Properties

Label 6018.2.a.v
Level 6018
Weight 2
Character orbit 6018.a
Self dual yes
Analytic conductor 48.054
Analytic rank 1
Dimension 9
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6018 = 2 \cdot 3 \cdot 17 \cdot 59 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6018.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.0539719364\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - 3 x^{8} - 21 x^{7} + 42 x^{6} + 121 x^{5} - 127 x^{4} - 141 x^{3} + 27 x^{2} + 26 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 5907 \nu^{8} - 25576 \nu^{7} - 96263 \nu^{6} + 398353 \nu^{5} + 298486 \nu^{4} - 1475547 \nu^{3} + 637856 \nu^{2} + 499745 \nu - 289639 \)\()/61504\)
\(\beta_{3}\)\(=\)\((\)\( 9203 \nu^{8} - 31080 \nu^{7} - 182951 \nu^{6} + 461489 \nu^{5} + 961014 \nu^{4} - 1597499 \nu^{3} - 788640 \nu^{2} + 560961 \nu + 75129 \)\()/61504\)
\(\beta_{4}\)\(=\)\((\)\( -6401 \nu^{8} + 21400 \nu^{7} + 124669 \nu^{6} - 306155 \nu^{5} - 617266 \nu^{4} + 963353 \nu^{3} + 294624 \nu^{2} - 161915 \nu + 20861 \)\()/30752\)
\(\beta_{5}\)\(=\)\((\)\( 12987 \nu^{8} - 49640 \nu^{7} - 236943 \nu^{6} + 752521 \nu^{5} + 1066438 \nu^{4} - 2690819 \nu^{3} - 286688 \nu^{2} + 1014297 \nu + 60241 \)\()/61504\)
\(\beta_{6}\)\(=\)\((\)\( -4883 \nu^{8} + 14312 \nu^{7} + 103367 \nu^{6} - 196913 \nu^{5} - 602150 \nu^{4} + 561227 \nu^{3} + 720192 \nu^{2} - 19745 \nu - 125641 \)\()/15376\)
\(\beta_{7}\)\(=\)\((\)\( -17371 \nu^{8} + 67112 \nu^{7} + 312239 \nu^{6} - 1016233 \nu^{5} - 1332326 \nu^{4} + 3591075 \nu^{3} - 85312 \nu^{2} - 1078105 \nu + 120975 \)\()/30752\)
\(\beta_{8}\)\(=\)\((\)\( 42985 \nu^{8} - 154744 \nu^{7} - 810245 \nu^{6} + 2291651 \nu^{5} + 3836354 \nu^{4} - 7760065 \nu^{3} - 1443360 \nu^{2} + 2036531 \nu - 159013 \)\()/61504\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{2} + 5\)
\(\nu^{3}\)\(=\)\(4 \beta_{8} + 2 \beta_{6} - 5 \beta_{5} + 3 \beta_{4} - 5 \beta_{2} + 9 \beta_{1} + 6\)
\(\nu^{4}\)\(=\)\(38 \beta_{8} + 14 \beta_{7} + 21 \beta_{6} - 23 \beta_{5} + 19 \beta_{4} - 33 \beta_{2} + 7 \beta_{1} + 69\)
\(\nu^{5}\)\(=\)\(119 \beta_{8} + 16 \beta_{7} + 70 \beta_{6} - 118 \beta_{5} + 69 \beta_{4} + 10 \beta_{3} - 147 \beta_{2} + 123 \beta_{1} + 181\)
\(\nu^{6}\)\(=\)\(762 \beta_{8} + 225 \beta_{7} + 452 \beta_{6} - 528 \beta_{5} + 364 \beta_{4} + 19 \beta_{3} - 807 \beta_{2} + 245 \beta_{1} + 1225\)
\(\nu^{7}\)\(=\)\(2963 \beta_{8} + 548 \beta_{7} + 1826 \beta_{6} - 2630 \beta_{5} + 1510 \beta_{4} + 260 \beta_{3} - 3651 \beta_{2} + 2063 \beta_{1} + 4466\)
\(\nu^{8}\)\(=\)\(16085 \beta_{8} + 4145 \beta_{7} + 9882 \beta_{6} - 12013 \beta_{5} + 7498 \beta_{4} + 761 \beta_{3} - 18509 \beta_{2} + 6440 \beta_{1} + 24615\)

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
4.72920
3.03878
1.53312
0.460972
0.0373069
