Properties

Label 6018.2.a.u
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - 4 x^{8} - 16 x^{7} + 37 x^{6} + 97 x^{5} - 72 x^{4} - 182 x^{3} + 24 x^{2} + 70 x - 19\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -7 \nu^{8} - 595 \nu^{7} + 3093 \nu^{6} + 7114 \nu^{5} - 28461 \nu^{4} - 34541 \nu^{3} + 63957 \nu^{2} + 48357 \nu - 30285 \)\()/1472\)
\(\beta_{3}\)\(=\)\((\)\( 77 \nu^{8} - 79 \nu^{7} - 2375 \nu^{6} + 498 \nu^{5} + 17199 \nu^{4} + 6063 \nu^{3} - 33031 \nu^{2} - 17463 \nu + 10767 \)\()/736\)
\(\beta_{4}\)\(=\)\((\)\( 163 \nu^{8} - 865 \nu^{7} - 1577 \nu^{6} + 8462 \nu^{5} + 6433 \nu^{4} - 23583 \nu^{3} - 7401 \nu^{2} + 17927 \nu - 3455 \)\()/1472\)
\(\beta_{5}\)\(=\)\((\)\( 211 \nu^{8} + 271 \nu^{7} - 8697 \nu^{6} - 5202 \nu^{5} + 68273 \nu^{4} + 47985 \nu^{3} - 136633 \nu^{2} - 84873 \nu + 49233 \)\()/1472\)
\(\beta_{6}\)\(=\)\((\)\( -205 \nu^{8} + 975 \nu^{7} + 2471 \nu^{6} - 9202 \nu^{5} - 11599 \nu^{4} + 20945 \nu^{3} + 14311 \nu^{2} - 10409 \nu + 2065 \)\()/736\)
\(\beta_{7}\)\(=\)\((\)\( 705 \nu^{8} - 3371 \nu^{7} - 8067 \nu^{6} + 29402 \nu^{5} + 39979 \nu^{4} - 56341 \nu^{3} - 59843 \nu^{2} + 10701 \nu + 3723 \)\()/1472\)
\(\beta_{8}\)\(=\)\((\)\( -511 \nu^{8} + 2197 \nu^{7} + 7197 \nu^{6} - 19430 \nu^{5} - 40405 \nu^{4} + 33899 \nu^{3} + 67389 \nu^{2} + 941 \nu - 13109 \)\()/736\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{8} + \beta_{7} + \beta_{4} + \beta_{3} + 3 \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(4 \beta_{8} + 4 \beta_{7} - \beta_{6} - 2 \beta_{5} + 3 \beta_{4} + 5 \beta_{3} - 3 \beta_{2} + 17 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(24 \beta_{8} + 23 \beta_{7} - 7 \beta_{6} - 10 \beta_{5} + 21 \beta_{4} + 26 \beta_{3} - 18 \beta_{2} + 79 \beta_{1} + 22\)
\(\nu^{5}\)\(=\)\(120 \beta_{8} + 113 \beta_{7} - 48 \beta_{6} - 63 \beta_{5} + 94 \beta_{4} + 133 \beta_{3} - 112 \beta_{2} + 404 \beta_{1} + 64\)
\(\nu^{6}\)\(=\)\(635 \beta_{8} + 592 \beta_{7} - 268 \beta_{6} - 326 \beta_{5} + 506 \beta_{4} + 674 \beta_{3} - 606 \beta_{2} + 2035 \beta_{1} + 328\)
\(\nu^{7}\)\(=\)\(3264 \beta_{8} + 3020 \beta_{7} - 1482 \beta_{6} - 1747 \beta_{5} + 2529 \beta_{4} + 3459 \beta_{3} - 3256 \beta_{2} + 10419 \beta_{1} + 1431\)
\(\nu^{8}\)\(=\)\(16912 \beta_{8} + 15604 \beta_{7} - 7834 \beta_{6} - 9049 \beta_{5} + 13096 \beta_{4} + 17715 \beta_{3} - 17051 \beta_{2} + 53374 \beta_{1} + 7168\)

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
−1.12576
0.443000
2.85528
0.342793
−2.14979
−1.32747
1.53430
−1.72522
5.15287
−1.00000 −1.00000 1.00000 −3.14251 1.00000 −1.86106 −1.00000 1.00000 3.14251
1.2 −1.00000 −1.00000 1.00000 −2.03963 1.00000 −4.77480 −1.00000 1.00000 2.03963
