Properties

Label 6042.2.a.ba
Level 6042
Weight 2
Character orbit 6042.a
Self dual yes
Analytic conductor 48.246
Analytic rank 0
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: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - x^{8} - 25 x^{7} + 62 x^{6} + 76 x^{5} - 360 x^{4} + 182 x^{3} + 459 x^{2} - 595 x + 199\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - x^{8} - 25 x^{7} + 62 x^{6} + 76 x^{5} - 360 x^{4} + 182 x^{3} + 459 x^{2} - 595 x + 199\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( -10 \nu^{8} - 7 \nu^{7} + 239 \nu^{6} - 212 \nu^{5} - 1140 \nu^{4} + 1658 \nu^{3} + 1099 \nu^{2} - 2760 \nu + 1112 \)
\(\beta_{2}\)\(=\)\( -21 \nu^{8} - 13 \nu^{7} + 504 \nu^{6} - 486 \nu^{5} - 2384 \nu^{4} + 3702 \nu^{3} + 2176 \nu^{2} - 6126 \nu + 2578 \)
\(\beta_{3}\)\(=\)\( -30 \nu^{8} - 17 \nu^{7} + 723 \nu^{6} - 728 \nu^{5} - 3413 \nu^{4} + 5454 \nu^{3} + 3052 \nu^{2} - 8976 \nu + 3824 \)
\(\beta_{4}\)\(=\)\( -34 \nu^{8} - 21 \nu^{7} + 816 \nu^{6} - 788 \nu^{5} - 3858 \nu^{4} + 5998 \nu^{3} + 3512 \nu^{2} - 9919 \nu + 4186 \)
\(\beta_{5}\)\(=\)\( 58 \nu^{8} + 34 \nu^{7} - 1396 \nu^{6} + 1382 \nu^{5} + 6599 \nu^{4} - 10417 \nu^{3} - 5959 \nu^{2} + 17178 \nu - 7279 \)
\(\beta_{6}\)\(=\)\( -66 \nu^{8} - 37 \nu^{7} + 1592 \nu^{6} - 1608 \nu^{5} - 7520 \nu^{4} + 12025 \nu^{3} + 6728 \nu^{2} - 19783 \nu + 8433 \)
\(\beta_{7}\)\(=\)\( 68 \nu^{8} + 36 \nu^{7} - 1645 \nu^{6} + 1700 \nu^{5} + 7769 \nu^{4} - 12598 \nu^{3} - 6893 \nu^{2} + 20670 \nu - 8850 \)
\(\beta_{8}\)\(=\)\( -98 \nu^{8} - 57 \nu^{7} + 2360 \nu^{6} - 2343 \nu^{5} - 11157 \nu^{4} + 17632 \nu^{3} + 10069 \nu^{2} - 29060 \nu + 12319 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(2 \beta_{8} + 2 \beta_{7} + \beta_{6} - \beta_{5} - 3 \beta_{4} - \beta_{3} - 2 \beta_{2} - \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{8} - 6 \beta_{7} - 9 \beta_{6} + \beta_{5} + 7 \beta_{4} + 5 \beta_{3} + 2 \beta_{2} + \beta_{1} + 24\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(14 \beta_{8} + 18 \beta_{7} + 17 \beta_{6} - 3 \beta_{5} - 27 \beta_{4} - 11 \beta_{3} - 6 \beta_{2} - 7 \beta_{1} - 26\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-86 \beta_{8} - 170 \beta_{7} - 219 \beta_{6} + 19 \beta_{5} + 229 \beta_{4} + 115 \beta_{3} + 38 \beta_{2} + 39 \beta_{1} + 376\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(590 \beta_{8} + 898 \beta_{7} + 1005 \beta_{6} - 85 \beta_{5} - 1291 \beta_{4} - 569 \beta_{3} - 202 \beta_{2} - 281 \beta_{1} - 1580\)\()/4\)
\(\nu^{6}\)\(=\)\(-648 \beta_{8} - 1133 \beta_{7} - 1378 \beta_{6} + 104 \beta_{5} + 1576 \beta_{4} + 731 \beta_{3} + 236 \beta_{2} + 297 \beta_{1} + 2200\)
\(\nu^{7}\)\(=\)\((\)\(14474 \beta_{8} + 23410 \beta_{7} + 27309 \beta_{6} - 2005 \beta_{5} - 33163 \beta_{4} - 14901 \beta_{3} - 4914 \beta_{2} - 6749 \beta_{1} - 42976\)\()/4\)