−0.542287
−0.625332
−2.43329
−3.19847
−1.00000 −1.00000 1.00000 −3.72920 1.00000 −3.41012 −1.00000 1.00000 3.72920
1.2 −1.00000 −1.00000 1.00000 −2.03878 1.00000 −3.87855 −1.00000 1.00000 2.03878
1.3 −1.00000 −1.00000 1.00000 −0.533116 1.00000 0.275545 −1.00000 1.00000 0.533116
1.4 −1.00000 −1.00000 1.00000 0.539028 1.00000 0.211506 −1.00000 1.00000 −0.539028
1.5 −1.00000 −1.00000 1.00000 0.962693 1.00000 0.389187 −1.00000 1.00000 −0.962693
1.6 −1.00000 −1.00000 1.00000 1.54229 1.00000 2.97024 −1.00000 1.00000 −1.54229
1.7 −1.00000 −1.00000 1.00000 1.62533 1.00000 −4.68222 −1.00000 1.00000 −1.62533
1.8 −1.00000 −1.00000 1.00000 3.43329 1.00000 −0.177683 −1.00000 1.00000 −3.43329
1.9 −1.00000 −1.00000 1.00000 4.19847 1.00000 −2.69790 −1.00000 1.00000 −4.19847
\(n\): 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 6018.2.a.v 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6018.2.a.v 9 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(17\) \(1\)
\(59\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6018))\):

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{9} \)
$3$ \( ( 1 + T )^{9} \)
$5$ \( 1 - 6 T + 36 T^{2} - 135 T^{3} + 475 T^{4} - 1339 T^{5} + 3607 T^{6} - 8598 T^{7} + 20449 T^{8} - 45204 T^{9} + 102245 T^{10} - 214950 T^{11} + 450875 T^{12} - 836875 T^{13} + 1484375 T^{14} - 2109375 T^{15} + 2812500 T^{16} - 2343750 T^{17} + 1953125 T^{18} \)
$7$ \( 1 + 11 T + 91 T^{2} + 545 T^{3} + 2776 T^{4} + 11887 T^{5} + 45333 T^{6} + 152805 T^{7} + 468746 T^{8} + 1295394 T^{9} + 3281222 T^{10} + 7487445 T^{11} + 15549219 T^{12} + 28540687 T^{13} + 46656232 T^{14} + 64118705 T^{15} + 74942413 T^{16} + 63412811 T^{17} + 40353607 T^{18} \)
$11$ \( 1 - T + 55 T^{2} - 78 T^{3} + 1586 T^{4} - 2543 T^{5} + 30833 T^{6} - 50189 T^{7} + 442475 T^{8} - 665306 T^{9} + 4867225 T^{10} - 6072869 T^{11} + 41038723 T^{12} - 37232063 T^{13} + 255426886 T^{14} - 138181758 T^{15} + 1071794405 T^{16} - 214358881 T^{17} + 2357947691 T^{18} \)
$13$ \( 1 - 4 T + 54 T^{2} - 290 T^{3} + 1874 T^{4} - 8878 T^{5} + 46560 T^{6} - 186326 T^{7} + 796527 T^{8} - 2883916 T^{9} + 10354851 T^{10} - 31489094 T^{11} + 102292320 T^{12} - 253564558 T^{13} + 695803082 T^{14} - 1399774610 T^{15} + 3388419918 T^{16} - 3262922884 T^{17} + 10604499373 T^{18} \)
$17$ \( ( 1 + T )^{9} \)
$19$ \( 1 + 13 T + 159 T^{2} + 1170 T^{3} + 7957 T^{4} + 37685 T^{5} + 168393 T^{6} + 496718 T^{7} + 1691010 T^{8} + 4329036 T^{9} + 32129190 T^{10} + 179315198 T^{11} + 1155007587 T^{12} + 4911146885 T^{13} + 19702319743 T^{14} + 55043680770 T^{15} + 142125606501 T^{16} + 220786319533 T^{17} + 322687697779 T^{18} \)
