Properties

Label 6042.2.a.z
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: \(x^{9} - 4 x^{8} - 11 x^{7} + 51 x^{6} + 25 x^{5} - 180 x^{4} + 29 x^{3} + 119 x^{2} - 8 x - 14\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - 4 x^{8} - 11 x^{7} + 51 x^{6} + 25 x^{5} - 180 x^{4} + 29 x^{3} + 119 x^{2} - 8 x - 14\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{8} - 9 \nu^{7} + 34 \nu^{6} - 119 \nu^{5} - 96 \nu^{4} + 1732 \nu^{3} - 755 \nu^{2} - 3266 \nu - 862 \)\()/716\)
\(\beta_{2}\)\(=\)\((\)\( 9 \nu^{8} - 81 \nu^{7} + 306 \nu^{6} + 361 \nu^{5} - 4444 \nu^{4} + 1984 \nu^{3} + 14685 \nu^{2} - 6482 \nu - 4894 \)\()/1432\)
\(\beta_{3}\)\(=\)\((\)\( 119 \nu^{8} - 355 \nu^{7} - 1682 \nu^{6} + 4455 \nu^{5} + 7192 \nu^{4} - 14420 \nu^{3} - 8937 \nu^{2} + 5146 \nu + 3390 \)\()/1432\)
\(\beta_{4}\)\(=\)\((\)\( -119 \nu^{8} + 355 \nu^{7} + 1682 \nu^{6} - 4455 \nu^{5} - 7192 \nu^{4} + 14420 \nu^{3} + 7505 \nu^{2} - 3714 \nu + 906 \)\()/1432\)
\(\beta_{5}\)\(=\)\((\)\( 127 \nu^{8} - 427 \nu^{7} - 1410 \nu^{6} + 4935 \nu^{5} + 3560 \nu^{4} - 14884 \nu^{3} + 775 \nu^{2} + 4794 \nu + 3654 \)\()/1432\)
\(\beta_{6}\)\(=\)\((\)\( -42 \nu^{8} + 199 \nu^{7} + 362 \nu^{6} - 2520 \nu^{5} + 273 \nu^{4} + 8343 \nu^{3} - 5701 \nu^{2} - 2806 \nu + 1478 \)\()/358\)
\(\beta_{7}\)\(=\)\((\)\( 61 \nu^{8} - 191 \nu^{7} - 790 \nu^{6} + 2407 \nu^{5} + 2736 \nu^{4} - 8192 \nu^{3} - 1305 \nu^{2} + 4476 \nu + 402 \)\()/358\)
\(\beta_{8}\)\(=\)\((\)\( -36 \nu^{8} + 145 \nu^{7} + 387 \nu^{6} - 1802 \nu^{5} - 840 \nu^{4} + 6026 \nu^{3} - 1102 \nu^{2} - 3070 \nu + 244 \)\()/179\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} - \beta_{5} + \beta_{4} + \beta_{1} + 2\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{7} - \beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_{1} + 8\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{8} + 7 \beta_{7} - 2 \beta_{6} - 5 \beta_{5} + 5 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 9 \beta_{1} + 16\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(2 \beta_{8} + 11 \beta_{7} - 2 \beta_{6} - 5 \beta_{5} - 7 \beta_{4} - 22 \beta_{3} - 6 \beta_{2} + 15 \beta_{1} + 60\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(24 \beta_{8} + 63 \beta_{7} - 24 \beta_{6} - 29 \beta_{5} + 29 \beta_{4} - 44 \beta_{3} - 32 \beta_{2} + 83 \beta_{1} + 146\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(42 \beta_{8} + 133 \beta_{7} - 40 \beta_{6} - 21 \beta_{5} - 45 \beta_{4} - 246 \beta_{3} - 96 \beta_{2} + 187 \beta_{1} + 520\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(276 \beta_{8} + 633 \beta_{7} - 260 \beta_{6} - 161 \beta_{5} + 171 \beta_{4} - 638 \beta_{3} - 396 \beta_{2} + 819 \beta_{1} + 1432\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(640 \beta_{8} + 1625 \beta_{7} - 564 \beta_{6} - 27 \beta_{5} - 301 \beta_{4} - 2772 \beta_{3} - 1220 \beta_{2} + 2195 \beta_{1} + 4926\)\()/2\)

