Properties

Label 6018.2.a.x
Level 6018
Weight 2
Character orbit 6018.a
Self dual yes
Analytic conductor 48.054
Analytic rank 0
Dimension 10
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: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - x^{9} - 34 x^{8} + 30 x^{7} + 341 x^{6} - 276 x^{5} - 1032 x^{4} + 1176 x^{3} + 416 x^{2} - 896 x + 272\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 158 \nu^{9} + 225 \nu^{8} - 4779 \nu^{7} - 5750 \nu^{6} + 32040 \nu^{5} + 12263 \nu^{4} + 67732 \nu^{3} + 300692 \nu^{2} - 399092 \nu - 66416 \)\()/120304\)
\(\beta_{3}\)\(=\)\((\)\( 1042 \nu^{9} - 3275 \nu^{8} - 35705 \nu^{7} + 105416 \nu^{6} + 375768 \nu^{5} - 987205 \nu^{4} - 1375194 \nu^{3} + 2975172 \nu^{2} + 843124 \nu - 1593080 \)\()/120304\)
\(\beta_{4}\)\(=\)\((\)\( 5903 \nu^{9} - 5918 \nu^{8} - 207243 \nu^{7} + 177972 \nu^{6} + 2223525 \nu^{5} - 1593723 \nu^{4} - 7953632 \nu^{3} + 6020464 \nu^{2} + 6322324 \nu - 4079568 \)\()/120304\)
\(\beta_{5}\)\(=\)\((\)\( 11493 \nu^{9} - 1955 \nu^{8} - 393074 \nu^{7} + 21556 \nu^{6} + 3954805 \nu^{5} + 26618 \nu^{4} - 11927502 \nu^{3} + 4205884 \nu^{2} + 8148160 \nu - 4432912 \)\()/60152\)
\(\beta_{6}\)\(=\)\((\)\( -24821 \nu^{9} + 10767 \nu^{8} + 848932 \nu^{7} - 255106 \nu^{6} - 8588857 \nu^{5} + 1739438 \nu^{4} + 26641132 \nu^{3} - 12453916 \nu^{2} - 18090136 \nu + 10265344 \)\()/120304\)
\(\beta_{7}\)\(=\)\((\)\( 20873 \nu^{9} - 4492 \nu^{8} - 711529 \nu^{7} + 62998 \nu^{6} + 7121451 \nu^{5} - 52371 \nu^{4} - 21305118 \nu^{3} + 7212112 \nu^{2} + 14231612 \nu - 7071528 \)\()/60152\)
\(\beta_{8}\)\(=\)\((\)\( -73925 \nu^{9} + 15269 \nu^{8} + 2527192 \nu^{7} - 221448 \nu^{6} - 25415913 \nu^{5} + 470848 \nu^{4} + 76622382 \nu^{3} - 27434996 \nu^{2} - 51134312 \nu + 26070664 \)\()/120304\)
\(\beta_{9}\)\(=\)\((\)\( 89793 \nu^{9} - 16752 \nu^{8} - 3064731 \nu^{7} + 197524 \nu^{6} + 30729695 \nu^{5} + 289317 \nu^{4} - 92137756 \nu^{3} + 30417112 \nu^{2} + 62114404 \nu - 30442128 \)\()/120304\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} + 7\)
\(\nu^{3}\)\(=\)\(-2 \beta_{9} - \beta_{8} + \beta_{7} - 2 \beta_{6} + \beta_{5} - \beta_{4} - 3 \beta_{3} + 2 \beta_{2} + 15 \beta_{1}\)
\(\nu^{4}\)\(=\)\(6 \beta_{9} - 30 \beta_{7} - 22 \beta_{6} + 13 \beta_{5} - 22 \beta_{4} + 2 \beta_{3} - 22 \beta_{2} + 95\)
\(\nu^{5}\)\(=\)\(-38 \beta_{9} - 12 \beta_{8} + 26 \beta_{7} - 41 \beta_{6} + 24 \beta_{5} - 11 \beta_{4} - 72 \beta_{3} + 65 \beta_{2} + 250 \beta_{1} + 18\)
\(\nu^{6}\)\(=\)\(177 \beta_{9} + 11 \beta_{8} - 659 \beta_{7} - 426 \beta_{6} + 189 \beta_{5} - 422 \beta_{4} + 63 \beta_{3} - 394 \beta_{2} - 27 \beta_{1} + 1517\)
