Properties

Label 8046.2.a.o
Level 8046
Weight 2
Character orbit 8046.a
Self dual yes
Analytic conductor 64.248
Analytic rank 0
Dimension 12
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8046.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.2476334663\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + \beta_{1} q^{5} + ( 1 + \beta_{6} ) q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} + \beta_{1} q^{5} + ( 1 + \beta_{6} ) q^{7} + q^{8} + \beta_{1} q^{10} + ( 1 - \beta_{5} ) q^{11} + ( 1 - \beta_{5} + \beta_{9} ) q^{13} + ( 1 + \beta_{6} ) q^{14} + q^{16} + ( 1 + \beta_{7} ) q^{17} -\beta_{11} q^{19} + \beta_{1} q^{20} + ( 1 - \beta_{5} ) q^{22} + ( -\beta_{2} - \beta_{7} + \beta_{8} + \beta_{10} ) q^{23} + ( -1 + \beta_{1} - \beta_{2} + \beta_{4} - \beta_{7} + \beta_{8} - \beta_{11} ) q^{25} + ( 1 - \beta_{5} + \beta_{9} ) q^{26} + ( 1 + \beta_{6} ) q^{28} + ( 2 - \beta_{8} ) q^{29} + ( 1 + \beta_{1} - \beta_{4} - \beta_{6} - \beta_{8} + \beta_{11} ) q^{31} + q^{32} + ( 1 + \beta_{7} ) q^{34} + ( 1 + \beta_{1} + \beta_{4} + \beta_{6} - \beta_{7} - \beta_{10} - \beta_{11} ) q^{35} + ( 1 + \beta_{7} - \beta_{9} + \beta_{11} ) q^{37} -\beta_{11} q^{38} + \beta_{1} q^{40} + ( 2 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{9} - \beta_{10} ) q^{41} + ( \beta_{1} + \beta_{2} + \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{43} + ( 1 - \beta_{5} ) q^{44} + ( -\beta_{2} - \beta_{7} + \beta_{8} + \beta_{10} ) q^{46} + ( 3 - 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{6} + \beta_{7} + \beta_{9} - \beta_{10} ) q^{47} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{10} + \beta_{11} ) q^{49} + ( -1 + \beta_{1} - \beta_{2} + \beta_{4} - \beta_{7} + \beta_{8} - \beta_{11} ) q^{50} + ( 1 - \beta_{5} + \beta_{9} ) q^{52} + ( 2 - \beta_{1} - \beta_{3} + \beta_{6} - \beta_{7} + \beta_{10} - \beta_{11} ) q^{53} + ( \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{11} ) q^{55} + ( 1 + \beta_{6} ) q^{56} + ( 2 - \beta_{8} ) q^{58} + ( 2 - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} ) q^{59} + ( \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} ) q^{61} + ( 1 + \beta_{1} - \beta_{4} - \beta_{6} - \beta_{8} + \beta_{11} ) q^{62} + q^{64} + ( 1 - \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} ) q^{65} + ( 1 - \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} ) q^{67} + ( 1 + \beta_{7} ) q^{68} + ( 1 + \beta_{1} + \beta_{4} + \beta_{6} - \beta_{7} - \beta_{10} - \beta_{11} ) q^{70} + ( 4 + \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} - 2 \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{71} + ( -\beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} ) q^{73} + ( 1 + \beta_{7} - \beta_{9} + \beta_{11} ) q^{74} -\beta_{11} q^{76} + ( 1 + 2 \beta_{3} - \beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} + 2 \beta_{11} ) q^{77} + ( 1 + \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{9} - \beta_{10} ) q^{79} + \beta_{1} q^{80} + ( 2 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{9} - \beta_{10} ) q^{82} + ( 2 - 2 \beta_{3} + \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} ) q^{83} + ( 2 \beta_{1} - \beta_{3} - \beta_{6} - \beta_{8} - \beta_{9} + \beta_{10} ) q^{85} + ( \beta_{1} + \beta_{2} + \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{86} + ( 1 - \beta_{5} ) q^{88} + ( 3 - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{7} - 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{89} + ( 1 - \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{9} - \beta_{10} - \beta_{11} ) q^{91} + ( -\beta_{2} - \beta_{7} + \beta_{8} + \beta_{10} ) q^{92} + ( 3 - 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{6} + \beta_{7} + \beta_{9} - \beta_{10} ) q^{94} + ( 2 \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} ) q^{95} + ( \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{10} - \beta_{11} ) q^{97} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{10} + \beta_{11} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 12q^{2} + 12q^{4} + 3q^{5} + 6q^{7} + 12q^{8} + O(q^{10}) \) \( 12q + 12q^{2} + 12q^{4} + 3q^{5} + 6q^{7} + 12q^{8} + 3q^{10} + 10q^{11} + 5q^{13} + 6q^{14} + 12q^{16} + 8q^{17} + 2q^{19} + 3q^{20} + 10q^{22} + 9q^{23} + 7q^{25} + 5q^{26} + 6q^{28} + 19q^{29} + 10q^{31} + 12q^{32} + 8q^{34} + 20q^{35} + 11q^{37} + 2q^{38} + 3q^{40} + 8q^{41} + 13q^{43} + 10q^{44} + 9q^{46} + 11q^{47} + 2q^{49} + 7q^{50} + 5q^{52} + 24q^{53} + 3q^{55} + 6q^{56} + 19q^{58} + 10q^{59} + 10q^{62} + 12q^{64} + 28q^{65} + 21q^{67} + 8q^{68} + 20q^{70} + 37q^{71} - 2q^{73} + 11q^{74} + 2q^{76} + 2q^{77} + 7q^{79} + 3q^{80} + 8q^{82} + 22q^{83} + 15q^{85} + 13q^{86} + 10q^{88} + 40q^{89} + q^{91} + 9q^{92} + 11q^{94} + 11q^{95} + 7q^{97} + 2q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 3 x^{11} - 29 x^{10} + 76 x^{9} + 320 x^{8} - 724 x^{7} - 1643 x^{6} + 3265 x^{5} + 3921 x^{4} - 6927 x^{3} - 3639 x^{2} + 5508 x + 423\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-1293103 \nu^{11} + 6001945 \nu^{10} + 23580251 \nu^{9} - 131132508 \nu^{8} - 90139960 \nu^{7} + 1024446702 \nu^{6} - 573136895 \nu^{5} - 3491861196 \nu^{4} + 4215813919 \nu^{3} + 4581336765 \nu^{2} - 6050385741 \nu - 735740835\)\()/ 174942609 \)
\(\beta_{3}\)\(=\)\((\)\(-1915090 \nu^{11} - 1046362 \nu^{10} + 74238453 \nu^{9} + 35401589 \nu^{8} - 1005031345 \nu^{7} - 425190469 \nu^{6} + 5944816403 \nu^{5} + 1979959133 \nu^{4} - 15609041924 \nu^{3} - 3485888106 \nu^{2} + 14934980991 \nu + 1318582776\)\()/ 174942609 \)
\(\beta_{4}\)\(=\)\((\)\(-4181218 \nu^{11} + 5656554 \nu^{10} + 132467602 \nu^{9} - 109619327 \nu^{8} - 1543743823 \nu^{7} + 649333723 \nu^{6} + 8064151794 \nu^{5} - 1288637477 \nu^{4} - 19088618612 \nu^{3} - 59111379 \nu^{2} + 16748632692 \nu + 1376131821\)\()/ 174942609 \)