\(\nu^{8}\)\(=\)\((\)\(-70914 \beta_{8} - 119602 \beta_{7} - 142821 \beta_{6} + 10373 \beta_{5} + 167787 \beta_{4} + 76445 \beta_{3} + 24734 \beta_{2} + 32693 \beta_{1} + 225496\)\()/4\)

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
2.76936
−5.12868
1.36790
1.58250
−1.61632
−2.00915
1.74814
0.734954
1.55130
−1.00000 1.00000 1.00000 −4.25796 −1.00000 −1.07291 −1.00000 1.00000 4.25796
1.2 −1.00000 1.00000 1.00000 −2.21991 −1.00000 3.78890 −1.00000 1.00000 2.21991
1.3 −1.00000 1.00000 1.00000 −1.75621 −1.00000 2.24076 −1.00000 1.00000 1.75621
1.4 −1.00000 1.00000 1.00000 −1.57851 −1.00000 4.87562 −1.00000 1.00000 1.57851
1.5 −1.00000 1.00000 1.00000 0.436408 −1.00000 −4.48433 −1.00000 1.00000 −0.436408
1.6 −1.00000 1.00000 1.00000 1.50179 −1.00000 −2.36178 −1.00000 1.00000 −1.50179
1.7 −1.00000 1.00000 1.00000 2.77115 −1.00000 −0.160214 −1.00000 1.00000 −2.77115
1.8 −1.00000 1.00000 1.00000 3.47154 −1.00000 3.94390 −1.00000 1.00000 −3.47154
1.9 −1.00000 1.00000 1.00000 3.63169 −1.00000 3.23005 −1.00000 1.00000 −3.63169
\(n\): 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.ba 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6042.2.a.ba 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

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{9} \)
$3$ \( ( 1 - T )^{9} \)
$5$ \( 1 - 2 T + 15 T^{2} - 23 T^{3} + 122 T^{4} - 91 T^{5} + 551 T^{6} + 253 T^{7} + 1711 T^{8} + 3230 T^{9} + 8555 T^{10} + 6325 T^{11} + 68875 T^{12} - 56875 T^{13} + 381250 T^{14} - 359375 T^{15} + 1171875 T^{16} - 781250 T^{17} + 1953125 T^{18} \)
$7$ \( 1 - 10 T + 65 T^{2} - 304 T^{3} + 1208 T^{4} - 4078 T^{5} + 12490 T^{6} - 34851 T^{7} + 94698 T^{8} - 249754 T^{9} + 662886 T^{10} - 1707699 T^{11} + 4284070 T^{12} - 9791278 T^{13} + 20302856 T^{14} - 35765296 T^{15} + 53530295 T^{16} - 57648010 T^{17} + 40353607 T^{18} \)
$11$ \( 1 - 7 T + 63 T^{2} - 371 T^{3} + 2204 T^{4} - 10318 T^{5} + 48308 T^{6} - 190661 T^{7} + 737962 T^{8} - 2472022 T^{9} + 8117582 T^{10} - 23069981 T^{11} + 64297948 T^{12} - 151065838 T^{13} + 354956404 T^{14} - 657249131 T^{15} + 1227691773 T^{16} - 1500512167 T^{17} + 2357947691 T^{18} \)
$13$ \( 1 - 2 T + 67 T^{2} - 64 T^{3} + 1842 T^{4} + 1104 T^{5} + 26378 T^{6} + 85137 T^{7} + 246116 T^{8} + 1673402 T^{9} + 3199508 T^{10} + 14388153 T^{11} + 57952466 T^{12} + 31531344 T^{13} + 683921706 T^{14} - 308915776 T^{15} + 4204150639 T^{16} - 1631461442 T^{17} + 10604499373 T^{18} \)
$17$ \( 1 - 4 T + 95 T^{2} - 398 T^{3} + 4362 T^{4} - 18356 T^{5} + 129458 T^{6} - 527085 T^{7} + 2821392 T^{8} - 10550450 T^{9} + 47963664 T^{10} - 152327565 T^{11} + 636027154 T^{12} - 1533111476 T^{13} + 6193416234 T^{14} - 9606752462 T^{15} + 38982173935 T^{16} - 27903029764 T^{17} + 118587876497 T^{18} \)
$19$ \( ( 1 + T )^{9} \)