\(\nu^{7}\)\(=\)\(-625 \beta_{9} - 112 \beta_{8} + 472 \beta_{7} - 790 \beta_{6} + 458 \beta_{5} - 127 \beta_{4} - 1430 \beta_{3} + 1522 \beta_{2} + 4330 \beta_{1} + 367\)
\(\nu^{8}\)\(=\)\(3987 \beta_{9} + 398 \beta_{8} - 13224 \beta_{7} - 8058 \beta_{6} + 3027 \beta_{5} - 7867 \beta_{4} + 1342 \beta_{3} - 6830 \beta_{2} - 996 \beta_{1} + 25886\)
\(\nu^{9}\)\(=\)\(-10043 \beta_{9} - 692 \beta_{8} + 7656 \beta_{7} - 15141 \beta_{6} + 8213 \beta_{5} - 1726 \beta_{4} - 27140 \beta_{3} + 31757 \beta_{2} + 76804 \beta_{1} + 5520\)

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
−4.29748
−2.78078
−2.54849
−0.976160
0.580342
0.780661
0.931948
1.39621
3.58712
4.32662
−1.00000 −1.00000 1.00000 −4.29748 1.00000 0.914984 −1.00000 1.00000 4.29748
1.2 −1.00000 −1.00000 1.00000 −2.78078 1.00000 3.42043 −1.00000 1.00000 2.78078
1.3 −1.00000 −1.00000 1.00000 −2.54849 1.00000 4.63796 −1.00000 1.00000 2.54849
1.4 −1.00000 −1.00000 1.00000 −0.976160 1.00000 −3.87477 −1.00000 1.00000 0.976160
1.5 −1.00000 −1.00000 1.00000 0.580342 1.00000 4.77865 −1.00000 1.00000 −0.580342
1.6 −1.00000 −1.00000 1.00000 0.780661 1.00000 1.13694 −1.00000 1.00000 −0.780661
1.7 −1.00000 −1.00000 1.00000 0.931948 1.00000 −0.903548 −1.00000 1.00000 −0.931948
1.8 −1.00000 −1.00000 1.00000 1.39621 1.00000 −1.23174 −1.00000 1.00000 −1.39621
1.9 −1.00000 −1.00000 1.00000 3.58712 1.00000 −2.07087 −1.00000 1.00000 −3.58712
1.10 −1.00000 −1.00000 1.00000 4.32662 1.00000 3.19198 −1.00000 1.00000 −4.32662
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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.x 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6018.2.a.x 10 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6018))\):

\(T_{5}^{10} - \cdots\)
\(T_{7}^{10} - \cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{10} \)
$3$ \( ( 1 + T )^{10} \)
$5$ \( 1 - T + 16 T^{2} - 15 T^{3} + 106 T^{4} - 126 T^{5} + 398 T^{6} - 474 T^{7} + 901 T^{8} + 244 T^{9} + 2132 T^{10} + 1220 T^{11} + 22525 T^{12} - 59250 T^{13} + 248750 T^{14} - 393750 T^{15} + 1656250 T^{16} - 1171875 T^{17} + 6250000 T^{18} - 1953125 T^{19} + 9765625 T^{20} \)
$7$ \( 1 - 10 T + 75 T^{2} - 418 T^{3} + 2009 T^{4} - 8287 T^{5} + 30980 T^{6} - 104534 T^{7} + 328204 T^{8} - 953199 T^{9} + 2612702 T^{10} - 6672393 T^{11} + 16081996 T^{12} - 35855162 T^{13} + 74382980 T^{14} - 139279609 T^{15} + 236356841 T^{16} - 344240974 T^{17} + 432360075 T^{18} - 403536070 T^{19} + 282475249 T^{20} \)