\(\beta_{5}\)\(=\)\((\)\(-4389972 \nu^{11} + 4775951 \nu^{10} + 138192352 \nu^{9} - 72357264 \nu^{8} - 1596499871 \nu^{7} + 185639731 \nu^{6} + 8187073056 \nu^{5} + 784010463 \nu^{4} - 18702206507 \nu^{3} - 2790478839 \nu^{2} + 15392930346 \nu + 1169371344\)\()/ 174942609 \)
\(\beta_{6}\)\(=\)\((\)\(6533895 \nu^{11} - 11760004 \nu^{10} - 202559541 \nu^{9} + 251802455 \nu^{8} + 2345517923 \nu^{7} - 1816569674 \nu^{6} - 12292593347 \nu^{5} + 5268511082 \nu^{4} + 28827085403 \nu^{3} - 4670185662 \nu^{2} - 24116897136 \nu - 1422840759\)\()/ 174942609 \)
\(\beta_{7}\)\(=\)\((\)\(-6869077 \nu^{11} + 5839142 \nu^{10} + 230548446 \nu^{9} - 91270315 \nu^{8} - 2856540142 \nu^{7} + 274308218 \nu^{6} + 15985954088 \nu^{5} + 676618097 \nu^{4} - 40668047126 \nu^{3} - 3142259250 \nu^{2} + 37843096842 \nu + 2092764192\)\()/ 174942609 \)
\(\beta_{8}\)\(=\)\((\)\(9158236 \nu^{11} - 12464738 \nu^{10} - 290204756 \nu^{9} + 242859302 \nu^{8} + 3388516467 \nu^{7} - 1488120542 \nu^{6} - 17686820067 \nu^{5} + 3338509649 \nu^{4} + 41281617595 \nu^{3} - 819687051 \nu^{2} - 34744019517 \nu - 3556778949\)\()/ 174942609 \)
\(\beta_{9}\)\(=\)\((\)\(-10768077 \nu^{11} + 13709156 \nu^{10} + 343745338 \nu^{9} - 248488722 \nu^{8} - 4077208325 \nu^{7} + 1283221540 \nu^{6} + 21816188427 \nu^{5} - 1622126526 \nu^{4} - 52604128721 \nu^{3} - 2624930826 \nu^{2} + 45986566413 \nu + 4392624474\)\()/ 174942609 \)
\(\beta_{10}\)\(=\)\((\)\(-11759125 \nu^{11} + 19783265 \nu^{10} + 355999031 \nu^{9} - 376860878 \nu^{8} - 4047108810 \nu^{7} + 2253401432 \nu^{6} + 20885745918 \nu^{5} - 4769542871 \nu^{4} - 48828823276 \nu^{3} + 1245833994 \nu^{2} + 41813328450 \nu + 3637207113\)\()/ 174942609 \)
\(\beta_{11}\)\(=\)\((\)\(13139198 \nu^{11} - 18649271 \nu^{10} - 411865851 \nu^{9} + 355642798 \nu^{8} + 4791452746 \nu^{7} - 2137541739 \nu^{6} - 25035485466 \nu^{5} + 4865115271 \nu^{4} + 58645232190 \nu^{3} - 2492818554 \nu^{2} - 49613155317 \nu - 2837900049\)\()/ 174942609 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{11} + \beta_{8} - \beta_{7} + \beta_{4} - \beta_{2} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(-3 \beta_{11} - \beta_{10} + 2 \beta_{9} + 2 \beta_{8} - 3 \beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - 3 \beta_{2} + 10 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-18 \beta_{11} - 2 \beta_{10} + 6 \beta_{9} + 16 \beta_{8} - 19 \beta_{7} + 4 \beta_{6} - 5 \beta_{5} + 18 \beta_{4} - 3 \beta_{3} - 17 \beta_{2} + 20 \beta_{1} + 27\)
\(\nu^{5}\)\(=\)\(-62 \beta_{11} - 20 \beta_{10} + 42 \beta_{9} + 47 \beta_{8} - 61 \beta_{7} + 23 \beta_{6} - 26 \beta_{5} + 53 \beta_{4} - 35 \beta_{3} - 56 \beta_{2} + 122 \beta_{1} + 13\)
\(\nu^{6}\)\(=\)\(-296 \beta_{11} - 61 \beta_{10} + 153 \beta_{9} + 256 \beta_{8} - 304 \beta_{7} + 94 \beta_{6} - 119 \beta_{5} + 295 \beta_{4} - 81 \beta_{3} - 254 \beta_{2} + 336 \beta_{1} + 248\)
