Properties

Label 8046.2.a.m
Level 8046
Weight 2
Character orbit 8046.a
Self dual yes
Analytic conductor 64.248
Analytic rank 1
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: \(1\)
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.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} -\beta_{1} q^{5} -\beta_{2} q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} -\beta_{1} q^{5} -\beta_{2} q^{7} + q^{8} -\beta_{1} q^{10} + ( \beta_{1} + \beta_{4} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{11} + ( -1 - \beta_{4} + \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{13} -\beta_{2} q^{14} + q^{16} + ( -1 + 2 \beta_{2} - \beta_{3} + \beta_{6} - \beta_{7} - \beta_{9} - \beta_{11} ) q^{17} + ( -2 + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} - \beta_{11} ) q^{19} -\beta_{1} q^{20} + ( \beta_{1} + \beta_{4} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{22} + ( -2 + \beta_{1} + \beta_{3} - \beta_{6} + \beta_{9} + \beta_{10} ) q^{23} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{25} + ( -1 - \beta_{4} + \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{26} -\beta_{2} q^{28} + ( -3 - 2 \beta_{1} - 2 \beta_{4} - \beta_{6} - \beta_{10} - \beta_{11} ) q^{29} + ( -1 + 2 \beta_{1} + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{31} + q^{32} + ( -1 + 2 \beta_{2} - \beta_{3} + \beta_{6} - \beta_{7} - \beta_{9} - \beta_{11} ) q^{34} + ( -3 + 2 \beta_{1} + \beta_{3} - \beta_{8} - \beta_{9} + \beta_{11} ) q^{35} + ( -2 + \beta_{1} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{11} ) q^{37} + ( -2 + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} - \beta_{11} ) q^{38} -\beta_{1} q^{40} + ( -2 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} ) q^{41} + ( -1 + \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{43} + ( \beta_{1} + \beta_{4} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{44} + ( -2 + \beta_{1} + \beta_{3} - \beta_{6} + \beta_{9} + \beta_{10} ) q^{46} + ( \beta_{1} + 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{47} + ( -\beta_{1} - \beta_{3} - \beta_{4} - 3 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{11} ) q^{49} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{50} + ( -1 - \beta_{4} + \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{52} + ( -2 - \beta_{3} - \beta_{4} - \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{11} ) q^{53} + ( -3 - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{10} ) q^{55} -\beta_{2} q^{56} + ( -3 - 2 \beta_{1} - 2 \beta_{4} - \beta_{6} - \beta_{10} - \beta_{11} ) q^{58} + ( \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{11} ) q^{59} + ( -2 + \beta_{1} - 2 \beta_{2} + \beta_{3} - 3 \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} ) q^{61} + ( -1 + 2 \beta_{1} + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{62} + q^{64} + ( -1 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} ) q^{65} + ( -3 + \beta_{3} - \beta_{5} - \beta_{8} - 2 \beta_{10} - \beta_{11} ) q^{67} + ( -1 + 2 \beta_{2} - \beta_{3} + \beta_{6} - \beta_{7} - \beta_{9} - \beta_{11} ) q^{68} + ( -3 + 2 \beta_{1} + \beta_{3} - \beta_{8} - \beta_{9} + \beta_{11} ) q^{70} + ( -3 + \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{71} + ( -1 + \beta_{2} - \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{73} + ( -2 + \beta_{1} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{11} ) q^{74} + ( -2 + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} - \beta_{11} ) q^{76} + ( -1 + \beta_{2} - \beta_{4} + 2 \beta_{7} + \beta_{8} - \beta_{9} - \beta_{11} ) q^{77} + ( -\beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{9} ) q^{79} -\beta_{1} q^{80} + ( -2 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} ) q^{82} + ( -\beta_{2} + \beta_{4} - 2 \beta_{5} + \beta_{8} + \beta_{9} + \beta_{11} ) q^{83} + ( -2 - 2 \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} + 3 \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{85} + ( -1 + \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{86} + ( \beta_{1} + \beta_{4} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{88} + ( -2 - 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + 4 \beta_{6} - \beta_{8} - 3 \beta_{11} ) q^{89} + ( 1 - 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{4} + 3 \beta_{6} - 2 \beta_{7} + 2 \beta_{9} ) q^{91} + ( -2 + \beta_{1} + \beta_{3} - \beta_{6} + \beta_{9} + \beta_{10} ) q^{92} + ( \beta_{1} + 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{94} + ( 4 \beta_{1} + \beta_{2} + 2 \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} + 3 \beta_{8} + 4 \beta_{9} + 3 \beta_{10} - \beta_{11} ) q^{95} + ( -1 + \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{11} ) q^{97} + ( -\beta_{1} - \beta_{3} - \beta_{4} - 3 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{11} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 12q^{2} + 12q^{4} - 5q^{5} - 6q^{7} + 12q^{8} + O(q^{10}) \) \( 12q + 12q^{2} + 12q^{4} - 5q^{5} - 6q^{7} + 12q^{8} - 5q^{10} - 10q^{11} - q^{13} - 6q^{14} + 12q^{16} - 6q^{17} - 10q^{19} - 5q^{20} - 10q^{22} - 15q^{23} + 7q^{25} - q^{26} - 6q^{28} - 33q^{29} - 6q^{31} + 12q^{32} - 6q^{34} - 16q^{35} - 13q^{37} - 10q^{38} - 5q^{40} - 20q^{41} - 11q^{43} - 10q^{44} - 15q^{46} - 15q^{47} + 2q^{49} + 7q^{50} - q^{52} - 4q^{53} - 17q^{55} - 6q^{56} - 33q^{58} - 10q^{59} - 12q^{61} - 6q^{62} + 12q^{64} - 40q^{65} - 19q^{67} - 6q^{68} - 16q^{70} - 47q^{71} - 2q^{73} - 13q^{74} - 10q^{76} + 6q^{77} - 15q^{79} - 5q^{80} - 20q^{82} - 18q^{83} - 25q^{85} - 11q^{86} - 10q^{88} - 24q^{89} - 3q^{91} - 15q^{92} - 15q^{94} + 3q^{95} - 25q^{97} + 2q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 5 x^{11} - 21 x^{10} + 116 x^{9} + 106 x^{8} - 774 x^{7} - 63 x^{6} + 2013 x^{5} - 417 x^{4} - 2249 x^{3} + 761 x^{2} + 910 x - 375\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-32197 \nu^{11} + 5673372 \nu^{10} - 27408721 \nu^{9} - 109537587 \nu^{8} + 609086107 \nu^{7} + 410288818 \nu^{6} - 3495027303 \nu^{5} + 447932378 \nu^{4} + 6463643167 \nu^{3} - 2256579470 \nu^{2} - 3729321842 \nu + 1592878652\)\()/5855999\)