1.3 −1.00000 −1.00000 1.00000 −1.56975 1.00000 1.56796 −1.00000 1.00000 1.56975
1.4 −1.00000 −1.00000 1.00000 −1.11819 1.00000 −1.49540 −1.00000 1.00000 1.11819
1.5 −1.00000 −1.00000 1.00000 −1.03642 1.00000 2.58814 −1.00000 1.00000 1.03642
1.6 −1.00000 −1.00000 1.00000 2.06239 1.00000 1.71252 −1.00000 1.00000 −2.06239
1.7 −1.00000 −1.00000 1.00000 2.21366 1.00000 −2.01100 −1.00000 1.00000 −2.21366
1.8 −1.00000 −1.00000 1.00000 3.17259 1.00000 −4.61193 −1.00000 1.00000 −3.17259
1.9 −1.00000 −1.00000 1.00000 3.45787 1.00000 3.88559 −1.00000 1.00000 −3.45787
\(n\): 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.u 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6018.2.a.u 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

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{9} \)
$3$ \( ( 1 + T )^{9} \)
$5$ \( 1 - 2 T + 22 T^{2} - 47 T^{3} + 287 T^{4} - 547 T^{5} + 2505 T^{6} - 4362 T^{7} + 16355 T^{8} - 24936 T^{9} + 81775 T^{10} - 109050 T^{11} + 313125 T^{12} - 341875 T^{13} + 896875 T^{14} - 734375 T^{15} + 1718750 T^{16} - 781250 T^{17} + 1953125 T^{18} \)
$7$ \( 1 + 5 T + 35 T^{2} + 145 T^{3} + 642 T^{4} + 2271 T^{5} + 7935 T^{6} + 23809 T^{7} + 72650 T^{8} + 189556 T^{9} + 508550 T^{10} + 1166641 T^{11} + 2721705 T^{12} + 5452671 T^{13} + 10790094 T^{14} + 17059105 T^{15} + 28824005 T^{16} + 28824005 T^{17} + 40353607 T^{18} \)
$11$ \( 1 + T + 45 T^{2} + 18 T^{3} + 998 T^{4} + 169 T^{5} + 15749 T^{6} + 2895 T^{7} + 202333 T^{8} + 42538 T^{9} + 2225663 T^{10} + 350295 T^{11} + 20961919 T^{12} + 2474329 T^{13} + 160728898 T^{14} + 31888098 T^{15} + 876922695 T^{16} + 214358881 T^{17} + 2357947691 T^{18} \)
$13$ \( 1 + 4 T + 78 T^{2} + 284 T^{3} + 2884 T^{4} + 9600 T^{5} + 68334 T^{6} + 205772 T^{7} + 1169167 T^{8} + 3124392 T^{9} + 15199171 T^{10} + 34775468 T^{11} + 150129798 T^{12} + 274185600 T^{13} + 1070809012 T^{14} + 1370813756 T^{15} + 4894384326 T^{16} + 3262922884 T^{17} + 10604499373 T^{18} \)
$17$ \( ( 1 - T )^{9} \)
$19$ \( 1 + 7 T + 115 T^{2} + 634 T^{3} + 6157 T^{4} + 27955 T^{5} + 208097 T^{6} + 807510 T^{7} + 5097894 T^{8} + 17385180 T^{9} + 96859986 T^{10} + 291511110 T^{11} + 1427337323 T^{12} + 3643123555 T^{13} + 15245341543 T^{14} + 29827088554 T^{15} + 102795249985 T^{16} + 118884941287 T^{17} + 322687697779 T^{18} \)
$23$ \( 1 + 8 T + 144 T^{2} + 962 T^{3} + 9987 T^{4} + 57911 T^{5} + 441863 T^{6} + 2236974 T^{7} + 13818492 T^{8} + 60708394 T^{9} + 317825316 T^{10} + 1183359246 T^{11} + 5376147121 T^{12} + 16205872151 T^{13} + 64279757541 T^{14} + 142410525218 T^{15} + 490294864368 T^{16} + 626487882248 T^{17} + 1801152661463 T^{18} \)
