Properties

Label 4008.2.a.l
Level 4008
Weight 2
Character orbit 4008.a
Self dual yes
Analytic conductor 32.004
Analytic rank 0
Dimension 12
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4008 = 2^{3} \cdot 3 \cdot 167 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4008.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.0040411301\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + \beta_{1} q^{5} + ( 1 - \beta_{9} ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + \beta_{1} q^{5} + ( 1 - \beta_{9} ) q^{7} + q^{9} + ( \beta_{1} - \beta_{10} ) q^{11} + ( 1 - \beta_{5} ) q^{13} + \beta_{1} q^{15} + ( \beta_{5} + \beta_{6} - \beta_{7} ) q^{17} + ( 1 - \beta_{11} ) q^{19} + ( 1 - \beta_{9} ) q^{21} + ( \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{23} + ( 2 + \beta_{4} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{25} + q^{27} + ( \beta_{1} + \beta_{7} - \beta_{10} ) q^{29} + ( 3 - \beta_{1} - \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{10} + \beta_{11} ) q^{31} + ( \beta_{1} - \beta_{10} ) q^{33} + ( 1 + 2 \beta_{1} + \beta_{3} - \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{35} + ( -\beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{7} - \beta_{11} ) q^{37} + ( 1 - \beta_{5} ) q^{39} + ( -1 + \beta_{2} - 2 \beta_{4} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{41} + ( 2 + \beta_{3} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{43} + \beta_{1} q^{45} + ( 2 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} + \beta_{10} ) q^{47} + ( 3 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{49} + ( \beta_{5} + \beta_{6} - \beta_{7} ) q^{51} + ( 3 + \beta_{3} + \beta_{4} - \beta_{6} - \beta_{10} ) q^{53} + ( 4 + \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} + \beta_{11} ) q^{55} + ( 1 - \beta_{11} ) q^{57} + ( -1 + \beta_{1} + \beta_{3} - 2 \beta_{4} - \beta_{8} + \beta_{9} ) q^{59} + ( \beta_{1} + \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} - \beta_{10} - \beta_{11} ) q^{61} + ( 1 - \beta_{9} ) q^{63} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{65} + ( -1 - \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} ) q^{67} + ( \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{69} + ( \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} + 2 \beta_{9} - \beta_{11} ) q^{71} + ( 2 - \beta_{1} + \beta_{2} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{73} + ( 2 + \beta_{4} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{75} + ( 3 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{77} + ( 5 + \beta_{2} + 2 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} ) q^{79} + q^{81} + ( 2 - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} ) q^{83} + ( 2 - 3 \beta_{1} + 2 \beta_{5} - 2 \beta_{7} + \beta_{10} ) q^{85} + ( \beta_{1} + \beta_{7} - \beta_{10} ) q^{87} + ( -2 - 4 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} + \beta_{9} ) q^{89} + ( 1 - \beta_{2} + 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{91} + ( 3 - \beta_{1} - \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{10} + \beta_{11} ) q^{93} + ( -1 + 2 \beta_{1} + \beta_{3} + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} - 2 \beta_{11} ) q^{95} + ( -1 + \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{11} ) q^{97} + ( \beta_{1} - \beta_{10} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 12q^{3} + 4q^{5} + 11q^{7} + 12q^{9} + O(q^{10}) \) \( 12q + 12q^{3} + 4q^{5} + 11q^{7} + 12q^{9} + q^{11} + 8q^{13} + 4q^{15} + 3q^{17} + 12q^{19} + 11q^{21} + 7q^{23} + 18q^{25} + 12q^{27} + 5q^{29} + 33q^{31} + q^{33} + 15q^{35} + 8q^{37} + 8q^{39} - 6q^{41} + 16q^{43} + 4q^{45} + 18q^{47} + 25q^{49} + 3q^{51} + 20q^{53} + 39q^{55} + 12q^{57} + 4q^{59} + 10q^{61} + 11q^{63} + 9q^{67} + 7q^{69} + 11q^{71} + 22q^{73} + 18q^{75} + 24q^{77} + 56q^{79} + 12q^{81} + 26q^{83} + 15q^{85} + 5q^{87} - 15q^{89} + 11q^{91} + 33q^{93} + 3q^{95} + 8q^{97} + q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 4 x^{11} - 31 x^{10} + 131 x^{9} + 309 x^{8} - 1453 x^{7} - 1072 x^{6} + 6350 x^{5} + 1411 x^{4} - 11022 x^{3} - 2450 x^{2} + 6960 x + 3008\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-2612041 \nu^{11} + 172266458 \nu^{10} - 431149861 \nu^{9} - 5604811029 \nu^{8} + 15598783254 \nu^{7} + 61532588360 \nu^{6} - 170251220041 \nu^{5} - 269226343929 \nu^{4} + 657292058507 \nu^{3} + 528815192804 \nu^{2} - 748850317708 \nu - 506734859797\)\()/ 6887737669 \)
\(\beta_{3}\)\(=\)\((\)\(48270257 \nu^{11} - 393971708 \nu^{10} - 803884765 \nu^{9} + 12731441603 \nu^{8} - 7245605191 \nu^{7} - 136791943933 \nu^{6} + 180010099322 \nu^{5} + 557383751450 \nu^{4} - 796092987441 \nu^{3} - 886580239346 \nu^{2} + 843664925302 \nu + 623931970190\)\()/ 13775475338 \)
\(\beta_{4}\)\(=\)\((\)\(433554181 \nu^{11} - 1027661208 \nu^{10} - 15151502491 \nu^{9} + 32370295155 \nu^{8} + 187572944809 \nu^{7} - 331978409981 \nu^{6} - 1009649126248 \nu^{5} + 1181123521726 \nu^{4} + 2516794688379 \nu^{3} - 933458223074 \nu^{2} - 2492427824318 \nu - 810005934996\)\()/ 27550950676 \)
\(\beta_{5}\)\(=\)\((\)\(-924523625 \nu^{11} + 3335040372 \nu^{10} + 28727151823 \nu^{9} - 105506939067 \nu^{8} - 286439821885 \nu^{7} + 1096386897181 \nu^{6} + 971661553200 \nu^{5} - 4121786159918 \nu^{4} - 965164697859 \nu^{3} + 4869923492678 \nu^{2} + 619274447522 \nu - 1194286516536\)\()/ 55101901352 \)
\(\beta_{6}\)\(=\)\((\)\(497566513 \nu^{11} - 583717328 \nu^{10} - 19421435387 \nu^{9} + 18043299379 \nu^{8} + 279234229797 \nu^{7} - 177472297861 \nu^{6} - 1817022049764 \nu^{5} + 523553323290 \nu^{4} + 5297394080111 \nu^{3} + 403005890338 \nu^{2} - 5364621239478 \nu - 2291097916892\)\()/ 27550950676 \)