\(\beta_{3}\)\(=\)\((\)\(-337779 \nu^{11} - 12051765 \nu^{10} + 77892894 \nu^{9} + 222792211 \nu^{8} - 1586491494 \nu^{7} - 664646379 \nu^{6} + 8970420017 \nu^{5} - 2097844062 \nu^{4} - 16685502287 \nu^{3} + 6698494966 \nu^{2} + 9660687071 \nu - 4301878920\)\()/29279995\)
\(\beta_{4}\)\(=\)\((\)\(-4626967 \nu^{11} + 27982820 \nu^{10} + 68317507 \nu^{9} - 610246892 \nu^{8} + 137663433 \nu^{7} + 3479105328 \nu^{6} - 3268300794 \nu^{5} - 6124911496 \nu^{4} + 8040696879 \nu^{3} + 2382541943 \nu^{2} - 5435421687 \nu + 1309434950\)\()/29279995\)
\(\beta_{5}\)\(=\)\((\)\(-5834388 \nu^{11} + 38266875 \nu^{10} + 69519048 \nu^{9} - 818338273 \nu^{8} + 526825367 \nu^{7} + 4419720832 \nu^{6} - 6005098721 \nu^{5} - 6551037399 \nu^{4} + 13075477756 \nu^{3} + 594551152 \nu^{2} - 8170221548 \nu + 2800717475\)\()/29279995\)
\(\beta_{6}\)\(=\)\((\)\(-11071809 \nu^{11} + 33647510 \nu^{10} + 329349654 \nu^{9} - 795142464 \nu^{8} - 3330885579 \nu^{7} + 5470397091 \nu^{6} + 13696849342 \nu^{5} - 15262968602 \nu^{4} - 22956311877 \nu^{3} + 17416329301 \nu^{2} + 12966000711 \nu - 6906022130\)\()/29279995\)
\(\beta_{7}\)\(=\)\((\)\(-15600107 \nu^{11} + 74241930 \nu^{10} + 327537312 \nu^{9} - 1640249962 \nu^{8} - 1697104197 \nu^{7} + 9679352383 \nu^{6} + 1916697621 \nu^{5} - 19399390581 \nu^{4} + 804575664 \nu^{3} + 13154737748 \nu^{2} - 1591143977 \nu - 1289750025\)\()/29279995\)
\(\beta_{8}\)\(=\)\((\)\(15915726 \nu^{11} - 100688050 \nu^{10} - 207068641 \nu^{9} + 2153825106 \nu^{8} - 1047579084 \nu^{7} - 11652307114 \nu^{6} + 14014610012 \nu^{5} + 17633361118 \nu^{4} - 30654436132 \nu^{3} - 3005054784 \nu^{2} + 18959758726 \nu - 6112777815\)\()/29279995\)
\(\beta_{9}\)\(=\)\((\)\(-3317302 \nu^{11} + 16732001 \nu^{10} + 64707528 \nu^{9} - 366093260 \nu^{8} - 254489192 \nu^{7} + 2107854700 \nu^{6} - 179845605 \nu^{5} - 3967835268 \nu^{4} + 1171309316 \nu^{3} + 2366997820 \nu^{2} - 842422122 \nu - 38973036\)\()/5855999\)
\(\beta_{10}\)\(=\)\((\)\(17078923 \nu^{11} - 69687500 \nu^{10} - 418005468 \nu^{9} + 1573561298 \nu^{8} + 3159684708 \nu^{7} - 9796137762 \nu^{6} - 9615020874 \nu^{5} + 22410652829 \nu^{4} + 13297311074 \nu^{3} - 19695978447 \nu^{2} - 6697290927 \nu + 5205009495\)\()/29279995\)
\(\beta_{11}\)\(=\)\((\)\(5445786 \nu^{11} - 29080114 \nu^{10} - 98087514 \nu^{9} + 632612308 \nu^{8} + 238661630 \nu^{7} - 3586733161 \nu^{6} + 1341737333 \nu^{5} + 6428622786 \nu^{4} - 3951467976 \nu^{3} - 3278532055 \nu^{2} + 2627858133 \nu - 395658693\)\()/5855999\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{3} + \beta_{2} + \beta_{1} + 5\)
\(\nu^{3}\)\(=\)\(-\beta_{11} + 3 \beta_{10} - \beta_{9} + 2 \beta_{8} + 4 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} + 14 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(2 \beta_{11} - 13 \beta_{10} - 11 \beta_{9} + 19 \beta_{8} + 21 \beta_{7} - \beta_{6} - \beta_{5} + 2 \beta_{4} - 16 \beta_{3} + 17 \beta_{2} + 25 \beta_{1} + 60\)