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
−0.690250
−2.30368
2.71772
1.09225
0.407650
2.32668
3.37216
−2.56034
−0.362196
−1.00000 −1.00000 1.00000 −4.37472 1.00000 −0.773393 −1.00000 1.00000 4.37472
1.2 −1.00000 −1.00000 1.00000 −2.37696 1.00000 0.603136 −1.00000 1.00000 2.37696
1.3 −1.00000 −1.00000 1.00000 −2.02880 1.00000 −1.70084 −1.00000 1.00000 2.02880
1.4 −1.00000 −1.00000 1.00000 −0.410706 1.00000 4.57610 −1.00000 1.00000 0.410706
1.5 −1.00000 −1.00000 1.00000 −0.186875 1.00000 −2.28090 −1.00000 1.00000 0.186875
1.6 −1.00000 −1.00000 1.00000 0.442981 1.00000 −2.93910 −1.00000 1.00000 −0.442981
1.7 −1.00000 −1.00000 1.00000 0.912007 1.00000 3.48340 −1.00000 1.00000 −0.912007
1.8 −1.00000 −1.00000 1.00000 3.19328 1.00000 3.82171 −1.00000 1.00000 −3.19328
1.9 −1.00000 −1.00000 1.00000 3.82978 1.00000 −0.790105 −1.00000 1.00000 −3.82978
\(n\): 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.z 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6042.2.a.z 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

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{9} \)
$3$ \( ( 1 + T )^{9} \)
$5$ \( 1 + T + 18 T^{2} + 18 T^{3} + 141 T^{4} + 197 T^{5} + 725 T^{6} + 1824 T^{7} + 3415 T^{8} + 11648 T^{9} + 17075 T^{10} + 45600 T^{11} + 90625 T^{12} + 123125 T^{13} + 440625 T^{14} + 281250 T^{15} + 1406250 T^{16} + 390625 T^{17} + 1953125 T^{18} \)
$7$ \( 1 - 4 T + 38 T^{2} - 153 T^{3} + 804 T^{4} - 2780 T^{5} + 11178 T^{6} - 32911 T^{7} + 107771 T^{8} - 273808 T^{9} + 754397 T^{10} - 1612639 T^{11} + 3834054 T^{12} - 6674780 T^{13} + 13512828 T^{14} - 18000297 T^{15} + 31294634 T^{16} - 23059204 T^{17} + 40353607 T^{18} \)
$11$ \( 1 - 12 T + 130 T^{2} - 949 T^{3} + 6296 T^{4} - 33826 T^{5} + 167392 T^{6} - 710051 T^{7} + 2794025 T^{8} - 9610148 T^{9} + 30734275 T^{10} - 85916171 T^{11} + 222798752 T^{12} - 495246466 T^{13} + 1013977096 T^{14} - 1681211389 T^{15} + 2533332230 T^{16} - 2572306572 T^{17} + 2357947691 T^{18} \)
$13$ \( 1 + T + 55 T^{2} + 85 T^{3} + 1550 T^{4} + 2920 T^{5} + 31630 T^{6} + 63651 T^{7} + 510328 T^{8} + 984590 T^{9} + 6634264 T^{10} + 10757019 T^{11} + 69491110 T^{12} + 83398120 T^{13} + 575504150 T^{14} + 410278765 T^{15} + 3451168435 T^{16} + 815730721 T^{17} + 10604499373 T^{18} \)
$17$ \( 1 - 12 T + 154 T^{2} - 1245 T^{3} + 9878 T^{4} - 61316 T^{5} + 368832 T^{6} - 1860731 T^{7} + 9079727 T^{8} - 38054384 T^{9} + 154355359 T^{10} - 537751259 T^{11} + 1812071616 T^{12} - 5121173636 T^{13} + 14025347446 T^{14} - 30051273405 T^{15} + 63192155642 T^{16} - 83709089292 T^{17} + 118587876497 T^{18} \)
$19$ \( ( 1 + T )^{9} \)
