Properties

Label 6045.2.a.u
Level 6045
Weight 2
Character orbit 6045.a
Self dual yes
Analytic conductor 48.270
Analytic rank 1
Dimension 9
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6045.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.2695680219\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - 3 x^{8} - 7 x^{7} + 20 x^{6} + 20 x^{5} - 38 x^{4} - 27 x^{3} + 13 x^{2} + 6 x - 1\)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - 3 x^{8} - 7 x^{7} + 20 x^{6} + 20 x^{5} - 38 x^{4} - 27 x^{3} + 13 x^{2} + 6 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 4 \nu + 1 \)
\(\beta_{4}\)\(=\)\( \nu^{5} - 3 \nu^{4} - 3 \nu^{3} + 10 \nu^{2} + 2 \nu - 3 \)
\(\beta_{5}\)\(=\)\( \nu^{5} - 2 \nu^{4} - 5 \nu^{3} + 7 \nu^{2} + 6 \nu - 2 \)
\(\beta_{6}\)\(=\)\( \nu^{6} - 3 \nu^{5} - 3 \nu^{4} + 11 \nu^{3} + \nu^{2} - 6 \nu + 1 \)
\(\beta_{7}\)\(=\)\( -\nu^{8} + 4 \nu^{7} + 3 \nu^{6} - 23 \nu^{5} + 2 \nu^{4} + 38 \nu^{3} - 7 \nu^{2} - 12 \nu + 3 \)
\(\beta_{8}\)\(=\)\( \nu^{8} - 3 \nu^{7} - 7 \nu^{6} + 20 \nu^{5} + 20 \nu^{4} - 38 \nu^{3} - 26 \nu^{2} + 12 \nu + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 5 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{5} - \beta_{4} + 2 \beta_{3} + 5 \beta_{2} + 9 \beta_{1} + 7\)
\(\nu^{5}\)\(=\)\(3 \beta_{5} - 2 \beta_{4} + 9 \beta_{3} + 8 \beta_{2} + 30 \beta_{1} + 7\)
\(\nu^{6}\)\(=\)\(\beta_{6} + 12 \beta_{5} - 9 \beta_{4} + 22 \beta_{3} + 27 \beta_{2} + 67 \beta_{1} + 28\)
\(\nu^{7}\)\(=\)\(\beta_{8} + \beta_{7} + 4 \beta_{6} + 35 \beta_{5} - 20 \beta_{4} + 71 \beta_{3} + 55 \beta_{2} + 193 \beta_{1} + 39\)
\(\nu^{8}\)\(=\)\(4 \beta_{8} + 3 \beta_{7} + 19 \beta_{6} + 109 \beta_{5} - 63 \beta_{4} + 185 \beta_{3} + 158 \beta_{2} + 472 \beta_{1} + 120\)

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
2.67506
2.37659
1.89284
0.529693
0.139045
−0.503427
−0.839112
−1.57676
−1.69393
−2.67506 1.00000 5.15596 1.00000 −2.67506 1.28643 −8.44238 1.00000 −2.67506
1.2 −2.37659 1.00000 3.64818 1.00000 −2.37659 −1.73113 −3.91706 1.00000 −2.37659
1.3 −1.89284 1.00000 1.58283 1.00000 −1.89284 −4.37246 0.789632 1.00000 −1.89284
1.4 −0.529693 1.00000 −1.71943 1.00000 −0.529693 −2.25073 1.97015 1.00000 −0.529693
1.5 −0.139045 1.00000 −1.98067 1.00000 −0.139045 −0.572827 0.553494 1.00000 −0.139045
1.6 0.503427 1.00000 −1.74656 1.00000 0.503427 1.63268 −1.88612 1.00000 0.503427
1.7 0.839112 1.00000 −1.29589 1.00000 0.839112 2.06151 −2.76562 1.00000 0.839112
1.8 1.57676 1.00000 0.486159 1.00000 1.57676 −0.0992003 −2.38696 1.00000 1.57676