\(\nu^{5}\)\(=\)\(-15 \beta_{11} + 51 \beta_{10} - 23 \beta_{9} + 44 \beta_{8} + 79 \beta_{7} + 27 \beta_{6} - 16 \beta_{5} + 23 \beta_{4} - 13 \beta_{3} + 27 \beta_{2} + 222 \beta_{1} + 49\)
\(\nu^{6}\)\(=\)\(34 \beta_{11} - 162 \beta_{10} - 138 \beta_{9} + 341 \beta_{8} + 378 \beta_{7} - 11 \beta_{6} - 24 \beta_{5} + 61 \beta_{4} - 258 \beta_{3} + 280 \beta_{2} + 506 \beta_{1} + 923\)
\(\nu^{7}\)\(=\)\(-223 \beta_{11} + 787 \beta_{10} - 406 \beta_{9} + 914 \beta_{8} + 1435 \beta_{7} + 498 \beta_{6} - 222 \beta_{5} + 450 \beta_{4} - 384 \beta_{3} + 578 \beta_{2} + 3665 \beta_{1} + 1285\)
\(\nu^{8}\)\(=\)\(451 \beta_{11} - 1964 \beta_{10} - 1962 \beta_{9} + 6113 \beta_{8} + 6668 \beta_{7} + 77 \beta_{6} - 348 \beta_{5} + 1390 \beta_{4} - 4316 \beta_{3} + 4693 \beta_{2} + 9664 \beta_{1} + 15112\)
\(\nu^{9}\)\(=\)\(-3384 \beta_{11} + 12202 \beta_{10} - 6570 \beta_{9} + 18455 \beta_{8} + 25796 \beta_{7} + 8546 \beta_{6} - 2832 \beta_{5} + 8544 \beta_{4} - 8748 \beta_{3} + 11538 \beta_{2} + 61610 \beta_{1} + 28417\)
\(\nu^{10}\)\(=\)\(5285 \beta_{11} - 22019 \beta_{10} - 29192 \beta_{9} + 110188 \beta_{8} + 117443 \beta_{7} + 6505 \beta_{6} - 3650 \beta_{5} + 29179 \beta_{4} - 73887 \beta_{3} + 79863 \beta_{2} + 180002 \beta_{1} + 253681\)
\(\nu^{11}\)\(=\)\(-52344 \beta_{11} + 193515 \beta_{10} - 102076 \beta_{9} + 364000 \beta_{8} + 462496 \beta_{7} + 145903 \beta_{6} - 31755 \beta_{5} + 161388 \beta_{4} - 182355 \beta_{3} + 222808 \beta_{2} + 1046987 \beta_{1} + 583486\)

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
4.25305
4.11015
1.99250
1.53008
1.09665
1.06735
0.448758
−1.12518
−1.17050
−1.20126
−2.26764
−3.73396
1.00000 0 1.00000 −4.25305 0 2.10378 1.00000 0 −4.25305
1.2 1.00000 0 1.00000 −4.11015 0 −3.78475 1.00000 0 −4.11015
1.3 1.00000 0 1.00000 −1.99250 0 −0.0206178 1.00000 0 −1.99250
1.4 1.00000 0 1.00000 −1.53008 0 5.04291 1.00000 0 −1.53008
1.5 1.00000 0 1.00000 −1.09665 0 −2.72925 1.00000 0 −1.09665
1.6 1.00000 0 1.00000 −1.06735 0 2.41300 1.00000 0 −1.06735
1.7 1.00000 0 1.00000 −0.448758 0 −1.52942 1.00000 0 −0.448758
1.8 1.00000 0 1.00000 1.12518 0 −0.686663 1.00000 0 1.12518
1.9 1.00000 0 1.00000 1.17050 0 0.506529 1.00000 0 1.17050
1.10 1.00000 0 1.00000 1.20126 0 −4.29383 1.00000 0 1.20126
1.11 1.00000 0 1.00000 2.26764 0 −0.397853 1.00000 0 2.26764
1.12 1.00000 0 1.00000 3.73396 0 −2.62384 1.00000 0 3.73396
\(n\): 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.m yes 12
3.b odd 2 1 8046.2.a.l 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8046.2.a.l 12 3.b odd 2 1
8046.2.a.m 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

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{12} \)
$3$ \( \)
$5$ \( 1 + 5 T + 39 T^{2} + 159 T^{3} + 706 T^{4} + 2429 T^{5} + 8052 T^{6} + 23802 T^{7} + 66268 T^{8} + 171524 T^{9} + 429546 T^{10} + 999575 T^{11} + 2324685 T^{12} + 4997875 T^{13} + 10738650 T^{14} + 21440500 T^{15} + 41417500 T^{16} + 74381250 T^{17} + 125812500 T^{18} + 189765625 T^{19} + 275781250 T^{20} + 310546875 T^{21} + 380859375 T^{22} + 244140625 T^{23} + 244140625 T^{24} \)