\(\nu^{7}\)\(=\)\(-1096 \beta_{11} - 361 \beta_{10} + 776 \beta_{9} + 897 \beta_{8} - 1076 \beta_{7} + 428 \beta_{6} - 516 \beta_{5} + 1042 \beta_{4} - 538 \beta_{3} - 905 \beta_{2} + 1656 \beta_{1} + 333\)
\(\nu^{8}\)\(=\)\(-4821 \beta_{11} - 1304 \beta_{10} + 3055 \beta_{9} + 4200 \beta_{8} - 4846 \beta_{7} + 1764 \beta_{6} - 2258 \beta_{5} + 4857 \beta_{4} - 1606 \beta_{3} - 3827 \beta_{2} + 5403 \beta_{1} + 2803\)
\(\nu^{9}\)\(=\)\(-18627 \beta_{11} - 6354 \beta_{10} + 13806 \beta_{9} + 15966 \beta_{8} - 18280 \beta_{7} + 7495 \beta_{6} - 9404 \beta_{5} + 18623 \beta_{4} - 8326 \beta_{3} - 14361 \beta_{2} + 24022 \beta_{1} + 6252\)
\(\nu^{10}\)\(=\)\(-78834 \beta_{11} - 24603 \beta_{10} + 55891 \beta_{9} + 69732 \beta_{8} - 78265 \beta_{7} + 30911 \beta_{6} - 40046 \beta_{5} + 80628 \beta_{4} - 28744 \beta_{3} - 59098 \beta_{2} + 86165 \beta_{1} + 36254\)
\(\nu^{11}\)\(=\)\(-312025 \beta_{11} - 109982 \beta_{10} + 240193 \beta_{9} + 275200 \beta_{8} - 305996 \beta_{7} + 128162 \beta_{6} - 164919 \beta_{5} + 320329 \beta_{4} - 131955 \beta_{3} - 228964 \beta_{2} + 364213 \beta_{1} + 106224\)

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
−2.89416
−2.72416
−2.41916
−1.76721
−1.46524
−0.0737297
1.41673
1.51858
1.84210
2.15566
3.34249
4.06809
1.00000 0 1.00000 −2.89416 0 0.647278 1.00000 0 −2.89416
1.2 1.00000 0 1.00000 −2.72416 0 −2.55489 1.00000 0 −2.72416
1.3 1.00000 0 1.00000 −2.41916 0 0.521062 1.00000 0 −2.41916
1.4 1.00000 0 1.00000 −1.76721 0 2.53166 1.00000 0 −1.76721
1.5 1.00000 0 1.00000 −1.46524 0 −3.53999 1.00000 0 −1.46524
1.6 1.00000 0 1.00000 −0.0737297 0 2.82077 1.00000 0 −0.0737297
1.7 1.00000 0 1.00000 1.41673 0 −1.27536 1.00000 0 1.41673
1.8 1.00000 0 1.00000 1.51858 0 4.35237 1.00000 0 1.51858
1.9 1.00000 0 1.00000 1.84210 0 2.73011 1.00000 0 1.84210
1.10 1.00000 0 1.00000 2.15566 0 −3.96511 1.00000 0 2.15566
1.11 1.00000 0 1.00000 3.34249 0 1.09959 1.00000 0 3.34249
1.12 1.00000 0 1.00000 4.06809 0 2.63250 1.00000 0 4.06809
\(n\): 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 8046.2.a.o yes 12
3.b odd 2 1 8046.2.a.j 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8046.2.a.j 12 3.b odd 2 1
8046.2.a.o yes 12 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(149\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8046))\):

\(T_{5}^{12} - \cdots\)
\(T_{11}^{12} - \cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{12} \)
$3$ \( \)
$5$ \( 1 - 3 T + 31 T^{2} - 89 T^{3} + 520 T^{4} - 1429 T^{5} + 6032 T^{6} - 15550 T^{7} + 53006 T^{8} - 126152 T^{9} + 367406 T^{10} - 795897 T^{11} + 2044683 T^{12} - 3979485 T^{13} + 9185150 T^{14} - 15769000 T^{15} + 33128750 T^{16} - 48593750 T^{17} + 94250000 T^{18} - 111640625 T^{19} + 203125000 T^{20} - 173828125 T^{21} + 302734375 T^{22} - 146484375 T^{23} + 244140625 T^{24} \)