\(\beta_{7}\)\(=\)\((\)\(-608147925 \nu^{11} + 2540267068 \nu^{10} + 17866911943 \nu^{9} - 80927081243 \nu^{8} - 156734313797 \nu^{7} + 851051501325 \nu^{6} + 322943033664 \nu^{5} - 3288682206998 \nu^{4} + 476137596409 \nu^{3} + 4250845840470 \nu^{2} - 751582083438 \nu - 1570311572784\)\()/ 27550950676 \)
\(\beta_{8}\)\(=\)\((\)\(1276718751 \nu^{11} - 4447291652 \nu^{10} - 40378997177 \nu^{9} + 141144188077 \nu^{8} + 420342586963 \nu^{7} - 1472464513643 \nu^{6} - 1640879715920 \nu^{5} + 5556472146210 \nu^{4} + 2723364140149 \nu^{3} - 6468711211946 \nu^{2} - 2707550657158 \nu + 1003995641592\)\()/ 55101901352 \)
\(\beta_{9}\)\(=\)\((\)\(-672160257 \nu^{11} + 2096323188 \nu^{10} + 22136844839 \nu^{9} - 66600085467 \nu^{8} - 248395598785 \nu^{7} + 696545389205 \nu^{6} + 1130315957180 \nu^{5} - 2631112008562 \nu^{4} - 2304461795323 \nu^{3} + 2886830776382 \nu^{2} + 2120611331722 \nu + 103637063844\)\()/ 27550950676 \)
\(\beta_{10}\)\(=\)\((\)\(825574067 \nu^{11} - 1232246264 \nu^{10} - 31068710789 \nu^{9} + 37574547953 \nu^{8} + 427577508547 \nu^{7} - 358636986823 \nu^{6} - 2660357207468 \nu^{5} + 975184226114 \nu^{4} + 7596696525145 \nu^{3} + 904825188458 \nu^{2} - 7836145638226 \nu - 3936944027888\)\()/ 27550950676 \)
\(\beta_{11}\)\(=\)\((\)\(-1494430261 \nu^{11} + 4167534940 \nu^{10} + 50046579515 \nu^{9} - 131337919583 \nu^{8} - 578035323361 \nu^{7} + 1352380306305 \nu^{6} + 2776874943512 \nu^{5} - 4915467010946 \nu^{4} - 6107654784727 \nu^{3} + 4699818876386 \nu^{2} + 5874998351882 \nu + 1214478641340\)\()/ 27550950676 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{9} + \beta_{7} - \beta_{6} + \beta_{4} + 7\)
\(\nu^{3}\)\(=\)\(-\beta_{10} + \beta_{7} - 2 \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} + 12 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(\beta_{11} - 13 \beta_{9} + 2 \beta_{8} + 15 \beta_{7} - 15 \beta_{6} + 19 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_{1} + 78\)
\(\nu^{5}\)\(=\)\(2 \beta_{11} - 15 \beta_{10} + \beta_{9} - 3 \beta_{8} + 11 \beta_{7} - \beta_{6} - 31 \beta_{5} + 23 \beta_{4} + 12 \beta_{3} + 30 \beta_{2} + 144 \beta_{1} + 31\)
\(\nu^{6}\)\(=\)\(26 \beta_{11} + 6 \beta_{10} - 157 \beta_{9} + 34 \beta_{8} + 188 \beta_{7} - 202 \beta_{6} + 4 \beta_{5} + 292 \beta_{4} - 38 \beta_{3} - 41 \beta_{2} - 46 \beta_{1} + 976\)
\(\nu^{7}\)\(=\)\(49 \beta_{11} - 199 \beta_{10} + 29 \beta_{9} - 57 \beta_{8} + 96 \beta_{7} - 32 \beta_{6} - 401 \beta_{5} + 406 \beta_{4} + 108 \beta_{3} + 394 \beta_{2} + 1764 \beta_{1} + 484\)
\(\nu^{8}\)\(=\)\(465 \beta_{11} + 136 \beta_{10} - 1897 \beta_{9} + 446 \beta_{8} + 2264 \beta_{7} - 2645 \beta_{6} + 150 \beta_{5} + 4221 \beta_{4} - 533 \beta_{3} - 623 \beta_{2} - 730 \beta_{1} + 12688\)