$7$ \( 1 + 6 T + 59 T^{2} + 248 T^{3} + 1398 T^{4} + 4348 T^{5} + 17885 T^{6} + 40132 T^{7} + 133511 T^{8} + 179525 T^{9} + 588556 T^{10} + 128130 T^{11} + 2459561 T^{12} + 896910 T^{13} + 28839244 T^{14} + 61577075 T^{15} + 320559911 T^{16} + 674498524 T^{17} + 2104152365 T^{18} + 3580764964 T^{19} + 8059191798 T^{20} + 10007694536 T^{21} + 16666039691 T^{22} + 11863960458 T^{23} + 13841287201 T^{24} \)
$11$ \( 1 + 10 T + 115 T^{2} + 724 T^{3} + 5023 T^{4} + 24667 T^{5} + 134846 T^{6} + 571308 T^{7} + 2675581 T^{8} + 10038817 T^{9} + 41178796 T^{10} + 137774274 T^{11} + 503919571 T^{12} + 1515517014 T^{13} + 4982634316 T^{14} + 13361665427 T^{15} + 39173181421 T^{16} + 92009724708 T^{17} + 238887914606 T^{18} + 480690047057 T^{19} + 1076724659263 T^{20} + 1707154128284 T^{21} + 2982803829115 T^{22} + 2853116706110 T^{23} + 3138428376721 T^{24} \)
$13$ \( 1 + T + 81 T^{2} - 7 T^{3} + 3249 T^{4} - 3204 T^{5} + 89567 T^{6} - 140528 T^{7} + 1937232 T^{8} - 3458175 T^{9} + 34186509 T^{10} - 59592236 T^{11} + 492567871 T^{12} - 774699068 T^{13} + 5777520021 T^{14} - 7597610475 T^{15} + 55329283152 T^{16} - 52177062704 T^{17} + 432322801703 T^{18} - 201046248468 T^{19} + 2650309112529 T^{20} - 74231495611 T^{21} + 11166537839769 T^{22} + 1792160394037 T^{23} + 23298085122481 T^{24} \)
$17$ \( 1 + 6 T + 79 T^{2} + 387 T^{3} + 3639 T^{4} + 15822 T^{5} + 119551 T^{6} + 471461 T^{7} + 3119918 T^{8} + 11308494 T^{9} + 67422746 T^{10} + 226504061 T^{11} + 1241062419 T^{12} + 3850569037 T^{13} + 19485173594 T^{14} + 55558631022 T^{15} + 260578671278 T^{16} + 669407201077 T^{17} + 2885670511519 T^{18} + 6492378484206 T^{19} + 25384781327799 T^{20} + 45893508204339 T^{21} + 159263518135471 T^{22} + 205631377845798 T^{23} + 582622237229761 T^{24} \)
$19$ \( 1 + 10 T + 127 T^{2} + 934 T^{3} + 6719 T^{4} + 38183 T^{5} + 196112 T^{6} + 876536 T^{7} + 3480005 T^{8} + 12122933 T^{9} + 40733640 T^{10} + 118097532 T^{11} + 509348397 T^{12} + 2243853108 T^{13} + 14704844040 T^{14} + 83151197447 T^{15} + 453517731605 T^{16} + 2170389913064 T^{17} + 9226261814672 T^{18} + 34130704610237 T^{19} + 114112560072479 T^{20} + 301390309725586 T^{21} + 778645414740727 T^{22} + 1164902588982190 T^{23} + 2213314919066161 T^{24} \)
$23$ \( 1 + 15 T + 268 T^{2} + 2999 T^{3} + 32777 T^{4} + 289696 T^{5} + 2422679 T^{6} + 17672115 T^{7} + 121520833 T^{8} + 751073331 T^{9} + 4381416603 T^{10} + 23264664391 T^{11} + 116738365747 T^{12} + 535087280993 T^{13} + 2317769382987 T^{14} + 9138309218277 T^{15} + 34006511427553 T^{16} + 113743793675445 T^{17} + 358643439526631 T^{18} + 986364312694112 T^{19} + 2566799164555337 T^{20} + 5401656831727537 T^{21} + 11102305005257932 T^{22} + 14292146368708905 T^{23} + 21914624432020321 T^{24} \)