$7$ \( 1 - 6 T + 59 T^{2} - 252 T^{3} + 1506 T^{4} - 5272 T^{5} + 24633 T^{6} - 75456 T^{7} + 299403 T^{8} - 822563 T^{9} + 2869172 T^{10} - 7125598 T^{11} + 22245057 T^{12} - 49879186 T^{13} + 140589428 T^{14} - 282139109 T^{15} + 718866603 T^{16} - 1268188992 T^{17} + 2898047817 T^{18} - 4341718696 T^{19} + 8681790306 T^{20} - 10169108964 T^{21} + 16666039691 T^{22} - 11863960458 T^{23} + 13841287201 T^{24} \)
$11$ \( 1 - 10 T + 119 T^{2} - 852 T^{3} + 6179 T^{4} - 34957 T^{5} + 194642 T^{6} - 921276 T^{7} + 4266253 T^{8} - 17433663 T^{9} + 69581584 T^{10} - 249077190 T^{11} + 870352803 T^{12} - 2739849090 T^{13} + 8419371664 T^{14} - 23204205453 T^{15} + 62462210173 T^{16} - 148372421076 T^{17} + 344820176162 T^{18} - 681213036647 T^{19} + 1324523525699 T^{20} - 2008971432732 T^{21} + 3086553527519 T^{22} - 2853116706110 T^{23} + 3138428376721 T^{24} \)
$13$ \( 1 - 5 T + 83 T^{2} - 389 T^{3} + 3299 T^{4} - 14252 T^{5} + 84327 T^{6} - 330070 T^{7} + 1590748 T^{8} - 5592589 T^{9} + 24482767 T^{10} - 78779092 T^{11} + 332956381 T^{12} - 1024128196 T^{13} + 4137587623 T^{14} - 12286918033 T^{15} + 45433353628 T^{16} - 122552680510 T^{17} + 407030322543 T^{18} - 894291864284 T^{19} + 2691095648579 T^{20} - 4125150256097 T^{21} + 11442254823467 T^{22} - 8960801970185 T^{23} + 23298085122481 T^{24} \)
$17$ \( 1 - 8 T + 163 T^{2} - 1083 T^{3} + 12073 T^{4} - 68054 T^{5} + 547343 T^{6} - 2664751 T^{7} + 17308162 T^{8} - 74034864 T^{9} + 413119042 T^{10} - 1576525489 T^{11} + 7818022337 T^{12} - 26800933313 T^{13} + 119391403138 T^{14} - 363733286832 T^{15} + 1445594998402 T^{16} - 3783565360607 T^{17} + 13211529429167 T^{18} - 27925188052342 T^{19} + 84218319585193 T^{20} - 128430670246251 T^{21} + 328607005773187 T^{22} - 274175170461064 T^{23} + 582622237229761 T^{24} \)
$19$ \( 1 - 2 T + 143 T^{2} - 226 T^{3} + 10027 T^{4} - 12501 T^{5} + 459424 T^{6} - 451004 T^{7} + 15440053 T^{8} - 12114979 T^{9} + 404393020 T^{10} - 265891932 T^{11} + 8522163709 T^{12} - 5051946708 T^{13} + 145985880220 T^{14} - 83096640961 T^{15} + 2012163147013 T^{16} - 1116730553396 T^{17} + 21614006832544 T^{18} - 11174290609239 T^{19} + 170294186612107 T^{20} - 72927419698054 T^{21} + 876742474865543 T^{22} - 232980517796438 T^{23} + 2213314919066161 T^{24} \)
$23$ \( 1 - 9 T + 164 T^{2} - 1157 T^{3} + 13061 T^{4} - 78224 T^{5} + 689511 T^{6} - 3640497 T^{7} + 27275253 T^{8} - 129726161 T^{9} + 856762847 T^{10} - 3689448901 T^{11} + 21824414143 T^{12} - 84857324723 T^{13} + 453227546063 T^{14} - 1578378200887 T^{15} + 7632734074773 T^{16} - 23431487382471 T^{17} + 102072373860279 T^{18} - 266339065766128 T^{19} + 1022819778755141 T^{20} - 2083933629312691 T^{21} + 6793947839038436 T^{22} - 8575287821225343 T^{23} + 21914624432020321 T^{24} \)