$23$ \( 1 - 11 T + 186 T^{2} - 1480 T^{3} + 14895 T^{4} - 95265 T^{5} + 718295 T^{6} - 3831708 T^{7} + 23398985 T^{8} - 105303104 T^{9} + 538176655 T^{10} - 2026973532 T^{11} + 8739495265 T^{12} - 26659052865 T^{13} + 95869328985 T^{14} - 219093115720 T^{15} + 633297533142 T^{16} - 861420838091 T^{17} + 1801152661463 T^{18} \)
$29$ \( 1 - 7 T + 152 T^{2} - 1024 T^{3} + 11598 T^{4} - 70476 T^{5} + 587392 T^{6} - 3107984 T^{7} + 22026585 T^{8} - 101811162 T^{9} + 638770965 T^{10} - 2613814544 T^{11} + 14325903488 T^{12} - 49846335756 T^{13} + 237888306102 T^{14} - 609099080704 T^{15} + 2621981198968 T^{16} - 3501724890727 T^{17} + 14507145975869 T^{18} \)
$31$ \( 1 + 12 T + 254 T^{2} + 2480 T^{3} + 29455 T^{4} + 236744 T^{5} + 2036883 T^{6} + 13602480 T^{7} + 92589857 T^{8} + 513923480 T^{9} + 2870285567 T^{10} + 13071983280 T^{11} + 60680781453 T^{12} + 218638055624 T^{13} + 843271642705 T^{14} + 2201009128880 T^{15} + 6988203984194 T^{16} + 10234692449292 T^{17} + 26439622160671 T^{18} \)
$37$ \( 1 + 9 T + 241 T^{2} + 1763 T^{3} + 27470 T^{4} + 171864 T^{5} + 1989422 T^{6} + 10764093 T^{7} + 100997566 T^{8} + 470700878 T^{9} + 3736909942 T^{10} + 14736043317 T^{11} + 100770192566 T^{12} + 322100806104 T^{13} + 1904878498790 T^{14} + 4523375659067 T^{15} + 22878582389053 T^{16} + 31612315085289 T^{17} + 129961739795077 T^{18} \)
$41$ \( 1 + 4 T + 234 T^{2} + 833 T^{3} + 28007 T^{4} + 87198 T^{5} + 2177373 T^{6} + 5926383 T^{7} + 120832265 T^{8} + 285048700 T^{9} + 4954122865 T^{10} + 9962249823 T^{11} + 150066724533 T^{12} + 246400707678 T^{13} + 3244784621407 T^{14} + 3956836832753 T^{15} + 45572500088154 T^{16} + 31939700916484 T^{17} + 327381934393961 T^{18} \)
$43$ \( 1 - 23 T + 488 T^{2} - 6590 T^{3} + 82757 T^{4} - 813497 T^{5} + 7608593 T^{6} - 59917394 T^{7} + 457786639 T^{8} - 3035426000 T^{9} + 19684825477 T^{10} - 110787261506 T^{11} + 604936403651 T^{12} - 2781184357097 T^{13} + 12165977717351 T^{14} - 41657782492910 T^{15} + 132647482220216 T^{16} - 268828606384823 T^{17} + 502592611936843 T^{18} \)
$47$ \( 1 - 35 T + 818 T^{2} - 13847 T^{3} + 193227 T^{4} - 2263832 T^{5} + 23203575 T^{6} - 209101393 T^{7} + 1688272269 T^{8} - 12186761370 T^{9} + 79348796643 T^{10} - 461904977137 T^{11} + 2409064767225 T^{12} - 11046777997592 T^{13} + 44315647667589 T^{14} - 149259794660663 T^{15} + 414417712538734 T^{16} - 833395033161635 T^{17} + 1119130473102767 T^{18} \)
$53$ \( ( 1 - T )^{9} \)
$59$ \( 1 - 14 T + 355 T^{2} - 3767 T^{3} + 55803 T^{4} - 469223 T^{5} + 5270813 T^{6} - 37409639 T^{7} + 364776252 T^{8} - 2355300378 T^{9} + 21521798868 T^{10} - 130222953359 T^{11} + 1082514303127 T^{12} - 5685744480503 T^{13} + 39894920657097 T^{14} - 158894070225647 T^{15} + 883471277110745 T^{16} - 2055626126460494 T^{17} + 8662995818654939 T^{18} \)