$23$ \( 1 + 6 T + 132 T^{2} + 764 T^{3} + 8943 T^{4} + 46939 T^{5} + 397909 T^{6} + 1833964 T^{7} + 12541318 T^{8} + 49942456 T^{9} + 288450314 T^{10} + 970166956 T^{11} + 4841358803 T^{12} + 13135456699 T^{13} + 57560215449 T^{14} + 113099419196 T^{15} + 449436959004 T^{16} + 469865911686 T^{17} + 1801152661463 T^{18} \)
$29$ \( 1 - 10 T + 154 T^{2} - 1535 T^{3} + 14289 T^{4} - 112663 T^{5} + 856855 T^{6} - 5526672 T^{7} + 34557483 T^{8} - 191504420 T^{9} + 1002167007 T^{10} - 4647931152 T^{11} + 20897836595 T^{12} - 79684399303 T^{13} + 293083808061 T^{14} - 913053797735 T^{15} + 2656480951586 T^{16} - 5002464129610 T^{17} + 14507145975869 T^{18} \)
$31$ \( 1 - T + 135 T^{2} + 36 T^{3} + 9101 T^{4} + 8582 T^{5} + 441967 T^{6} + 429396 T^{7} + 17228920 T^{8} + 14081542 T^{9} + 534096520 T^{10} + 412649556 T^{11} + 13166638897 T^{12} + 7925657222 T^{13} + 260553903251 T^{14} + 31950132516 T^{15} + 3714202904985 T^{16} - 852891037441 T^{17} + 26439622160671 T^{18} \)
$37$ \( 1 + 2 T + 188 T^{2} + 423 T^{3} + 18227 T^{4} + 44165 T^{5} + 1209239 T^{6} + 2807696 T^{7} + 59254779 T^{8} + 122227240 T^{9} + 2192426823 T^{10} + 3843735824 T^{11} + 61251583067 T^{12} + 82772320565 T^{13} + 1263932304239 T^{14} + 1085302271007 T^{15} + 17847192901004 T^{16} + 7024958907842 T^{17} + 129961739795077 T^{18} \)
$41$ \( 1 - 20 T + 392 T^{2} - 5020 T^{3} + 61165 T^{4} - 593867 T^{5} + 5495533 T^{6} - 43187704 T^{7} + 324961130 T^{8} - 2125066040 T^{9} + 13323406330 T^{10} - 72598530424 T^{11} + 378757629893 T^{12} - 1678126207787 T^{13} + 7086344534165 T^{14} - 23845523289820 T^{15} + 76343675361352 T^{16} - 159698504582420 T^{17} + 327381934393961 T^{18} \)
$43$ \( 1 + 17 T + 227 T^{2} + 2140 T^{3} + 17914 T^{4} + 131075 T^{5} + 879969 T^{6} + 5320599 T^{7} + 32526199 T^{8} + 196873042 T^{9} + 1398626557 T^{10} + 9837787551 T^{11} + 69963695283 T^{12} + 448119341075 T^{13} + 2633509247902 T^{14} + 13527716924860 T^{15} + 61702824721289 T^{16} + 198699404719217 T^{17} + 502592611936843 T^{18} \)
$47$ \( 1 - 4 T + 147 T^{2} + 207 T^{3} + 6029 T^{4} + 68694 T^{5} + 197879 T^{6} + 3753473 T^{7} + 18796588 T^{8} + 136181628 T^{9} + 883439636 T^{10} + 8291421857 T^{11} + 20544391417 T^{12} + 335204806614 T^{13} + 1382721047203 T^{14} + 2231297573103 T^{15} + 74473598708061 T^{16} - 95245146647044 T^{17} + 1119130473102767 T^{18} \)
$53$ \( 1 - 16 T + 317 T^{2} - 3576 T^{3} + 41298 T^{4} - 345801 T^{5} + 2982442 T^{6} - 20147904 T^{7} + 153700154 T^{8} - 1006557198 T^{9} + 8146108162 T^{10} - 56595462336 T^{11} + 444017017634 T^{12} - 2728536220281 T^{13} + 17270637469914 T^{14} - 79259755397304 T^{15} + 372383431328329 T^{16} - 996155046581776 T^{17} + 3299763591802133 T^{18} \)
$59$ \( ( 1 + T )^{9} \)
$61$ \( 1 + 9 T + 281 T^{2} + 3052 T^{3} + 49442 T^{4} + 467431 T^{5} + 5890869 T^{6} + 48018117 T^{7} + 488641059 T^{8} + 3485876686 T^{9} + 29807104599 T^{10} + 178675413357 T^{11} + 1337115336489 T^{12} + 6471975304471 T^{13} + 41758530314042 T^{14} + 157240182549772 T^{15} + 883110736921901 T^{16} + 1725365816975529 T^{17} + 11694146092834141 T^{18} \)