\(\nu^{9}\)\(=\)\(810 \beta_{11} - 2629 \beta_{10} + 581 \beta_{9} - 879 \beta_{8} + 663 \beta_{7} - 715 \beta_{6} - 4895 \beta_{5} + 6479 \beta_{4} + 759 \beta_{3} + 5122 \beta_{2} + 22012 \beta_{1} + 7620\)
\(\nu^{10}\)\(=\)\(7141 \beta_{11} + 2177 \beta_{10} - 22913 \beta_{9} + 5288 \beta_{8} + 26926 \beta_{7} - 34360 \beta_{6} + 3630 \beta_{5} + 59518 \beta_{4} - 6696 \beta_{3} - 8532 \beta_{2} - 9985 \beta_{1} + 167606\)
\(\nu^{11}\)\(=\)\(11158 \beta_{11} - 35307 \beta_{10} + 10456 \beta_{9} - 13224 \beta_{8} + 1478 \beta_{7} - 13396 \beta_{6} - 57692 \beta_{5} + 98497 \beta_{4} + 2215 \beta_{3} + 67138 \beta_{2} + 278574 \beta_{1} + 119224\)

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
−3.64003
−3.49351
−2.09997
−1.14709
−0.716151
−0.619387
1.53322
1.56937
2.31305
2.93317
3.59055
3.77676
0 1.00000 0 −3.64003 0 −1.45249 0 1.00000 0
1.2 0 1.00000 0 −3.49351 0 4.58171 0 1.00000 0
1.3 0 1.00000 0 −2.09997 0 2.97416 0 1.00000 0
1.4 0 1.00000 0 −1.14709 0 −3.32442 0 1.00000 0
1.5 0 1.00000 0 −0.716151 0 −3.33002 0 1.00000 0
1.6 0 1.00000 0 −0.619387 0 0.954126 0 1.00000 0
1.7 0 1.00000 0 1.53322 0 −0.915565 0 1.00000 0
1.8 0 1.00000 0 1.56937 0 3.21458 0 1.00000 0
1.9 0 1.00000 0 2.31305 0 4.58648 0 1.00000 0
1.10 0 1.00000 0 2.93317 0 1.35675 0 1.00000 0
1.11 0 1.00000 0 3.59055 0 4.10828 0 1.00000 0
1.12 0 1.00000 0 3.77676 0 −1.75359 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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 4008.2.a.l 12
4.b odd 2 1 8016.2.a.bf 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4008.2.a.l 12 1.a even 1 1 trivial
8016.2.a.bf 12 4.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(167\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4008))\):

\(T_{5}^{12} - \cdots\)
\(T_{7}^{12} - \cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( ( 1 - T )^{12} \)
$5$ \( 1 - 4 T + 29 T^{2} - 89 T^{3} + 409 T^{4} - 1058 T^{5} + 3913 T^{6} - 9105 T^{7} + 29926 T^{8} - 64597 T^{9} + 193020 T^{10} - 386495 T^{11} + 1053908 T^{12} - 1932475 T^{13} + 4825500 T^{14} - 8074625 T^{15} + 18703750 T^{16} - 28453125 T^{17} + 61140625 T^{18} - 82656250 T^{19} + 159765625 T^{20} - 173828125 T^{21} + 283203125 T^{22} - 195312500 T^{23} + 244140625 T^{24} \)
$7$ \( 1 - 11 T + 90 T^{2} - 550 T^{3} + 2935 T^{4} - 13552 T^{5} + 57133 T^{6} - 218439 T^{7} + 778294 T^{8} - 2566764 T^{9} + 7963941 T^{10} - 23067122 T^{11} + 63084108 T^{12} - 161469854 T^{13} + 390233109 T^{14} - 880400052 T^{15} + 1868683894 T^{16} - 3671304273 T^{17} + 6721640317 T^{18} - 11160654736 T^{19} + 16919690935 T^{20} - 22194483850 T^{21} + 25422772410 T^{22} - 21750594173 T^{23} + 13841287201 T^{24} \)