$29$ \( 1 - 19 T + 421 T^{2} - 5467 T^{3} + 72294 T^{4} - 725015 T^{5} + 7169556 T^{6} - 58779582 T^{7} + 470430092 T^{8} - 3247459554 T^{9} + 21808912434 T^{10} - 128624114567 T^{11} + 737186448151 T^{12} - 3730099322443 T^{13} + 18341295356994 T^{14} - 79202291062506 T^{15} + 332726265899852 T^{16} - 1205636764559718 T^{17} + 4264619110015476 T^{18} - 12506419072169635 T^{19} + 36164814178602534 T^{20} - 79310567050075823 T^{21} + 177117745219384621 T^{22} - 231809685548410751 T^{23} + 353814783205469041 T^{24} \)
$31$ \( 1 - 10 T + 238 T^{2} - 2081 T^{3} + 28109 T^{4} - 210477 T^{5} + 2139213 T^{6} - 13941262 T^{7} + 117503039 T^{8} - 680726339 T^{9} + 4999690949 T^{10} - 26064900855 T^{11} + 171532558799 T^{12} - 808011926505 T^{13} + 4804703001989 T^{14} - 20279518365149 T^{15} + 108516524080319 T^{16} - 399126494928562 T^{17} + 1898559411943053 T^{18} - 5790772480240947 T^{19} + 23973914171429069 T^{20} - 55020853716356351 T^{21} + 195071532301430638 T^{22} - 254084768964048310 T^{23} + 787662783788549761 T^{24} \)
$37$ \( 1 - 11 T + 359 T^{2} - 3519 T^{3} + 62339 T^{4} - 540523 T^{5} + 6839425 T^{6} - 52371342 T^{7} + 525133134 T^{8} - 3545120461 T^{9} + 29642085735 T^{10} - 175713509903 T^{11} + 1259572750809 T^{12} - 6501399866411 T^{13} + 40580015371215 T^{14} - 179570986711033 T^{15} + 984184039550574 T^{16} - 3631636087680294 T^{17} + 17548093344874825 T^{18} - 51312863023560559 T^{19} + 218964456677981219 T^{20} - 457335362338875963 T^{21} + 1726281789698007791 T^{22} - 1957093839574064543 T^{23} + 6582952005840035281 T^{24} \)
$41$ \( 1 - 8 T + 280 T^{2} - 1912 T^{3} + 37858 T^{4} - 239703 T^{5} + 3422732 T^{6} - 20732444 T^{7} + 233518280 T^{8} - 1353253845 T^{9} + 12753687572 T^{10} - 69492949936 T^{11} + 573947757762 T^{12} - 2849210947376 T^{13} + 21438948808532 T^{14} - 93267608251245 T^{15} + 659866848411080 T^{16} - 2401982199285244 T^{17} + 16258333789006412 T^{18} - 46683183712097343 T^{19} + 302293299324062818 T^{20} - 625954258561253432 T^{21} + 3758344606842672280 T^{22} - 4402632253729987528 T^{23} + 22563490300366186081 T^{24} \)
$43$ \( 1 - 13 T + 314 T^{2} - 3115 T^{3} + 43775 T^{4} - 356232 T^{5} + 3818346 T^{6} - 27341201 T^{7} + 251135363 T^{8} - 1668841681 T^{9} + 13812272101 T^{10} - 85899802546 T^{11} + 646688616071 T^{12} - 3693691509478 T^{13} + 25538891114749 T^{14} - 132684595531267 T^{15} + 858581830159763 T^{16} - 4019387388760043 T^{17} + 24137151312696954 T^{18} - 96830487471868824 T^{19} + 511650967151983775 T^{20} - 1565575986183265945 T^{21} + 6786005446371254186 T^{22} - 12080818613125895191 T^{23} + 39959630797262576401 T^{24} \)
$47$ \( 1 - 11 T + 380 T^{2} - 3348 T^{3} + 67142 T^{4} - 501867 T^{5} + 7617229 T^{6} - 49927967 T^{7} + 632089918 T^{8} - 3697798992 T^{9} + 40883587123 T^{10} - 215405763898 T^{11} + 2129597328809 T^{12} - 10124070903206 T^{13} + 90311843954707 T^{14} - 383916584746416 T^{15} + 3084397163156158 T^{16} - 11450729941110769 T^{17} + 82107751601303341 T^{18} - 254257425597404421 T^{19} + 1598737409043957062 T^{20} - 3746848823948063916 T^{21} + 19987670249615418620 T^{22} - 27193751365924135333 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \)