$29$ \( 1 + 33 T + 647 T^{2} + 9211 T^{3} + 106462 T^{4} + 1046559 T^{5} + 9064944 T^{6} + 70513256 T^{7} + 501529378 T^{8} + 3300509040 T^{9} + 20342854914 T^{10} + 118306330217 T^{11} + 653439497197 T^{12} + 3430883576293 T^{13} + 17108340982674 T^{14} + 80496114976560 T^{15} + 354722200001218 T^{16} + 1446307900291144 T^{17} + 5392040094759024 T^{18} + 18053013300070731 T^{19} + 53257233616653982 T^{20} + 133625321583729359 T^{21} + 272197579945230047 T^{22} + 402616822268292357 T^{23} + 353814783205469041 T^{24} \)
$31$ \( 1 + 6 T + 178 T^{2} + 999 T^{3} + 16153 T^{4} + 86715 T^{5} + 1009493 T^{6} + 5272578 T^{7} + 49204727 T^{8} + 249105881 T^{9} + 1979200121 T^{10} + 9495174993 T^{11} + 66874337227 T^{12} + 294350424783 T^{13} + 1902011316281 T^{14} + 7421113300871 T^{15} + 45441598683767 T^{16} + 150949431721278 T^{17} + 895928753443733 T^{18} + 2385756332635365 T^{19} + 13776748927784473 T^{20} + 26413182538510329 T^{21} + 145893835082582578 T^{22} + 152450861378428986 T^{23} + 787662783788549761 T^{24} \)
$37$ \( 1 + 13 T + 267 T^{2} + 2513 T^{3} + 31583 T^{4} + 238965 T^{5} + 2355489 T^{6} + 15309362 T^{7} + 131379734 T^{8} + 768853859 T^{9} + 6046087991 T^{10} + 32723836129 T^{11} + 239861350569 T^{12} + 1210781936773 T^{13} + 8277094459679 T^{14} + 38944754519927 T^{15} + 246226773653174 T^{16} + 1061611740225434 T^{17} + 6043540333409001 T^{18} + 22685396019087345 T^{19} + 110934638593186943 T^{20} + 326593852105028501 T^{21} + 1283892027435565683 T^{22} + 2312929083132985369 T^{23} + 6582952005840035281 T^{24} \)
$41$ \( 1 + 20 T + 468 T^{2} + 6832 T^{3} + 98146 T^{4} + 1131373 T^{5} + 12475664 T^{6} + 119187864 T^{7} + 1085210964 T^{8} + 8834724827 T^{9} + 68563015800 T^{10} + 482456685988 T^{11} + 3237627435202 T^{12} + 19780724125508 T^{13} + 115254429559800 T^{14} + 608898069801667 T^{15} + 3066546818843604 T^{16} + 13808653128344664 T^{17} + 59260704475691024 T^{18} + 220339727103568613 T^{19} + 783688471537309666 T^{20} + 2236673375779541552 T^{21} + 6281804557151323668 T^{22} + 11006580634324968820 T^{23} + 22563490300366186081 T^{24} \)
$43$ \( 1 + 11 T + 328 T^{2} + 2903 T^{3} + 49571 T^{4} + 361934 T^{5} + 4671520 T^{6} + 29010177 T^{7} + 318352639 T^{8} + 1738315065 T^{9} + 17316356409 T^{10} + 85944319606 T^{11} + 799628157817 T^{12} + 3695605743058 T^{13} + 32017943000241 T^{14} + 138208215872955 T^{15} + 1088384320565839 T^{16} + 4264740951924411 T^{17} + 29530373910664480 T^{18} + 98380397192400938 T^{19} + 579395775960959171 T^{20} + 1459026352452655229 T^{21} + 7088566198757233672 T^{22} + 10222231134183449777 T^{23} + 39959630797262576401 T^{24} \)
$47$ \( 1 + 15 T + 516 T^{2} + 6208 T^{3} + 121430 T^{4} + 1225645 T^{5} + 17536021 T^{6} + 151949323 T^{7} + 1743788176 T^{8} + 13120541600 T^{9} + 126243197873 T^{10} + 827657632458 T^{11} + 6834095624191 T^{12} + 38899908725526 T^{13} + 278871224101457 T^{14} + 1362213990536800 T^{15} + 8509130030451856 T^{16} + 34848818547080261 T^{17} + 189024546372865909 T^{18} + 620940094479873635 T^{19} + 2891404539337638230 T^{20} + 6947561977021977536 T^{21} + 27141152233688305284 T^{22} + 37082388226260184545 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \)