$11$ \( 1 - T + 57 T^{2} + 9 T^{3} + 1303 T^{4} + 2130 T^{5} + 15693 T^{6} + 54780 T^{7} + 142415 T^{8} + 536457 T^{9} + 1867410 T^{10} + 1451533 T^{11} + 25055730 T^{12} + 15966863 T^{13} + 225956610 T^{14} + 714024267 T^{15} + 2085098015 T^{16} + 8822373780 T^{17} + 27801106773 T^{18} + 41507674230 T^{19} + 279309621943 T^{20} + 21221529219 T^{21} + 1478433202257 T^{22} - 285311670611 T^{23} + 3138428376721 T^{24} \)
$13$ \( 1 - 8 T + 103 T^{2} - 624 T^{3} + 4818 T^{4} - 23799 T^{5} + 141509 T^{6} - 600471 T^{7} + 3036479 T^{8} - 11489293 T^{9} + 51865524 T^{10} - 178767981 T^{11} + 735638636 T^{12} - 2323983753 T^{13} + 8765273556 T^{14} - 25241976721 T^{15} + 86724876719 T^{16} - 222950679003 T^{17} + 683036914781 T^{18} - 1493351956083 T^{19} + 3930190613778 T^{20} - 6617207608752 T^{21} + 14199424660447 T^{22} - 14337283152296 T^{23} + 23298085122481 T^{24} \)
$17$ \( 1 - 3 T + 118 T^{2} - 387 T^{3} + 7251 T^{4} - 23602 T^{5} + 300910 T^{6} - 932520 T^{7} + 9256187 T^{8} - 26738797 T^{9} + 221263212 T^{10} - 584079475 T^{11} + 4205285970 T^{12} - 9929351075 T^{13} + 63945068268 T^{14} - 131367709661 T^{15} + 773085994427 T^{16} - 1324045049640 T^{17} + 7263235887790 T^{18} - 9684813360146 T^{19} + 50581217204691 T^{20} - 45893508204339 T^{21} + 237887280252982 T^{22} - 102815688922899 T^{23} + 582622237229761 T^{24} \)
$19$ \( 1 - 12 T + 199 T^{2} - 1674 T^{3} + 16320 T^{4} - 107611 T^{5} + 785745 T^{6} - 4319201 T^{7} + 26066607 T^{8} - 124863285 T^{9} + 660397808 T^{10} - 2851154001 T^{11} + 13691500992 T^{12} - 54171926019 T^{13} + 238403608688 T^{14} - 856437271815 T^{15} + 3397026290847 T^{16} - 10694769276899 T^{17} + 36966065766345 T^{18} - 96190431705529 T^{19} + 277171748829120 T^{20} - 540179206082046 T^{21} + 1220082185302399 T^{22} - 1397883106778628 T^{23} + 2213314919066161 T^{24} \)
$23$ \( 1 - 7 T + 167 T^{2} - 941 T^{3} + 13183 T^{4} - 60444 T^{5} + 655771 T^{6} - 2465474 T^{7} + 23482855 T^{8} - 73464215 T^{9} + 666819294 T^{10} - 1819767099 T^{11} + 16262816722 T^{12} - 41854643277 T^{13} + 352747406526 T^{14} - 893839103905 T^{15} + 6571465626055 T^{16} - 15868636321582 T^{17} + 97077642965419 T^{18} - 205801269318468 T^{19} + 1032373718959423 T^{20} - 1694884654436683 T^{21} + 6918227372679383 T^{22} - 6669668305397489 T^{23} + 21914624432020321 T^{24} \)
$29$ \( 1 - 5 T + 265 T^{2} - 939 T^{3} + 31695 T^{4} - 75812 T^{5} + 2339341 T^{6} - 3519780 T^{7} + 122806087 T^{8} - 110222539 T^{9} + 4959840978 T^{10} - 2882442317 T^{11} + 160066031378 T^{12} - 83590827193 T^{13} + 4171226262498 T^{14} - 2688217503671 T^{15} + 86858412019447 T^{16} - 72194732027220 T^{17} + 1391494582571461 T^{18} - 1307747622737908 T^{19} + 15855310058798895 T^{20} - 13622210071340991 T^{21} + 111487416824553265 T^{22} - 61002548828529145 T^{23} + 353814783205469041 T^{24} \)