$11$ \( 1 - 2 T + 46 T^{2} + 5 T^{3} + 940 T^{4} + 1701 T^{5} + 15490 T^{6} + 42545 T^{7} + 209859 T^{8} + 715659 T^{9} + 2346368 T^{10} + 7872249 T^{11} + 25392939 T^{12} + 56627395 T^{13} + 226789090 T^{14} + 273947751 T^{15} + 1665267340 T^{16} + 97435855 T^{17} + 9860508526 T^{18} - 4715895382 T^{19} + 25937424601 T^{20} \)
$13$ \( 1 + 39 T^{2} + 82 T^{3} + 968 T^{4} + 2496 T^{5} + 23358 T^{6} + 46212 T^{7} + 405807 T^{8} + 870802 T^{9} + 5459510 T^{10} + 11320426 T^{11} + 68581383 T^{12} + 101527764 T^{13} + 667127838 T^{14} + 926747328 T^{15} + 4672351112 T^{16} + 5145378394 T^{17} + 31813498119 T^{18} + 137858491849 T^{20} \)
$17$ \( ( 1 - T )^{10} \)
$19$ \( 1 - 15 T + 199 T^{2} - 1652 T^{3} + 12796 T^{4} - 76620 T^{5} + 457933 T^{6} - 2320711 T^{7} + 12250563 T^{8} - 56006184 T^{9} + 263591800 T^{10} - 1064117496 T^{11} + 4422453243 T^{12} - 15917756749 T^{13} + 59678286493 T^{14} - 189718705380 T^{15} + 601999093276 T^{16} - 1476676112828 T^{17} + 3379729045159 T^{18} - 4840315466685 T^{19} + 6131066257801 T^{20} \)
$23$ \( 1 - 19 T + 299 T^{2} - 3139 T^{3} + 28833 T^{4} - 211264 T^{5} + 1405874 T^{6} - 7996629 T^{7} + 43279532 T^{8} - 212465565 T^{9} + 1050593858 T^{10} - 4886707995 T^{11} + 22894872428 T^{12} - 97294985043 T^{13} + 393421186034 T^{14} - 1359767567552 T^{15} + 4268318787537 T^{16} - 10687747078133 T^{17} + 23414984599019 T^{18} - 34221900567797 T^{19} + 41426511213649 T^{20} \)
$29$ \( 1 + T + 152 T^{2} + 341 T^{3} + 11852 T^{4} + 38638 T^{5} + 633370 T^{6} + 2412694 T^{7} + 25925483 T^{8} + 99142794 T^{9} + 841964060 T^{10} + 2875141026 T^{11} + 21803331203 T^{12} + 58843193966 T^{13} + 447970566970 T^{14} + 792509775062 T^{15} + 7049846000492 T^{16} + 5882207821369 T^{17} + 76037454770072 T^{18} + 14507145975869 T^{19} + 420707233300201 T^{20} \)
$31$ \( 1 - 15 T + 234 T^{2} - 2397 T^{3} + 21704 T^{4} - 161656 T^{5} + 1059798 T^{6} - 6063172 T^{7} + 32048335 T^{8} - 162507738 T^{9} + 868334960 T^{10} - 5037739878 T^{11} + 30798449935 T^{12} - 180627957052 T^{13} + 978745708758 T^{14} - 4628074034056 T^{15} + 19262379892424 T^{16} - 65947736024067 T^{17} + 199576502761194 T^{18} - 396594332410065 T^{19} + 819628286980801 T^{20} \)
$37$ \( 1 - T + 184 T^{2} - 149 T^{3} + 14718 T^{4} - 10974 T^{5} + 667218 T^{6} - 592030 T^{7} + 19762489 T^{8} - 26578394 T^{9} + 575597964 T^{10} - 983400578 T^{11} + 27054847441 T^{12} - 29988095590 T^{13} + 1250473954098 T^{14} - 760980584118 T^{15} + 37762361287662 T^{16} - 14144849692817 T^{17} + 646296219521464 T^{18} - 129961739795077 T^{19} + 4808584372417849 T^{20} \)