1.9 1.69393 1.00000 0.869413 1.00000 1.69393 −0.954273 −1.91514 1.00000 1.69393
\(n\): 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 6045.2.a.u 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6045.2.a.u 9 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(13\) \(1\)
\(31\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6045))\):

\(T_{2}^{9} + \cdots\)
\(T_{7}^{9} + \cdots\)
\(T_{11}^{9} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T + 11 T^{2} + 28 T^{3} + 66 T^{4} + 134 T^{5} + 257 T^{6} + 435 T^{7} + 700 T^{8} + 1021 T^{9} + 1400 T^{10} + 1740 T^{11} + 2056 T^{12} + 2144 T^{13} + 2112 T^{14} + 1792 T^{15} + 1408 T^{16} + 768 T^{17} + 512 T^{18} \)
$3$ \( ( 1 - T )^{9} \)
$5$ \( ( 1 - T )^{9} \)
$7$ \( 1 + 5 T + 57 T^{2} + 237 T^{3} + 1472 T^{4} + 5166 T^{5} + 22744 T^{6} + 67486 T^{7} + 232183 T^{8} + 577104 T^{9} + 1625281 T^{10} + 3306814 T^{11} + 7801192 T^{12} + 12403566 T^{13} + 24739904 T^{14} + 27882813 T^{15} + 46941951 T^{16} + 28824005 T^{17} + 40353607 T^{18} \)
$11$ \( 1 + 44 T^{2} - 70 T^{3} + 1003 T^{4} - 2648 T^{5} + 17661 T^{6} - 49734 T^{7} + 251882 T^{8} - 635156 T^{9} + 2770702 T^{10} - 6017814 T^{11} + 23506791 T^{12} - 38769368 T^{13} + 161534153 T^{14} - 124009270 T^{15} + 857435524 T^{16} + 2357947691 T^{18} \)
$13$ \( ( 1 + T )^{9} \)
$17$ \( 1 + 5 T + 96 T^{2} + 429 T^{3} + 4610 T^{4} + 18537 T^{5} + 145338 T^{6} + 524605 T^{7} + 3312252 T^{8} + 10500230 T^{9} + 56308284 T^{10} + 151610845 T^{11} + 714045594 T^{12} + 1548228777 T^{13} + 6545540770 T^{14} + 10355017101 T^{15} + 39392512608 T^{16} + 34878787205 T^{17} + 118587876497 T^{18} \)
$19$ \( 1 + 12 T + 99 T^{2} + 570 T^{3} + 3011 T^{4} + 14274 T^{5} + 74779 T^{6} + 380688 T^{7} + 1995209 T^{8} + 9087580 T^{9} + 37908971 T^{10} + 137428368 T^{11} + 512909161 T^{12} + 1860201954 T^{13} + 7455534089 T^{14} + 26816152170 T^{15} + 88493302161 T^{16} + 203802756492 T^{17} + 322687697779 T^{18} \)
$23$ \( 1 + 21 T + 352 T^{2} + 4030 T^{3} + 39880 T^{4} + 321562 T^{5} + 2322771 T^{6} + 14473143 T^{7} + 82250359 T^{8} + 411866056 T^{9} + 1891758257 T^{10} + 7656292647 T^{11} + 28261154757 T^{12} + 89986231642 T^{13} + 256681358840 T^{14} + 596584632670 T^{15} + 1198498557344 T^{16} + 1644530690901 T^{17} + 1801152661463 T^{18} \)
$29$ \( 1 + 15 T + 172 T^{2} + 1279 T^{3} + 8642 T^{4} + 51148 T^{5} + 320390 T^{6} + 1821606 T^{7} + 10638354 T^{8} + 55321138 T^{9} + 308512266 T^{10} + 1531970646 T^{11} + 7813991710 T^{12} + 36176008588 T^{13} + 177257349658 T^{14} + 760779027559 T^{15} + 2966978725148 T^{16} + 7503696194415 T^{17} + 14507145975869 T^{18} \)