$31$ \( 1 - 33 T + 715 T^{2} - 11453 T^{3} + 151689 T^{4} - 1717753 T^{5} + 17166335 T^{6} - 153572178 T^{7} + 1245751554 T^{8} - 9228392127 T^{9} + 62798159170 T^{10} - 393831808170 T^{11} + 2281105342288 T^{12} - 12208786053270 T^{13} + 60349030962370 T^{14} - 274923029855457 T^{15} + 1150477720901634 T^{16} - 4396641073360878 T^{17} + 15235185501779135 T^{18} - 47259875427012583 T^{19} + 129374188578387849 T^{20} - 302812992606164963 T^{21} + 586034225191272715 T^{22} - 838479737581359423 T^{23} + 787662783788549761 T^{24} \)
$37$ \( 1 - 8 T + 199 T^{2} - 657 T^{3} + 15497 T^{4} - 11784 T^{5} + 1024693 T^{6} - 275281 T^{7} + 62717478 T^{8} + 2913275 T^{9} + 2925421190 T^{10} + 1550447139 T^{11} + 114197307556 T^{12} + 57366544143 T^{13} + 4004901609110 T^{14} + 147566118575 T^{15} + 117542651285958 T^{16} - 19089073826917 T^{17} + 2629081891217437 T^{18} - 1118677240135272 T^{19} + 54432894097413737 T^{20} - 85384863045365589 T^{21} + 956908290111151951 T^{22} - 1423340974235683304 T^{23} + 6582952005840035281 T^{24} \)
$41$ \( 1 + 6 T + 246 T^{2} + 1573 T^{3} + 32094 T^{4} + 205504 T^{5} + 2951176 T^{6} + 17984407 T^{7} + 208488291 T^{8} + 1180521459 T^{9} + 11680066058 T^{10} + 60818249171 T^{11} + 529803935660 T^{12} + 2493548216011 T^{13} + 19634191043498 T^{14} + 81362719475739 T^{15} + 589138081664451 T^{16} + 2083605072257807 T^{17} + 14018393633537416 T^{18} + 40022782299641024 T^{19} + 256268190303409374 T^{20} + 514971782801700653 T^{21} + 3301974190297490646 T^{22} + 3301974190297490646 T^{23} + 22563490300366186081 T^{24} \)
$43$ \( 1 - 16 T + 352 T^{2} - 4313 T^{3} + 57980 T^{4} - 594402 T^{5} + 6230176 T^{6} - 55553277 T^{7} + 491590559 T^{8} - 3876909777 T^{9} + 29940548104 T^{10} - 210459992337 T^{11} + 1441935098296 T^{12} - 9049779670491 T^{13} + 55360073444296 T^{14} - 308241465639939 T^{15} + 1680650294699759 T^{16} - 8166800755317711 T^{17} + 39383204355166624 T^{18} - 161569526079223014 T^{19} + 677681852095305980 T^{20} - 2167681935283603859 T^{21} + 7607241774276055648 T^{22} - 14868699831539563312 T^{23} + 39959630797262576401 T^{24} \)
$47$ \( 1 - 18 T + 425 T^{2} - 5660 T^{3} + 83071 T^{4} - 912185 T^{5} + 10346123 T^{6} - 97439810 T^{7} + 923239432 T^{8} - 7625094882 T^{9} + 62597370528 T^{10} - 457914714495 T^{11} + 3314420997080 T^{12} - 21521991581265 T^{13} + 138277591496352 T^{14} - 791660225933886 T^{15} + 4505113914781192 T^{16} - 22347333906528670 T^{17} + 111523087637319467 T^{18} - 462134011139541655 T^{19} + 1978027394279148031 T^{20} - 6334278477761661220 T^{21} + 22354631200227770825 T^{22} - 44498865871512221454 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \)