$53$ \( 1 + 4 T + 317 T^{2} + 1535 T^{3} + 53635 T^{4} + 288929 T^{5} + 6330481 T^{6} + 35530248 T^{7} + 570336844 T^{8} + 3173471249 T^{9} + 41004300501 T^{10} + 216160124956 T^{11} + 2401609762549 T^{12} + 11456486622668 T^{13} + 115181080107309 T^{14} + 472456879137373 T^{15} + 4500232031181964 T^{16} + 14858589578772264 T^{17} + 140311067004273049 T^{18} + 339408114921964573 T^{19} + 3339298495213347235 T^{20} + 5065137113416274155 T^{21} + 55439328105867636533 T^{22} + 37076143717488766388 T^{23} + \)\(49\!\cdots\!41\)\( T^{24} \)
$59$ \( 1 + 10 T + 219 T^{2} + 1772 T^{3} + 32138 T^{4} + 256033 T^{5} + 3645418 T^{6} + 26020690 T^{7} + 326852615 T^{8} + 2230252101 T^{9} + 25077488626 T^{10} + 155431115459 T^{11} + 1578741333065 T^{12} + 9170435812081 T^{13} + 87294737907106 T^{14} + 458046946251279 T^{15} + 3960591129749015 T^{16} + 18602823557746310 T^{17} + 153765676584506938 T^{18} + 637176905612663027 T^{19} + 4718836603727668298 T^{20} + 15350828590656551908 T^{21} + \)\(11\!\cdots\!19\)\( T^{22} + \)\(30\!\cdots\!90\)\( T^{23} + \)\(17\!\cdots\!81\)\( T^{24} \)
$61$ \( 1 + 12 T + 298 T^{2} + 3496 T^{3} + 54064 T^{4} + 546885 T^{5} + 6800109 T^{6} + 62808914 T^{7} + 667720626 T^{8} + 5627341982 T^{9} + 53783116351 T^{10} + 415772734409 T^{11} + 3577595975703 T^{12} + 25362136798949 T^{13} + 200126975942071 T^{14} + 1277299710416342 T^{15} + 9245153620016466 T^{16} + 53048176434227114 T^{17} + 350344161375605349 T^{18} + 1718718915877344585 T^{19} + 10364464169884999984 T^{20} + 40882734740548156936 T^{21} + \)\(21\!\cdots\!98\)\( T^{22} + \)\(52\!\cdots\!32\)\( T^{23} + \)\(26\!\cdots\!21\)\( T^{24} \)
$67$ \( 1 + 19 T + 607 T^{2} + 9305 T^{3} + 169538 T^{4} + 2182477 T^{5} + 29437067 T^{6} + 327561454 T^{7} + 3613890498 T^{8} + 35542644025 T^{9} + 337982347007 T^{10} + 2981324477278 T^{11} + 25176664348651 T^{12} + 199748739977626 T^{13} + 1517202755714423 T^{14} + 10689912244891075 T^{15} + 72823944705948258 T^{16} + 442248943130825578 T^{17} + 2662829456620458323 T^{18} + 13227363682250525071 T^{19} + 68843901917597801858 T^{20} + \)\(25\!\cdots\!35\)\( T^{21} + \)\(11\!\cdots\!43\)\( T^{22} + \)\(23\!\cdots\!77\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \)
$71$ \( 1 + 47 T + 1415 T^{2} + 30825 T^{3} + 552183 T^{4} + 8406194 T^{5} + 113856814 T^{6} + 1392488028 T^{7} + 15731393240 T^{8} + 164995119135 T^{9} + 1624120581131 T^{10} + 14986823940348 T^{11} + 130317347009003 T^{12} + 1064064499764708 T^{13} + 8187191849481371 T^{14} + 59053568084726985 T^{15} + 399761146700436440 T^{16} + 2512367771033709828 T^{17} + 14585090199740487694 T^{18} + 76455344504745473854 T^{19} + \)\(35\!\cdots\!63\)\( T^{20} + \)\(14\!\cdots\!75\)\( T^{21} + \)\(46\!\cdots\!15\)\( T^{22} + \)\(10\!\cdots\!37\)\( T^{23} + \)\(16\!\cdots\!41\)\( T^{24} \)