$23$ \( 1 - 15 T + 199 T^{2} - 1869 T^{3} + 16018 T^{4} - 115994 T^{5} + 773046 T^{6} - 4578867 T^{7} + 25217538 T^{8} - 125348622 T^{9} + 580003374 T^{10} - 2422220643 T^{11} + 9405650682 T^{12} - 32459876954 T^{13} + 103097342174 T^{14} - 276679076541 T^{15} + 677560263953 T^{16} - 1174664779215 T^{17} + 1801152661463 T^{18} \)
$29$ \( 1 - 5 T + 189 T^{2} - 754 T^{3} + 16497 T^{4} - 54579 T^{5} + 905138 T^{6} - 2550651 T^{7} + 35227319 T^{8} - 85815094 T^{9} + 1021592251 T^{10} - 2145097491 T^{11} + 22075410682 T^{12} - 38602689699 T^{13} + 338372425053 T^{14} - 448496784034 T^{15} + 3260226622401 T^{16} - 2501232064805 T^{17} + 14507145975869 T^{18} \)
$31$ \( 1 - T + 198 T^{2} - 142 T^{3} + 18928 T^{4} - 9904 T^{5} + 1149354 T^{6} - 461754 T^{7} + 48947231 T^{8} - 16199054 T^{9} + 1517364161 T^{10} - 443745594 T^{11} + 34240405014 T^{12} - 9146551984 T^{13} + 541892570128 T^{14} - 126025522702 T^{15} + 5447497593978 T^{16} - 852891037441 T^{17} + 26439622160671 T^{18} \)
$37$ \( 1 + 4 T + 195 T^{2} + 658 T^{3} + 19960 T^{4} + 58410 T^{5} + 1356172 T^{6} + 3440965 T^{7} + 66883884 T^{8} + 148089038 T^{9} + 2474703708 T^{10} + 4710681085 T^{11} + 68694180316 T^{12} + 109469744010 T^{13} + 1384105381720 T^{14} + 1688247977122 T^{15} + 18511716040935 T^{16} + 14049917815684 T^{17} + 129961739795077 T^{18} \)
$41$ \( 1 - 5 T + 315 T^{2} - 1369 T^{3} + 45208 T^{4} - 170362 T^{5} + 3922507 T^{6} - 12739029 T^{7} + 229032721 T^{8} - 632291654 T^{9} + 9390341561 T^{10} - 21414307749 T^{11} + 270343104947 T^{12} - 481402295482 T^{13} + 5237627134808 T^{14} - 6502892705929 T^{15} + 61347596272515 T^{16} - 39924626145605 T^{17} + 327381934393961 T^{18} \)
$43$ \( 1 - 27 T + 525 T^{2} - 7591 T^{3} + 94556 T^{4} - 1001202 T^{5} + 9482608 T^{6} - 79558113 T^{7} + 607521420 T^{8} - 4168110150 T^{9} + 26123421060 T^{10} - 147102950937 T^{11} + 753933714256 T^{12} - 3422910398802 T^{13} + 13900530336308 T^{14} - 47985466904959 T^{15} + 142704770831175 T^{16} - 315581407495227 T^{17} + 502592611936843 T^{18} \)
$47$ \( 1 + 252 T^{2} + 301 T^{3} + 31382 T^{4} + 56410 T^{5} + 2632137 T^{6} + 4987774 T^{7} + 163353922 T^{8} + 282222870 T^{9} + 7677634334 T^{10} + 11017992766 T^{11} + 273276359751 T^{12} + 275262805210 T^{13} + 7197305009674 T^{14} + 3244543814029 T^{15} + 127669026356676 T^{16} + 1119130473102767 T^{18} \)
$53$ \( ( 1 + T )^{9} \)
$59$ \( 1 - 7 T + 151 T^{2} - 120 T^{3} + 8685 T^{4} + 32869 T^{5} + 617438 T^{6} + 3000109 T^{7} + 35453595 T^{8} + 258722514 T^{9} + 2091762105 T^{10} + 10443379429 T^{11} + 126808799002 T^{12} + 398285538709 T^{13} + 6209117536815 T^{14} - 5061664036920 T^{15} + 375786374207669 T^{16} - 1027813063230247 T^{17} + 8662995818654939 T^{18} \)