$53$ \( 1 - 20 T + 549 T^{2} - 7632 T^{3} + 122313 T^{4} - 1328735 T^{5} + 16067905 T^{6} - 147185648 T^{7} + 1496194206 T^{8} - 12145564678 T^{9} + 109080630288 T^{10} - 798675767107 T^{11} + 6425851329756 T^{12} - 42329815656671 T^{13} + 306407490478992 T^{14} - 1808195232566606 T^{15} + 11805691954753086 T^{16} - 61552374627884464 T^{17} + 356134849006464745 T^{18} - 1560879806391316195 T^{19} + 7615169513284797993 T^{20} - 25183795732633879056 T^{21} + 96013221230666663901 T^{22} - \)\(18\!\cdots\!40\)\( T^{23} + \)\(49\!\cdots\!41\)\( T^{24} \)
$59$ \( 1 - 4 T + 377 T^{2} - 1836 T^{3} + 73773 T^{4} - 392035 T^{5} + 9904875 T^{6} - 53653804 T^{7} + 1009471726 T^{8} - 5306097190 T^{9} + 81672478638 T^{10} - 400645649097 T^{11} + 5344564122780 T^{12} - 23638093296723 T^{13} + 284301898138878 T^{14} - 1089760934785010 T^{15} + 12232133323235086 T^{16} - 38358408213383396 T^{17} + 417792913147399875 T^{18} - 975638484851016665 T^{19} + 10832121873383573133 T^{20} - 15905260323050468004 T^{21} + \)\(19\!\cdots\!77\)\( T^{22} - \)\(12\!\cdots\!36\)\( T^{23} + \)\(17\!\cdots\!81\)\( T^{24} \)
$61$ \( 1 - 10 T + 445 T^{2} - 4156 T^{3} + 94520 T^{4} - 789057 T^{5} + 12579213 T^{6} - 91911701 T^{7} + 1186381407 T^{8} - 7600064489 T^{9} + 87750053054 T^{10} - 509722415835 T^{11} + 5618056854064 T^{12} - 31093067365935 T^{13} + 326517947413934 T^{14} - 1725070237777709 T^{15} + 16426448326678287 T^{16} - 77628282683218001 T^{17} + 648085762926757893 T^{18} - 2479803233962222197 T^{19} + 18120175224503000120 T^{20} - 48600871161818689996 T^{21} + \)\(31\!\cdots\!45\)\( T^{22} - \)\(43\!\cdots\!10\)\( T^{23} + \)\(26\!\cdots\!21\)\( T^{24} \)
$67$ \( 1 - 9 T + 364 T^{2} - 2307 T^{3} + 64381 T^{4} - 335261 T^{5} + 8131027 T^{6} - 36716836 T^{7} + 807347350 T^{8} - 3186819725 T^{9} + 66922528995 T^{10} - 241334650304 T^{11} + 4818627301004 T^{12} - 16169421570368 T^{13} + 300415232658555 T^{14} - 958477460950175 T^{15} + 16268954138879350 T^{16} - 49572322133201452 T^{17} + 735519547792457563 T^{18} - 2031920233512194303 T^{19} + 26143043148774104221 T^{20} - 62765474852252442729 T^{21} + \)\(66\!\cdots\!36\)\( T^{22} - \)\(10\!\cdots\!47\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \)
$71$ \( 1 - 11 T + 396 T^{2} - 4588 T^{3} + 97063 T^{4} - 1016101 T^{5} + 16323044 T^{6} - 156812678 T^{7} + 2072326023 T^{8} - 18020382775 T^{9} + 206089443520 T^{10} - 1618074602797 T^{11} + 16319516903874 T^{12} - 114883296798587 T^{13} + 1038896884784320 T^{14} - 6449693219383025 T^{15} + 52661287824474663 T^{16} - 282926036256511978 T^{17} + 2090986570854975524 T^{18} - 9241560688061253491 T^{19} + 62678775003307299943 T^{20} - \)\(21\!\cdots\!28\)\( T^{21} + \)\(12\!\cdots\!96\)\( T^{22} - \)\(25\!\cdots\!81\)\( T^{23} + \)\(16\!\cdots\!41\)\( T^{24} \)
