7} + 3989280 T^{8} + 16648366 T^{9} + 67817760 T^{10} + 261631989 T^{11} + 986019448 T^{12} + 3429038176 T^{13} + 11043647746 T^{14} + 30123686112 T^{15} + 69347235737 T^{16} + 111612119056 T^{17} + 118587876497 T^{18} \)
$19$ \( ( 1 + T )^{9} \)
$23$ \( 1 + 46 T^{2} - 209 T^{3} + 1532 T^{4} - 7812 T^{5} + 67326 T^{6} - 234375 T^{7} + 1590143 T^{8} - 7529704 T^{9} + 36573289 T^{10} - 123984375 T^{11} + 819155442 T^{12} - 2186117892 T^{13} + 9860477476 T^{14} - 30939500801 T^{15} + 156621970562 T^{16} + 1801152661463 T^{18} \)
$29$ \( 1 + 2 T + 148 T^{2} + 452 T^{3} + 11158 T^{4} + 40174 T^{5} + 579852 T^{6} + 2038992 T^{7} + 22500190 T^{8} + 69939618 T^{9} + 652505510 T^{10} + 1714792272 T^{11} + 14142010428 T^{12} + 28414306894 T^{13} + 228863400542 T^{14} + 268860141092 T^{15} + 2552981693732 T^{16} + 1000492825922 T^{17} + 14507145975869 T^{18} \)
$31$ \( 1 + 10 T + 163 T^{2} + 1209 T^{3} + 10191 T^{4} + 53643 T^{5} + 300441 T^{6} + 924503 T^{7} + 4643316 T^{8} + 7780950 T^{9} + 143942796 T^{10} + 888447383 T^{11} + 8950437831 T^{12} + 49540437003 T^{13} + 291759677841 T^{14} + 1072991950329 T^{15} + 4484556100093 T^{16} + 8528910374410 T^{17} + 26439622160671 T^{18} \)
$37$ \( 1 + 14 T + 277 T^{2} + 2852 T^{3} + 32426 T^{4} + 260024 T^{5} + 2213082 T^{6} + 14674513 T^{7} + 105044438 T^{8} + 610422506 T^{9} + 3886644206 T^{10} + 20089408297 T^{11} + 112099242546 T^{12} + 487326839864 T^{13} + 2248547149682 T^{14} + 7317451718468 T^{15} + 26296129965841 T^{16} + 49174712354894 T^{17} + 129961739795077 T^{18} \)
$41$ \( 1 + 21 T + 365 T^{2} + 4359 T^{3} + 47586 T^{4} + 427874 T^{5} + 3668915 T^{6} + 27556397 T^{7} + 200665973 T^{8} + 1303747130 T^{9} + 8227304893 T^{10} + 46322303357 T^{11} + 252865290715 T^{12} + 1209069662114 T^{13} + 5513133180786 T^{14} + 20705704386519 T^{15} + 71085309966565 T^{16} + 167683429811541 T^{17} + 327381934393961 T^{18} \)
$43$ \( 1 + 4 T + 178 T^{2} + 247 T^{3} + 15626 T^{4} + 5768 T^{5} + 1051956 T^{6} - 40543 T^{7} + 54205619 T^{8} - 17058136 T^{9} + 2330841617 T^{10} - 74964007 T^{11} + 83637865692 T^{12} + 19719644168 T^{13} + 2297153930318 T^{14} + 1561376673103 T^{15} + 48383712777046 T^{16} + 46752801110404 T^{17} + 502592611936843 T^{18} \)
$47$ \( 1 + 8 T + 318 T^{2} + 1871 T^{3} + 44588 T^{4} + 196804 T^{5} + 3814163 T^{6} + 13087044 T^{7} + 232493014 T^{8} + 671128762 T^{9} + 10927171658 T^{10} + 28909280196 T^{11} + 395997845149 T^{12} + 960340739524 T^{13} + 10226035172116 T^{14} + 20167911880559 T^{15} + 161106152307234 T^{16} + 190490293294088 T^{17} + 1119130473102767 T^{18} \)
$53$ \( ( 1 - T )^{9} \)
$59$ \( 1 + 12 T + 446 T^{2} + 4422 T^{3} + 91338 T^{4} + 766258 T^{5} + 11414970 T^{6} + 81497346 T^{7} + 962393800 T^{8} + 5807367456 T^{9} + 56781234200 T^{10} + 283692261426 T^{11} + 2344395123630 T^{12} + 9285024805138 T^{13} + 65299755622062 T^{14} + 186522319760502 T^{15} + 1109938562229274 T^{16} + 1761965251251852 T^{17} + 8662995818654939 T^{18} \)