$31$ \( ( 1 + T )^{9} \)
$37$ \( 1 + 13 T + 230 T^{2} + 2369 T^{3} + 25949 T^{4} + 217240 T^{5} + 1835348 T^{6} + 13055096 T^{7} + 91633307 T^{8} + 562245230 T^{9} + 3390432359 T^{10} + 17872426424 T^{11} + 92965882244 T^{12} + 407142735640 T^{13} + 1799406340193 T^{14} + 6078205862921 T^{15} + 21834331740590 T^{16} + 45662232900973 T^{17} + 129961739795077 T^{18} \)
$41$ \( 1 + 11 T + 69 T^{2} + 258 T^{3} - 586 T^{4} - 15317 T^{5} - 39519 T^{6} + 72870 T^{7} + 4629282 T^{8} + 46527646 T^{9} + 189800562 T^{10} + 122494470 T^{11} - 2723688999 T^{12} - 43282181237 T^{13} - 67891733786 T^{14} + 1225526894178 T^{15} + 13438044897789 T^{16} + 87834177520331 T^{17} + 327381934393961 T^{18} \)
$43$ \( 1 + 26 T + 494 T^{2} + 6604 T^{3} + 76653 T^{4} + 748365 T^{5} + 6712464 T^{6} + 53471378 T^{7} + 399148131 T^{8} + 2695839498 T^{9} + 17163369633 T^{10} + 98868577922 T^{11} + 533687875248 T^{12} + 2558511010365 T^{13} + 11268638181279 T^{14} + 41746281575596 T^{15} + 134278393886858 T^{16} + 303893207217626 T^{17} + 502592611936843 T^{18} \)
$47$ \( 1 + 21 T + 553 T^{2} + 7851 T^{3} + 119484 T^{4} + 1281203 T^{5} + 14106542 T^{6} + 119696619 T^{7} + 1027292313 T^{8} + 7016410372 T^{9} + 48282738711 T^{10} + 264409831371 T^{11} + 1464583510066 T^{12} + 6251861936243 T^{13} + 27403058816388 T^{14} + 84627619547979 T^{15} + 280162585616039 T^{16} + 500037019896981 T^{17} + 1119130473102767 T^{18} \)
$53$ \( 1 - 18 T + 346 T^{2} - 3587 T^{3} + 39541 T^{4} - 300760 T^{5} + 2777314 T^{6} - 19902155 T^{7} + 179692251 T^{8} - 1204842118 T^{9} + 9523689303 T^{10} - 55905153395 T^{11} + 413478176378 T^{12} - 2373141065560 T^{13} + 16535867988713 T^{14} - 79503563369723 T^{15} + 406450054383602 T^{16} - 1120674427404498 T^{17} + 3299763591802133 T^{18} \)
$59$ \( 1 + 24 T + 608 T^{2} + 9483 T^{3} + 143623 T^{4} + 1690806 T^{5} + 19093649 T^{6} + 180916580 T^{7} + 1644078538 T^{8} + 12890317302 T^{9} + 97000633742 T^{10} + 629770614980 T^{11} + 3921434537971 T^{12} + 20488106682966 T^{13} + 102679572595277 T^{14} + 399998000517603 T^{15} + 1513100102769952 T^{16} + 3523930502503704 T^{17} + 8662995818654939 T^{18} \)
$61$ \( 1 + 4 T + 215 T^{2} + 1414 T^{3} + 27958 T^{4} + 233890 T^{5} + 2533608 T^{6} + 24913092 T^{7} + 184343481 T^{8} + 1802227950 T^{9} + 11244952341 T^{10} + 92701615332 T^{11} + 575080877448 T^{12} + 3238403751490 T^{13} + 23613223383358 T^{14} + 72849809346454 T^{15} + 675689709744515 T^{16} + 766829251989124 T^{17} + 11694146092834141 T^{18} \)