$29$ \( 1 - 6 T + 162 T^{2} - 885 T^{3} + 13643 T^{4} - 66193 T^{5} + 749731 T^{6} - 3220346 T^{7} + 29472239 T^{8} - 110106212 T^{9} + 854694931 T^{10} - 2708310986 T^{11} + 18285189359 T^{12} - 46817051233 T^{13} + 279833605807 T^{14} - 526418639085 T^{15} + 2794479962058 T^{16} - 3001478477766 T^{17} + 14507145975869 T^{18} \)
$31$ \( 1 + 17 T + 227 T^{2} + 2266 T^{3} + 20673 T^{4} + 159466 T^{5} + 1165811 T^{6} + 7644830 T^{7} + 47861988 T^{8} + 271606810 T^{9} + 1483721628 T^{10} + 7346681630 T^{11} + 34730675501 T^{12} + 147270199786 T^{13} + 591850438623 T^{14} + 2011083341146 T^{15} + 6245363403197 T^{16} + 14499147636497 T^{17} + 26439622160671 T^{18} \)
$37$ \( 1 - 2 T + 48 T^{2} - 27 T^{3} + 3551 T^{4} + 2193 T^{5} + 151893 T^{6} - 142272 T^{7} + 7156071 T^{8} + 4961232 T^{9} + 264774627 T^{10} - 194770368 T^{11} + 7693836129 T^{12} + 4110035073 T^{13} + 246240391307 T^{14} - 69274613043 T^{15} + 4556730102384 T^{16} - 7024958907842 T^{17} + 129961739795077 T^{18} \)
$41$ \( 1 - 14 T + 264 T^{2} - 2862 T^{3} + 30777 T^{4} - 271721 T^{5} + 2235801 T^{6} - 16697792 T^{7} + 118205270 T^{8} - 771056792 T^{9} + 4846416070 T^{10} - 28068988352 T^{11} + 154093640721 T^{12} - 767818604681 T^{13} + 3565706298177 T^{14} - 13594798337742 T^{15} + 51415128304584 T^{16} - 111788953207694 T^{17} + 327381934393961 T^{18} \)
$43$ \( 1 + 27 T + 623 T^{2} + 9796 T^{3} + 134988 T^{4} + 1519071 T^{5} + 15288387 T^{6} + 132641947 T^{7} + 1040179099 T^{8} + 7162763078 T^{9} + 44727701257 T^{10} + 245254960003 T^{11} + 1215533785209 T^{12} + 5193401453871 T^{13} + 19844375703684 T^{14} + 61924072428004 T^{15} + 169342994719661 T^{16} + 315581407495227 T^{17} + 502592611936843 T^{18} \)
$47$ \( 1 + 18 T + 507 T^{2} + 6567 T^{3} + 104857 T^{4} + 1053980 T^{5} + 12087715 T^{6} + 97507985 T^{7} + 870758740 T^{8} + 5697200500 T^{9} + 40925660780 T^{10} + 215395138865 T^{11} + 1254982834445 T^{12} + 5143086180380 T^{13} + 24048429398999 T^{14} + 70787107065543 T^{15} + 256857922074741 T^{16} + 428603159911698 T^{17} + 1119130473102767 T^{18} \)
$53$ \( 1 - 4 T + 385 T^{2} - 1356 T^{3} + 69288 T^{4} - 213625 T^{5} + 7667926 T^{6} - 20492876 T^{7} + 576286884 T^{8} - 1312745238 T^{9} + 30543204852 T^{10} - 57564488684 T^{11} + 1141577819102 T^{12} - 1685604003625 T^{13} + 28975929318984 T^{14} - 30054873690924 T^{15} + 452263788837245 T^{16} - 249038761645444 T^{17} + 3299763591802133 T^{18} \)
$59$ \( ( 1 - T )^{9} \)
$61$ \( 1 - 5 T + 267 T^{2} - 1488 T^{3} + 34062 T^{4} - 237001 T^{5} + 2959303 T^{6} - 25206437 T^{7} + 207842667 T^{8} - 1848437882 T^{9} + 12678402687 T^{10} - 93793152077 T^{11} + 671705554243 T^{12} - 3281478162841 T^{13} + 28768639204662 T^{14} - 76662317049168 T^{15} + 839112337217607 T^{16} - 958536564986405 T^{17} + 11694146092834141 T^{18} \)