$61$ \( 1 + 13 T + 259 T^{2} + 2270 T^{3} + 31025 T^{4} + 219131 T^{5} + 2605882 T^{6} + 15936701 T^{7} + 176122951 T^{8} + 981151638 T^{9} + 10743500011 T^{10} + 59300464421 T^{11} + 591485702242 T^{12} + 3034052984171 T^{13} + 26203600238525 T^{14} + 116951249799470 T^{15} + 813970394529439 T^{16} + 2492195068964653 T^{17} + 11694146092834141 T^{18} \)
$67$ \( 1 - 22 T + 518 T^{2} - 7105 T^{3} + 101100 T^{4} - 1091212 T^{5} + 12308690 T^{6} - 115764263 T^{7} + 1113420091 T^{8} - 9097321308 T^{9} + 74599146097 T^{10} - 519665776607 T^{11} + 3701998530470 T^{12} - 21989145048652 T^{13} + 136497648317700 T^{14} - 642706805310745 T^{15} + 3139448611557314 T^{16} - 8933488906246102 T^{17} + 27206534396294947 T^{18} \)
$71$ \( 1 + 13 T + 526 T^{2} + 5338 T^{3} + 125478 T^{4} + 1057116 T^{5} + 18409668 T^{6} + 131338126 T^{7} + 1842242075 T^{8} + 11146727134 T^{9} + 130799187325 T^{10} + 662075493166 T^{11} + 6589022683548 T^{12} + 26863094571996 T^{13} + 226391090504778 T^{14} + 683799315570298 T^{15} + 4784033203313666 T^{16} + 8394795906194893 T^{17} + 45848500718449031 T^{18} \)
$73$ \( 1 + 17 T + 497 T^{2} + 5433 T^{3} + 95518 T^{4} + 784314 T^{5} + 11404822 T^{6} + 80224631 T^{7} + 1057689490 T^{8} + 6617644778 T^{9} + 77211332770 T^{10} + 427517058599 T^{11} + 4436669639974 T^{12} + 22273137991674 T^{13} + 198015652420174 T^{14} + 822198851428137 T^{15} + 5490557063991209 T^{16} + 13709821562199377 T^{17} + 58871586708267913 T^{18} \)
$79$ \( 1 + 205 T^{2} + 183 T^{3} + 24756 T^{4} + 12939 T^{5} + 2209335 T^{6} - 1734015 T^{7} + 166196755 T^{8} - 192830726 T^{9} + 13129543645 T^{10} - 10821987615 T^{11} + 1089288319065 T^{12} + 503975098059 T^{13} + 76175608213644 T^{14} + 44485004360343 T^{15} + 3936801342162595 T^{16} + 119851595982618319 T^{18} \)
$83$ \( 1 + 24 T + 707 T^{2} + 11464 T^{3} + 202314 T^{4} + 2564064 T^{5} + 34411744 T^{6} + 362681133 T^{7} + 4001007548 T^{8} + 35709483102 T^{9} + 332083626484 T^{10} + 2498510325237 T^{11} + 19676187866528 T^{12} + 121686172376544 T^{13} + 796923068647902 T^{14} + 3748044440302216 T^{15} + 19185188049666289 T^{16} + 54055013571336984 T^{17} + 186940255267540403 T^{18} \)
$89$ \( 1 + 23 T + 406 T^{2} + 6208 T^{3} + 84232 T^{4} + 1114330 T^{5} + 13298214 T^{6} + 145868408 T^{7} + 1507642811 T^{8} + 14437010926 T^{9} + 134180210179 T^{10} + 1155423659768 T^{11} + 9374828625366 T^{12} + 69915561413530 T^{13} + 470356495508168 T^{14} + 3085259854285888 T^{15} + 17957921967584774 T^{16} + 90541542531147863 T^{17} + 350356403707485209 T^{18} \)
$97$ \( 1 + 29 T + 661 T^{2} + 9715 T^{3} + 127350 T^{4} + 1251218 T^{5} + 11966062 T^{6} + 89327821 T^{7} + 799936422 T^{8} + 6332583586 T^{9} + 77593832934 T^{10} + 840485467789 T^{11} + 10921101703726 T^{12} + 110769429914258 T^{13} + 1093597781728950 T^{14} + 8092323027885235 T^{15} + 53407666040032693 T^{16} + 227285574236931869 T^{17} + 760231058654565217 T^{18} \)
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