$67$ \( 1 + 8 T + 361 T^{2} + 2209 T^{3} + 60341 T^{4} + 272426 T^{5} + 6379568 T^{6} + 21228770 T^{7} + 508519117 T^{8} + 1405021510 T^{9} + 34070780839 T^{10} + 95295948530 T^{11} + 1918738010384 T^{12} + 5489689289546 T^{13} + 81467899081487 T^{14} + 199822566211321 T^{15} + 2187916889521603 T^{16} + 3248541420453128 T^{17} + 27206534396294947 T^{18} \)
$71$ \( 1 + 8 T + 247 T^{2} + 1057 T^{3} + 25643 T^{4} - 14888 T^{5} + 1291597 T^{6} - 13800489 T^{7} + 31878792 T^{8} - 1460258384 T^{9} + 2263394232 T^{10} - 69568265049 T^{11} + 462276773867 T^{12} - 378329106728 T^{13} + 46265853247693 T^{14} + 135402000104497 T^{15} + 2246494679122577 T^{16} + 5166028249966088 T^{17} + 45848500718449031 T^{18} \)
$73$ \( 1 + 20 T + 534 T^{2} + 6762 T^{3} + 98191 T^{4} + 815125 T^{5} + 8049077 T^{6} + 39012798 T^{7} + 344322498 T^{8} + 1131087832 T^{9} + 25135542354 T^{10} + 207899200542 T^{11} + 3131227787309 T^{12} + 23148116195125 T^{13} + 203556972788263 T^{14} + 1023322038166218 T^{15} + 5899310809197798 T^{16} + 16129201837881620 T^{17} + 58871586708267913 T^{18} \)
$79$ \( 1 + 29 T + 677 T^{2} + 11795 T^{3} + 182133 T^{4} + 2424297 T^{5} + 29519599 T^{6} + 322824554 T^{7} + 3261478800 T^{8} + 30087955658 T^{9} + 257656825200 T^{10} + 2014748041514 T^{11} + 14554313571361 T^{12} + 94426564518057 T^{13} + 560433513119067 T^{14} + 2867216537870195 T^{15} + 13001046383629643 T^{16} + 43996155487290269 T^{17} + 119851595982618319 T^{18} \)
$83$ \( 1 + 16 T + 560 T^{2} + 6742 T^{3} + 132344 T^{4} + 1305713 T^{5} + 18799240 T^{6} + 161145654 T^{7} + 1926967871 T^{8} + 14936632438 T^{9} + 159938333293 T^{10} + 1110132410406 T^{11} + 10749161041880 T^{12} + 61966946687873 T^{13} + 521308394857192 T^{14} + 2204231997253798 T^{15} + 15196188554191120 T^{16} + 36036675714224656 T^{17} + 186940255267540403 T^{18} \)
$89$ \( 1 - 11 T + 234 T^{2} - 2060 T^{3} + 33454 T^{4} - 239317 T^{5} + 3794919 T^{6} - 23446853 T^{7} + 360445499 T^{8} - 2272558860 T^{9} + 32079649411 T^{10} - 185722522613 T^{11} + 2675300252511 T^{12} - 15015284889397 T^{13} + 186809124806846 T^{14} - 1023781459379660 T^{15} + 10350132365553786 T^{16} - 43302476862722891 T^{17} + 350356403707485209 T^{18} \)
$97$ \( 1 - 17 T + 356 T^{2} - 3429 T^{3} + 39554 T^{4} - 153349 T^{5} + 1037720 T^{6} + 18846449 T^{7} - 131487679 T^{8} + 3109904788 T^{9} - 12754304863 T^{10} + 177326238641 T^{11} + 947099025560 T^{12} - 13575876712069 T^{13} + 339663656525378 T^{14} - 2856261004901541 T^{15} + 28764189274208228 T^{16} - 133236371104408337 T^{17} + 760231058654565217 T^{18} \)
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