$67$ \( 1 + 22 T + 569 T^{2} + 8923 T^{3} + 134769 T^{4} + 1652654 T^{5} + 18588196 T^{6} + 188369670 T^{7} + 1727722827 T^{8} + 14884634158 T^{9} + 115757429409 T^{10} + 845591448630 T^{11} + 5590641593548 T^{12} + 33302830725134 T^{13} + 181955010545283 T^{14} + 807160144093987 T^{15} + 3448544903428787 T^{16} + 8933488906246102 T^{17} + 27206534396294947 T^{18} \)
$71$ \( 1 - 16 T + 655 T^{2} - 8487 T^{3} + 188239 T^{4} - 2007554 T^{5} + 31343121 T^{6} - 276706657 T^{7} + 3343827312 T^{8} - 24314212844 T^{9} + 237411739152 T^{10} - 1394878257937 T^{11} + 11218047780231 T^{12} - 51015321838274 T^{13} + 339626328802889 T^{14} - 1087187109637527 T^{15} + 5957303703746105 T^{16} - 10332056499932176 T^{17} + 45848500718449031 T^{18} \)
$73$ \( 1 + 12 T + 362 T^{2} + 4748 T^{3} + 64151 T^{4} + 800069 T^{5} + 7986895 T^{6} + 80640506 T^{7} + 770032900 T^{8} + 6258925320 T^{9} + 56212401700 T^{10} + 429733256474 T^{11} + 3107037932215 T^{12} + 22720552278629 T^{13} + 132989615762543 T^{14} + 718534906420172 T^{15} + 3999158263913114 T^{16} + 9677521102728972 T^{17} + 58871586708267913 T^{18} \)
$79$ \( 1 + 9 T + 475 T^{2} + 3387 T^{3} + 102791 T^{4} + 559471 T^{5} + 13701201 T^{6} + 57000880 T^{7} + 1329287220 T^{8} + 4662605594 T^{9} + 105013690380 T^{10} + 355742492080 T^{11} + 6755226439839 T^{12} + 21791440767151 T^{13} + 316293704309609 T^{14} + 823337211849627 T^{15} + 9121856768425525 T^{16} + 13653979289159049 T^{17} + 119851595982618319 T^{18} \)
$83$ \( 1 - 10 T + 376 T^{2} - 3320 T^{3} + 72694 T^{4} - 617937 T^{5} + 9918590 T^{6} - 79421708 T^{7} + 1028885207 T^{8} - 7529751666 T^{9} + 85397472181 T^{10} - 547136146412 T^{11} + 5671320820330 T^{12} - 29326252503777 T^{13} + 286344620502242 T^{14} - 1085442039585080 T^{15} + 10203155172099752 T^{16} - 22522922321390410 T^{17} + 186940255267540403 T^{18} \)
$89$ \( 1 - 15 T + 696 T^{2} - 8180 T^{3} + 216540 T^{4} - 2103941 T^{5} + 40815863 T^{6} - 334484591 T^{7} + 5185712987 T^{8} - 35850759452 T^{9} + 461528455843 T^{10} - 2649452445311 T^{11} + 28773918123247 T^{12} - 132005973271781 T^{13} + 1209172233086460 T^{14} - 4065306960060980 T^{15} + 30785009087288184 T^{16} - 59048832085531215 T^{17} + 350356403707485209 T^{18} \)
$97$ \( 1 + 33 T + 1196 T^{2} + 25561 T^{3} + 540434 T^{4} + 8574681 T^{5} + 131840820 T^{6} + 1644606739 T^{7} + 19785644749 T^{8} + 198510330292 T^{9} + 1919207540653 T^{10} + 15474104807251 T^{11} + 120327556711860 T^{12} + 759110343734361 T^{13} + 4640890644451538 T^{14} + 21291597417990169 T^{15} + 96634748235823148 T^{16} + 258635308614439713 T^{17} + 760231058654565217 T^{18} \)
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