$61$ \( 1 - 5 T + 351 T^{2} - 1470 T^{3} + 60893 T^{4} - 222337 T^{5} + 6879394 T^{6} - 22133155 T^{7} + 560335029 T^{8} - 1580330634 T^{9} + 34180436769 T^{10} - 82357469755 T^{11} + 1561491729514 T^{12} - 3078442750417 T^{13} + 51430002556793 T^{14} - 75734950310670 T^{15} + 1103102735443371 T^{16} - 958536564986405 T^{17} + 11694146092834141 T^{18} \)
$67$ \( 1 - 19 T + 607 T^{2} - 8649 T^{3} + 159722 T^{4} - 1814070 T^{5} + 24462390 T^{6} - 227645407 T^{7} + 2428480718 T^{8} - 18655767246 T^{9} + 162708208106 T^{10} - 1021900232023 T^{11} + 7357381803570 T^{12} - 36555544072470 T^{13} + 215644682340254 T^{14} - 782374547379681 T^{15} + 3678851944431061 T^{16} - 7715285873576179 T^{17} + 27206534396294947 T^{18} \)
$71$ \( 1 + 11 T + 334 T^{2} + 2758 T^{3} + 53246 T^{4} + 388336 T^{5} + 5998412 T^{6} + 41576578 T^{7} + 541707531 T^{8} + 3442504058 T^{9} + 38461234701 T^{10} + 209587529698 T^{11} + 2146897637332 T^{12} + 9868270552816 T^{13} + 96067996023346 T^{14} + 353300583054118 T^{15} + 3037770132902594 T^{16} + 7103288843703371 T^{17} + 45848500718449031 T^{18} \)
$73$ \( 1 - 25 T + 617 T^{2} - 9709 T^{3} + 145754 T^{4} - 1774610 T^{5} + 20605810 T^{6} - 210833539 T^{7} + 2047487906 T^{8} - 17992552746 T^{9} + 149466617138 T^{10} - 1123531929331 T^{11} + 8016010388770 T^{12} - 50395802461010 T^{13} + 302158476966122 T^{14} - 1469304003039901 T^{15} + 6816244886282849 T^{16} - 20161502297352025 T^{17} + 58871586708267913 T^{18} \)
$79$ \( 1 - 28 T + 697 T^{2} - 10703 T^{3} + 152712 T^{4} - 1648163 T^{5} + 17604103 T^{6} - 155693933 T^{7} + 1480820123 T^{8} - 12396314634 T^{9} + 116984789717 T^{10} - 971685835853 T^{11} + 8679509339017 T^{12} - 64196082351203 T^{13} + 469903436804088 T^{14} - 2601765036441263 T^{15} + 13385124563352823 T^{16} - 42479046677383708 T^{17} + 119851595982618319 T^{18} \)
$83$ \( 1 + 8 T + 221 T^{2} + 1226 T^{3} + 32318 T^{4} + 143636 T^{5} + 2982958 T^{6} + 6129043 T^{7} + 237314108 T^{8} + 506463718 T^{9} + 19697070964 T^{10} + 42222977227 T^{11} + 1705616605946 T^{12} + 6816723395156 T^{13} + 127301915500474 T^{14} + 400828897750394 T^{15} + 5997067268707567 T^{16} + 18018337857112328 T^{17} + 186940255267540403 T^{18} \)
$89$ \( 1 - 22 T + 616 T^{2} - 8235 T^{3} + 134509 T^{4} - 1316330 T^{5} + 17534367 T^{6} - 149126805 T^{7} + 1864092435 T^{8} - 14564305056 T^{9} + 165904226715 T^{10} - 1181233422405 T^{11} + 12361185169623 T^{12} - 82589494095530 T^{13} + 751106252425541 T^{14} - 4092640931063835 T^{15} + 27246502295645864 T^{16} - 86604953725445782 T^{17} + 350356403707485209 T^{18} \)
$97$ \( 1 - 13 T + 685 T^{2} - 7967 T^{3} + 218294 T^{4} - 2254626 T^{5} + 42954494 T^{6} - 389449057 T^{7} + 5809329214 T^{8} - 45366455586 T^{9} + 563504933758 T^{10} - 3664326177313 T^{11} + 39203406902462 T^{12} - 199600418703906 T^{13} + 1874564854061558 T^{14} - 6636287963269343 T^{15} + 55346824867507405 T^{16} - 101886636726900493 T^{17} + 760231058654565217 T^{18} \)
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