$41$ \( 1 + 5 T + 325 T^{2} + 1517 T^{3} + 50061 T^{4} + 214156 T^{5} + 4814010 T^{6} + 18501559 T^{7} + 319916256 T^{8} + 1080481419 T^{9} + 15343917382 T^{10} + 44299738179 T^{11} + 537779226336 T^{12} + 1275145947839 T^{13} + 13603241711610 T^{14} + 24811300581356 T^{15} + 237794968408701 T^{16} + 295442233477477 T^{17} + 2595100699464325 T^{18} + 1636909671969805 T^{19} + 13422659310152401 T^{20} \)
$43$ \( 1 - 26 T + 512 T^{2} - 6855 T^{3} + 80190 T^{4} - 766511 T^{5} + 6782614 T^{6} - 52375395 T^{7} + 389955937 T^{8} - 2646857257 T^{9} + 17977206388 T^{10} - 113814862051 T^{11} + 721028527513 T^{12} - 4164210530265 T^{13} + 23188407525814 T^{14} - 112683588652373 T^{15} + 506910102899310 T^{16} - 1863316579138485 T^{17} + 5984358542131712 T^{18} - 13067407910357918 T^{19} + 21611482313284249 T^{20} \)
$47$ \( 1 - 14 T + 348 T^{2} - 3667 T^{3} + 54992 T^{4} - 484131 T^{5} + 5561478 T^{6} - 42351223 T^{7} + 402147183 T^{8} - 2671469577 T^{9} + 21718582076 T^{10} - 125559070119 T^{11} + 888343127247 T^{12} - 4397031025529 T^{13} + 27138238528518 T^{14} - 111033027583917 T^{15} + 592770609372368 T^{16} - 1857786982737821 T^{17} + 8286327758292828 T^{18} - 15667826623438738 T^{19} + 52599132235830049 T^{20} \)
$53$ \( 1 + 2 T + 405 T^{2} + 926 T^{3} + 78481 T^{4} + 186453 T^{5} + 9575990 T^{6} + 22056540 T^{7} + 813784196 T^{8} + 1710472535 T^{9} + 50276749610 T^{10} + 90655044355 T^{11} + 2285919806564 T^{12} + 3283711505580 T^{13} + 75559167151190 T^{14} + 77973804256329 T^{15} + 1739481225765049 T^{16} + 1087782515489062 T^{17} + 25215174616601205 T^{18} + 6599527183604266 T^{19} + 174887470365513049 T^{20} \)
$59$ \( ( 1 + T )^{10} \)
$61$ \( 1 - 4 T + 478 T^{2} - 1863 T^{3} + 108950 T^{4} - 399955 T^{5} + 15556186 T^{6} - 52174469 T^{7} + 1537278929 T^{8} - 4568319325 T^{9} + 109690163744 T^{10} - 278667478825 T^{11} + 5720214894809 T^{12} - 11842613148089 T^{13} + 215388477922426 T^{14} - 337800513566455 T^{15} + 5613144786630950 T^{16} - 5854929903507123 T^{17} + 91636095612700318 T^{18} - 46776584371336564 T^{19} + 713342911662882601 T^{20} \)
$67$ \( 1 - 15 T + 429 T^{2} - 5446 T^{3} + 91406 T^{4} - 1000859 T^{5} + 12769198 T^{6} - 121934859 T^{7} + 1290952545 T^{8} - 10813832283 T^{9} + 98756472682 T^{10} - 724526762961 T^{11} + 5795085974505 T^{12} - 36673493997417 T^{13} + 257313653970958 T^{14} - 1351284864466913 T^{15} + 8268438880539614 T^{16} - 33006635402589058 T^{17} + 174203033671798989 T^{18} - 408098015944424205 T^{19} + 1822837804551761449 T^{20} \)