$53$ \( 1 - 24 T + 603 T^{2} - 9679 T^{3} + 149521 T^{4} - 1854987 T^{5} + 21952133 T^{6} - 225761694 T^{7} + 2221765972 T^{8} - 19721645009 T^{9} + 168377353725 T^{10} - 1321000801602 T^{11} + 10002935068151 T^{12} - 70013042484906 T^{13} + 472971986613525 T^{14} - 2936099344004893 T^{15} + 17530802188512532 T^{16} - 94412522922845142 T^{17} + 486555003363838157 T^{18} - 2179073893152817119 T^{19} + 9309131169997108081 T^{20} - 31938411805052845307 T^{21} + \)\(10\!\cdots\!47\)\( T^{22} - \)\(22\!\cdots\!28\)\( T^{23} + \)\(49\!\cdots\!41\)\( T^{24} \)
$59$ \( 1 - 10 T + 471 T^{2} - 4552 T^{3} + 110446 T^{4} - 1016167 T^{5} + 16993766 T^{6} - 146376854 T^{7} + 1901091247 T^{8} - 15078221015 T^{9} + 162209617518 T^{10} - 1164196223873 T^{11} + 10798200421241 T^{12} - 68687577208507 T^{13} + 564651678580158 T^{14} - 3096749953839685 T^{15} + 23036208933839167 T^{16} - 104648369735775346 T^{17} + 716806118450282006 T^{18} - 2528885513374068773 T^{19} + 16216834511646837166 T^{20} - 39433956966517282328 T^{21} + \)\(24\!\cdots\!71\)\( T^{22} - \)\(30\!\cdots\!90\)\( T^{23} + \)\(17\!\cdots\!81\)\( T^{24} \)
$61$ \( 1 + 518 T^{2} - 300 T^{3} + 126924 T^{4} - 139383 T^{5} + 19680217 T^{6} - 29906974 T^{7} + 2183730146 T^{8} - 3899695138 T^{9} + 185807889183 T^{10} - 341309649871 T^{11} + 12608534769183 T^{12} - 20819888642131 T^{13} + 691391155649943 T^{14} - 885156702118378 T^{15} + 30235580388422786 T^{16} - 25259319614503174 T^{17} + 1013932147345716337 T^{18} - 438044924713115043 T^{19} + 24332258994866893644 T^{20} - 3508243827850242300 T^{21} + \)\(36\!\cdots\!18\)\( T^{22} + \)\(26\!\cdots\!21\)\( T^{24} \)
$67$ \( 1 - 21 T + 799 T^{2} - 12751 T^{3} + 276774 T^{4} - 3570999 T^{5} + 56863559 T^{6} - 615634366 T^{7} + 7907297962 T^{8} - 73592160483 T^{9} + 799798334931 T^{10} - 6491628578230 T^{11} + 61208138531903 T^{12} - 434939114741410 T^{13} + 3590294725505259 T^{14} - 22133798963348529 T^{15} + 159340918015315402 T^{16} - 831183414268627162 T^{17} + 5143785551511479471 T^{18} - 21642795081896827677 T^{19} + \)\(11\!\cdots\!34\)\( T^{20} - \)\(34\!\cdots\!97\)\( T^{21} + \)\(14\!\cdots\!51\)\( T^{22} - \)\(25\!\cdots\!43\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \)
$71$ \( 1 - 37 T + 1111 T^{2} - 22875 T^{3} + 415447 T^{4} - 6245078 T^{5} + 86287478 T^{6} - 1055184800 T^{7} + 12132103572 T^{8} - 127257281273 T^{9} + 1267733176411 T^{10} - 11660132106472 T^{11} + 102111645443079 T^{12} - 827869379559512 T^{13} + 6390642942287851 T^{14} - 45546780797700703 T^{15} + 308297145830624532 T^{16} - 1903795386889064800 T^{17} + 11053450430627041238 T^{18} - 56799734808524149498 T^{19} + \)\(26\!\cdots\!67\)\( T^{20} - \)\(10\!\cdots\!25\)\( T^{21} + \)\(36\!\cdots\!11\)\( T^{22} - \)\(85\!\cdots\!27\)\( T^{23} + \)\(16\!\cdots\!41\)\( T^{24} \)