$67$ \( 1 + 20 T + 322 T^{2} + 3931 T^{3} + 40781 T^{4} + 393806 T^{5} + 3501928 T^{6} + 29623749 T^{7} + 246047197 T^{8} + 1964890708 T^{9} + 16485162199 T^{10} + 132981009261 T^{11} + 1053250371064 T^{12} + 7935632356526 T^{13} + 55059451988567 T^{14} + 355591900306339 T^{15} + 1951549136914006 T^{16} + 8121353551132820 T^{17} + 27206534396294947 T^{18} \)
$71$ \( 1 + 13 T + 426 T^{2} + 4681 T^{3} + 88394 T^{4} + 846234 T^{5} + 11948845 T^{6} + 99424923 T^{7} + 1152524519 T^{8} + 8282865646 T^{9} + 81829240849 T^{10} + 501201036843 T^{11} + 4276623062795 T^{12} + 21504228459354 T^{13} + 159483049252294 T^{14} + 599637429034201 T^{15} + 3874521187474566 T^{16} + 8394795906194893 T^{17} + 45848500718449031 T^{18} \)
$73$ \( 1 - 15 T + 409 T^{2} - 3834 T^{3} + 59954 T^{4} - 318375 T^{5} + 3850208 T^{6} + 54673 T^{7} + 91328903 T^{8} + 1281695164 T^{9} + 6667009919 T^{10} + 291352417 T^{11} + 1497796365536 T^{12} - 9041289978375 T^{13} + 124288934286722 T^{14} - 580215423592026 T^{15} + 4518385994310673 T^{16} - 12096901378411215 T^{17} + 58871586708267913 T^{18} \)
$79$ \( 1 + 15 T + 445 T^{2} + 5783 T^{3} + 93019 T^{4} + 1021063 T^{5} + 12321962 T^{6} + 114544383 T^{7} + 1205592346 T^{8} + 9887775904 T^{9} + 95241795334 T^{10} + 714871494303 T^{11} + 6075207822518 T^{12} + 39770486556103 T^{13} + 286224709178581 T^{14} + 1405774755277943 T^{15} + 8545739498840755 T^{16} + 22756632148598415 T^{17} + 119851595982618319 T^{18} \)
$83$ \( 1 + 5 T + 332 T^{2} + 1845 T^{3} + 61639 T^{4} + 349470 T^{5} + 8329386 T^{6} + 44405270 T^{7} + 869454263 T^{8} + 4179329088 T^{9} + 72164703829 T^{10} + 305907905030 T^{11} + 4762634632782 T^{12} + 16585259439870 T^{13} + 242798526193877 T^{14} + 603204988865805 T^{15} + 9009168928556164 T^{16} + 11261461160695205 T^{17} + 186940255267540403 T^{18} \)
$89$ \( 1 - 31 T + 545 T^{2} - 6249 T^{3} + 52948 T^{4} - 292404 T^{5} + 990786 T^{6} - 2183976 T^{7} + 73780943 T^{8} - 970405202 T^{9} + 6566503927 T^{10} - 17299273896 T^{11} + 698473415634 T^{12} - 18346082237364 T^{13} + 295664779705652 T^{14} - 3105636087215289 T^{15} + 24106077518063305 T^{16} - 122034252976764511 T^{17} + 350356403707485209 T^{18} \)
$97$ \( 1 + 11 T + 312 T^{2} + 2960 T^{3} + 46950 T^{4} + 246022 T^{5} + 2985909 T^{6} - 8069923 T^{7} - 5718697 T^{8} - 2885202234 T^{9} - 554713609 T^{10} - 75929905507 T^{11} + 2725158524757 T^{12} + 21780150770182 T^{13} + 403175625066150 T^{14} + 2465597134589840 T^{15} + 25209064757171256 T^{16} + 86211769538146571 T^{17} + 760231058654565217 T^{18} \)
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