$71$ \( 1 - 14 T + 480 T^{2} - 6135 T^{3} + 111962 T^{4} - 1278779 T^{5} + 17030166 T^{6} - 169954891 T^{7} + 1869416669 T^{8} - 16137639849 T^{9} + 153117235508 T^{10} - 1145772429279 T^{11} + 9423729428429 T^{12} - 60828724992701 T^{13} + 432765145769046 T^{14} - 2307210605242429 T^{15} + 14342363988363002 T^{16} - 55798562171728785 T^{17} + 309961694997965280 T^{18} - 641879010058286434 T^{19} + 3255243551009881201 T^{20} \)
$73$ \( 1 - 43 T + 1265 T^{2} - 26511 T^{3} + 461617 T^{4} - 6754708 T^{5} + 87998606 T^{6} - 1021936865 T^{7} + 10878475704 T^{8} - 105354721041 T^{9} + 942066932846 T^{10} - 7690894635993 T^{11} + 57971397026616 T^{12} - 397550813411705 T^{13} + 2499005620852046 T^{14} - 14002993273809844 T^{15} + 69858451536849313 T^{16} - 292877582139780567 T^{17} + 1020172016246012465 T^{18} - 2531478228455520259 T^{19} + 4297625829703557649 T^{20} \)
$79$ \( 1 + 428 T^{2} + 28 T^{3} + 97162 T^{4} + 852 T^{5} + 15015246 T^{6} + 209960 T^{7} + 1729283261 T^{8} + 36203184 T^{9} + 154471209180 T^{10} + 2860051536 T^{11} + 10792456831901 T^{12} + 103518468440 T^{13} + 584845047934926 T^{14} + 2621652051948 T^{15} + 23618863353331402 T^{16} + 537709451612452 T^{17} + 649322570640008108 T^{18} + 9468276082626847201 T^{20} \)
$83$ \( 1 + 4 T + 454 T^{2} + 1670 T^{3} + 98929 T^{4} + 340815 T^{5} + 14368903 T^{6} + 44475310 T^{7} + 1598660102 T^{8} + 4310958037 T^{9} + 145321537494 T^{10} + 357809517071 T^{11} + 11013169442678 T^{12} + 25430404078970 T^{13} + 681924010991863 T^{14} + 1342484136744045 T^{15} + 32343884197021801 T^{16} + 45317205152677090 T^{17} + 1022540673391124614 T^{18} + 747761021070161612 T^{19} + 15516041187205853449 T^{20} \)
$89$ \( 1 + 22 T + 768 T^{2} + 12056 T^{3} + 247577 T^{4} + 3070135 T^{5} + 47437575 T^{6} + 491580966 T^{7} + 6308467804 T^{8} + 56747582645 T^{9} + 635334397110 T^{10} + 5050534855405 T^{11} + 49969373475484 T^{12} + 346549342020054 T^{13} + 2976339763105575 T^{14} + 17143816356455615 T^{15} + 123041137072251497 T^{16} + 533252973500497624 T^{17} + 3023300202779198208 T^{18} + 7707840881564674598 T^{19} + 31181719929966183601 T^{20} \)
$97$ \( 1 - 37 T + 1088 T^{2} - 22621 T^{3} + 400557 T^{4} - 5971412 T^{5} + 79694671 T^{6} - 955015727 T^{7} + 10701319044 T^{8} - 112000135835 T^{9} + 1132020155870 T^{10} - 10864013175995 T^{11} + 100688710884996 T^{12} - 871617068608271 T^{13} + 7055311923161551 T^{14} - 51278546658732884 T^{15} + 333652767378345453 T^{16} - 1827737993179394173 T^{17} + 8527127750682133568 T^{18} - 28128549170218913029 T^{19} + 73742412689492826049 T^{20} \)
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