$73$ \( 1 + 2 T + 502 T^{2} + 1073 T^{3} + 125653 T^{4} + 313232 T^{5} + 20818320 T^{6} + 60283783 T^{7} + 2565761663 T^{8} + 8188988785 T^{9} + 250651940189 T^{10} + 809922763303 T^{11} + 20087838486865 T^{12} + 59124361721119 T^{13} + 1335724189267181 T^{14} + 3185655850174345 T^{15} + 72863118054434783 T^{16} + 124972598055876319 T^{17} + 3150524349836814480 T^{18} + 3460398732933791504 T^{19} + \)\(10\!\cdots\!93\)\( T^{20} + 63169212537971470649 T^{21} + \)\(21\!\cdots\!98\)\( T^{22} + \)\(62\!\cdots\!54\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \)
$79$ \( 1 - 7 T + 700 T^{2} - 3679 T^{3} + 228154 T^{4} - 879730 T^{5} + 46628737 T^{6} - 127937112 T^{7} + 6779437914 T^{8} - 13119162148 T^{9} + 752285146813 T^{10} - 1100360658298 T^{11} + 66267884019181 T^{12} - 86928492005542 T^{13} + 4695011601259933 T^{14} - 6468258586287772 T^{15} + 264059655884771034 T^{16} - 393669709149179688 T^{17} + 11334861031487906977 T^{18} - 16894254852393657070 T^{19} + \)\(34\!\cdots\!94\)\( T^{20} - \)\(44\!\cdots\!01\)\( T^{21} + \)\(66\!\cdots\!00\)\( T^{22} - \)\(52\!\cdots\!53\)\( T^{23} + \)\(59\!\cdots\!41\)\( T^{24} \)
$83$ \( 1 - 22 T + 719 T^{2} - 10556 T^{3} + 199296 T^{4} - 2156997 T^{5} + 30929344 T^{6} - 264149042 T^{7} + 3374891639 T^{8} - 24768853573 T^{9} + 314751114974 T^{10} - 2148093090825 T^{11} + 27214698406967 T^{12} - 178291726538475 T^{13} + 2168320431055886 T^{14} - 14162508477944951 T^{15} + 160166690743878119 T^{16} - 1040493812247514006 T^{17} + 10112051275418239936 T^{18} - 58532380576472470119 T^{19} + \)\(44\!\cdots\!36\)\( T^{20} - \)\(19\!\cdots\!68\)\( T^{21} + \)\(11\!\cdots\!31\)\( T^{22} - \)\(28\!\cdots\!74\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \)
$89$ \( 1 - 40 T + 1430 T^{2} - 35157 T^{3} + 772387 T^{4} - 14151415 T^{5} + 235920178 T^{6} - 3483529268 T^{7} + 47407646992 T^{8} - 586085327676 T^{9} + 6728255791535 T^{10} - 70958208182375 T^{11} + 697284512185649 T^{12} - 6315280528231375 T^{13} + 53294514124748735 T^{14} - 413171987366422044 T^{15} + 2974462012814989072 T^{16} - 19452234524843453332 T^{17} + \)\(11\!\cdots\!58\)\( T^{18} - \)\(62\!\cdots\!35\)\( T^{19} + \)\(30\!\cdots\!47\)\( T^{20} - \)\(12\!\cdots\!13\)\( T^{21} + \)\(44\!\cdots\!30\)\( T^{22} - \)\(11\!\cdots\!60\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} \)
$97$ \( 1 - 7 T + 517 T^{2} - 4705 T^{3} + 141436 T^{4} - 1468706 T^{5} + 27804818 T^{6} - 301005861 T^{7} + 4295214504 T^{8} - 46235717659 T^{9} + 541114581696 T^{10} - 5592020727862 T^{11} + 57075181715471 T^{12} - 542426010602614 T^{13} + 5091347099177664 T^{14} - 42198091142992507 T^{15} + 380252251779891624 T^{16} - 2584839747758246277 T^{17} + 23160634996145947922 T^{18} - \)\(11\!\cdots\!78\)\( T^{19} + \)\(11\!\cdots\!96\)\( T^{20} - \)\(35\!\cdots\!85\)\( T^{21} + \)\(38\!\cdots\!33\)\( T^{22} - \)\(50\!\cdots\!71\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \)
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