$73$ \( 1 - 22 T + 633 T^{2} - 9692 T^{3} + 162972 T^{4} - 1961473 T^{5} + 25060151 T^{6} - 256579429 T^{7} + 2800657223 T^{8} - 25744486769 T^{9} + 255423588848 T^{10} - 2167713986391 T^{11} + 20005755817240 T^{12} - 158243121006543 T^{13} + 1361152304970992 T^{14} - 10015043009416073 T^{15} + 79533738777144743 T^{16} - 531907525608060397 T^{17} + 3792458562270509639 T^{18} - 21669173915448749881 T^{19} + \)\(13\!\cdots\!32\)\( T^{20} - \)\(57\!\cdots\!96\)\( T^{21} + \)\(27\!\cdots\!17\)\( T^{22} - \)\(69\!\cdots\!94\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \)
$79$ \( 1 - 56 T + 1860 T^{2} - 44299 T^{3} + 844526 T^{4} - 13509480 T^{5} + 189081904 T^{6} - 2376082387 T^{7} + 27488453199 T^{8} - 296941458635 T^{9} + 3028562934604 T^{10} - 29215410638157 T^{11} + 267020165736708 T^{12} - 2308017440414403 T^{13} + 18901261274863564 T^{14} - 146403719823941765 T^{15} + 1070677478665759119 T^{16} - 7311339513469544413 T^{17} + 45963438928425991984 T^{18} - \)\(25\!\cdots\!20\)\( T^{19} + \)\(12\!\cdots\!86\)\( T^{20} - \)\(53\!\cdots\!81\)\( T^{21} + \)\(17\!\cdots\!60\)\( T^{22} - \)\(41\!\cdots\!24\)\( T^{23} + \)\(59\!\cdots\!41\)\( T^{24} \)
$83$ \( 1 - 26 T + 973 T^{2} - 17670 T^{3} + 376595 T^{4} - 5194783 T^{5} + 80296787 T^{6} - 880200858 T^{7} + 10936562682 T^{8} - 99486639658 T^{9} + 1085115323418 T^{10} - 8825068327103 T^{11} + 92536860445288 T^{12} - 732480671149549 T^{13} + 7475359463026602 T^{14} - 56885167230128846 T^{15} + 519030902398976922 T^{16} - 3467146953665471694 T^{17} + 26252261522111065403 T^{18} - \)\(14\!\cdots\!41\)\( T^{19} + \)\(84\!\cdots\!95\)\( T^{20} - \)\(33\!\cdots\!10\)\( T^{21} + \)\(15\!\cdots\!77\)\( T^{22} - \)\(33\!\cdots\!42\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \)
$89$ \( 1 + 15 T + 264 T^{2} + 2377 T^{3} + 35623 T^{4} + 309093 T^{5} + 4329853 T^{6} + 33743512 T^{7} + 444605126 T^{8} + 3224059839 T^{9} + 42312991591 T^{10} + 313857383284 T^{11} + 4118090623756 T^{12} + 27933307112276 T^{13} + 335161206392311 T^{14} + 2272862240639991 T^{15} + 27895521965327366 T^{16} + 188425777026044888 T^{17} + 2151855933611358733 T^{18} + 13671595996863745197 T^{19} + \)\(14\!\cdots\!63\)\( T^{20} + \)\(83\!\cdots\!93\)\( T^{21} + \)\(82\!\cdots\!64\)\( T^{22} + \)\(41\!\cdots\!35\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} \)
$97$ \( 1 - 8 T + 622 T^{2} - 5261 T^{3} + 199675 T^{4} - 1747832 T^{5} + 43601123 T^{6} - 383408405 T^{7} + 7204409176 T^{8} - 61549406631 T^{9} + 947657007183 T^{10} - 7599297626119 T^{11} + 101521370408824 T^{12} - 737131869733543 T^{13} + 8916504780584847 T^{14} - 56174481598134663 T^{15} + 637801164381082456 T^{16} - 3292458431128660085 T^{17} + 36318514842465935267 T^{18} - \)\(14\!\cdots\!16\)\( T^{19} + \)\(15\!\cdots\!75\)\( T^{20} - \)\(39\!\cdots\!37\)\( T^{21} + \)\(45\!\cdots\!78\)\( T^{22} - \)\(57\!\cdots\!24\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \)
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