$73$ \( 1 + 2 T + 486 T^{2} + 1029 T^{3} + 117469 T^{4} + 238412 T^{5} + 18645928 T^{6} + 34124103 T^{7} + 2191985511 T^{8} + 3513470157 T^{9} + 205786863273 T^{10} + 292339731915 T^{11} + 16232626831117 T^{12} + 21340800429795 T^{13} + 1096638194381817 T^{14} + 1366799620065669 T^{15} + 62248532809886151 T^{16} + 70741708565906079 T^{17} + 2821767087320401192 T^{18} + 2633832375734953964 T^{19} + 94734060534705800989 T^{20} + 60578862722807682477 T^{21} + \)\(20\!\cdots\!14\)\( T^{22} + \)\(62\!\cdots\!54\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \)
$79$ \( 1 + 15 T + 564 T^{2} + 6783 T^{3} + 152314 T^{4} + 1542062 T^{5} + 26410551 T^{6} + 232068714 T^{7} + 3357776408 T^{8} + 26270103298 T^{9} + 339911306201 T^{10} + 2426868197790 T^{11} + 28970653055967 T^{12} + 191722587625410 T^{13} + 2121386462000441 T^{14} + 12952185459942622 T^{15} + 130785663071489048 T^{16} + 714088521421400886 T^{17} + 6420073641497602071 T^{18} + 29613618299014319858 T^{19} + \)\(23\!\cdots\!54\)\( T^{20} + \)\(81\!\cdots\!77\)\( T^{21} + \)\(53\!\cdots\!64\)\( T^{22} + \)\(11\!\cdots\!85\)\( T^{23} + \)\(59\!\cdots\!41\)\( T^{24} \)
$83$ \( 1 + 18 T + 863 T^{2} + 13116 T^{3} + 348984 T^{4} + 4560335 T^{5} + 87697456 T^{6} + 996405958 T^{7} + 15277660667 T^{8} + 151747795747 T^{9} + 1945449095534 T^{10} + 16894314638903 T^{11} + 185758185266123 T^{12} + 1402228115028949 T^{13} + 13402198819133726 T^{14} + 86767416886789889 T^{15} + 725052124063560107 T^{16} + 3924883565489350994 T^{17} + 28671839008151449264 T^{18} + \)\(12\!\cdots\!45\)\( T^{19} + \)\(78\!\cdots\!44\)\( T^{20} + \)\(24\!\cdots\!48\)\( T^{21} + \)\(13\!\cdots\!87\)\( T^{22} + \)\(23\!\cdots\!06\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \)
$89$ \( 1 + 24 T + 578 T^{2} + 8123 T^{3} + 131247 T^{4} + 1610393 T^{5} + 23145950 T^{6} + 257958908 T^{7} + 3198014560 T^{8} + 31793265584 T^{9} + 360076976807 T^{10} + 3357276169589 T^{11} + 35144002539569 T^{12} + 298797579093421 T^{13} + 2852169733288247 T^{14} + 22413266645486896 T^{15} + 200650600245028960 T^{16} + 1440457877671121692 T^{17} + 11503104111518757950 T^{18} + 71229832096415632897 T^{19} + \)\(51\!\cdots\!07\)\( T^{20} + \)\(28\!\cdots\!07\)\( T^{21} + \)\(18\!\cdots\!78\)\( T^{22} + \)\(66\!\cdots\!36\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} \)
$97$ \( 1 + 25 T + 1103 T^{2} + 21705 T^{3} + 551572 T^{4} + 8957656 T^{5} + 167995324 T^{6} + 2313751659 T^{7} + 34991740282 T^{8} + 414911018767 T^{9} + 5273346067898 T^{10} + 54184093348156 T^{11} + 590821306410541 T^{12} + 5255857054771132 T^{13} + 49616913152852282 T^{14} + 378678084231134191 T^{15} + 3097793608104197242 T^{16} + 19868972766031236363 T^{17} + \)\(13\!\cdots\!96\)\( T^{18} + \)\(72\!\cdots\!28\)\( T^{19} + \)\(43\!\cdots\!92\)\( T^{20} + \)\(16\!\cdots\!85\)\( T^{21} + \)\(81\!\cdots\!47\)\( T^{22} + \)\(17\!\cdots\!25\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \)
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