$61$ \( 1 - 14 T + 418 T^{2} - 4419 T^{3} + 77909 T^{4} - 685740 T^{5} + 9158979 T^{6} - 69482597 T^{7} + 761575889 T^{8} - 4986790796 T^{9} + 46456129229 T^{10} - 258544743437 T^{11} + 2078914212399 T^{12} - 9494647007340 T^{13} + 65801653214609 T^{14} - 227668534301259 T^{15} + 1313666505456778 T^{16} - 2683902381961934 T^{17} + 11694146092834141 T^{18} \)
$67$ \( 1 + 10 T + 316 T^{2} + 3381 T^{3} + 52944 T^{4} + 539910 T^{5} + 5980494 T^{6} + 56470739 T^{7} + 502506209 T^{8} + 4352382816 T^{9} + 33667916003 T^{10} + 253497147371 T^{11} + 1798711316922 T^{12} + 10879791739110 T^{13} + 71481023665008 T^{14} + 305839790113389 T^{15} + 1915184867282068 T^{16} + 4060676775566410 T^{17} + 27206534396294947 T^{18} \)
$71$ \( 1 - 4 T + 347 T^{2} - 888 T^{3} + 61392 T^{4} - 117856 T^{5} + 7414408 T^{6} - 11385864 T^{7} + 669936618 T^{8} - 875285560 T^{9} + 47565499878 T^{10} - 57396140424 T^{11} + 2653698181688 T^{12} - 2994919075936 T^{13} + 110765248316592 T^{14} - 113753052121848 T^{15} + 3156006694961677 T^{16} - 2583014124983044 T^{17} + 45848500718449031 T^{18} \)
$73$ \( 1 + 5 T + 387 T^{2} + 1657 T^{3} + 78808 T^{4} + 296890 T^{5} + 10683452 T^{6} + 35592151 T^{7} + 1046082720 T^{8} + 3033081026 T^{9} + 76364038560 T^{10} + 189670572679 T^{11} + 4156044446684 T^{12} + 8431153770490 T^{13} + 163374626101144 T^{14} + 250760812960873 T^{15} + 4275343226890539 T^{16} + 4032300459470405 T^{17} + 58871586708267913 T^{18} \)
$79$ \( 1 + 14 T + 370 T^{2} + 3233 T^{3} + 58794 T^{4} + 447314 T^{5} + 7154740 T^{6} + 50178255 T^{7} + 682819339 T^{8} + 4325003040 T^{9} + 53942727781 T^{10} + 313162489455 T^{11} + 3527565854860 T^{12} + 17422916532434 T^{13} + 180912453922806 T^{14} + 785901743699393 T^{15} + 7105446324878830 T^{16} + 21239523338691854 T^{17} + 119851595982618319 T^{18} \)
$83$ \( 1 - 37 T + 975 T^{2} - 19221 T^{3} + 321998 T^{4} - 4669300 T^{5} + 60411650 T^{6} - 699529371 T^{7} + 7354919590 T^{8} - 70078464318 T^{9} + 610458325970 T^{10} - 4819057836819 T^{11} + 34542596118550 T^{12} - 221597138245300 T^{13} + 1268363208964714 T^{14} - 6284120916525549 T^{15} + 26457649714886325 T^{16} - 83334812589144517 T^{17} + 186940255267540403 T^{18} \)
$89$ \( 1 + 20 T + 625 T^{2} + 10919 T^{3} + 196203 T^{4} + 2803097 T^{5} + 38033347 T^{6} + 446055007 T^{7} + 4909103832 T^{8} + 47869278570 T^{9} + 436910241048 T^{10} + 3533201710447 T^{11} + 26812330601243 T^{12} + 175872587520377 T^{13} + 1095609216072147 T^{14} + 5426538716003159 T^{15} + 27644584309705625 T^{16} + 78731776114041620 T^{17} + 350356403707485209 T^{18} \)
$97$ \( 1 + 2 T + 504 T^{2} + 161 T^{3} + 129388 T^{4} - 33272 T^{5} + 22693242 T^{6} - 9390905 T^{7} + 2906526433 T^{8} - 1285370468 T^{9} + 281933064001 T^{10} - 88359025145 T^{11} + 20711509255866 T^{12} - 2945546237432 T^{13} + 1111098781172716 T^{14} + 134108492793569 T^{15} + 40722335376968952 T^{16} + 15674867188753922 T^{17} + 760231058654565217 T^{18} \)
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