Properties

Label 8046.2.a.n
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.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} -\beta_{1} q^{5} + ( -1 - \beta_{9} ) q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} -\beta_{1} q^{5} + ( -1 - \beta_{9} ) q^{7} + q^{8} -\beta_{1} q^{10} + ( -1 - \beta_{6} + \beta_{7} + \beta_{9} ) q^{11} + ( 1 + \beta_{1} - \beta_{4} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{11} ) q^{13} + ( -1 - \beta_{9} ) q^{14} + q^{16} + ( -1 - \beta_{2} + \beta_{5} + \beta_{6} ) q^{17} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{11} ) q^{19} -\beta_{1} q^{20} + ( -1 - \beta_{6} + \beta_{7} + \beta_{9} ) q^{22} + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{11} ) q^{23} + ( 1 + \beta_{1} - \beta_{3} - \beta_{5} + \beta_{6} - \beta_{8} + \beta_{10} ) q^{25} + ( 1 + \beta_{1} - \beta_{4} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{11} ) q^{26} + ( -1 - \beta_{9} ) q^{28} + ( -2 + \beta_{1} + \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - 2 \beta_{10} ) q^{29} + ( -2 + 2 \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{11} ) q^{31} + q^{32} + ( -1 - \beta_{2} + \beta_{5} + \beta_{6} ) q^{34} + ( -4 + \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{10} + \beta_{11} ) q^{35} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{8} - \beta_{11} ) q^{37} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{11} ) q^{38} -\beta_{1} q^{40} + ( -3 - 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{10} - \beta_{11} ) q^{41} + ( -4 + \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} + 3 \beta_{11} ) q^{43} + ( -1 - \beta_{6} + \beta_{7} + \beta_{9} ) q^{44} + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{11} ) q^{46} + ( \beta_{1} - 2 \beta_{2} + \beta_{5} - \beta_{7} + \beta_{9} + \beta_{10} ) q^{47} + ( 3 - \beta_{2} - 2 \beta_{4} - \beta_{7} + \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{49} + ( 1 + \beta_{1} - \beta_{3} - \beta_{5} + \beta_{6} - \beta_{8} + \beta_{10} ) q^{50} + ( 1 + \beta_{1} - \beta_{4} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{11} ) q^{52} + ( -1 + \beta_{2} - 3 \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{11} ) q^{53} + ( -1 + 3 \beta_{1} - \beta_{2} - \beta_{4} + 3 \beta_{5} + \beta_{6} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{55} + ( -1 - \beta_{9} ) q^{56} + ( -2 + \beta_{1} + \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - 2 \beta_{10} ) q^{58} + ( -2 + \beta_{4} + 3 \beta_{7} + 2 \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{59} + ( -\beta_{2} - \beta_{4} + \beta_{6} - 4 \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{10} ) q^{61} + ( -2 + 2 \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{11} ) q^{62} + q^{64} + ( 1 + 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} + 3 \beta_{9} + \beta_{10} - \beta_{11} ) q^{65} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{11} ) q^{67} + ( -1 - \beta_{2} + \beta_{5} + \beta_{6} ) q^{68} + ( -4 + \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{10} + \beta_{11} ) q^{70} + ( -1 - \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{8} - \beta_{9} ) q^{71} + ( 1 - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{73} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{8} - \beta_{11} ) q^{74} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{11} ) q^{76} + ( -3 + 2 \beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{5} + \beta_{6} - 2 \beta_{7} - 2 \beta_{9} - 2 \beta_{10} + 3 \beta_{11} ) q^{77} + ( -2 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{9} + \beta_{11} ) q^{79} -\beta_{1} q^{80} + ( -3 - 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{10} - \beta_{11} ) q^{82} + ( -2 + 2 \beta_{2} + \beta_{3} + \beta_{4} - 3 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} + 3 \beta_{11} ) q^{83} + ( -2 - \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - \beta_{7} - 2 \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{85} + ( -4 + \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} + 3 \beta_{11} ) q^{86} + ( -1 - \beta_{6} + \beta_{7} + \beta_{9} ) q^{88} + ( -1 + 2 \beta_{2} + \beta_{5} - \beta_{6} + 3 \beta_{7} + 3 \beta_{9} + \beta_{10} ) q^{89} + ( -5 - \beta_{1} - 3 \beta_{2} - 3 \beta_{7} + \beta_{8} - 5 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{91} + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{11} ) q^{92} + ( \beta_{1} - 2 \beta_{2} + \beta_{5} - \beta_{7} + \beta_{9} + \beta_{10} ) q^{94} + ( -2 - \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} - 4 \beta_{7} - 2 \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{95} + ( 3 - \beta_{1} + \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - 3 \beta_{9} - \beta_{10} ) q^{97} + ( 3 - \beta_{2} - 2 \beta_{4} - \beta_{7} + \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 12q^{2} + 12q^{4} - 3q^{5} - 6q^{7} + 12q^{8} + O(q^{10}) \) \( 12q + 12q^{2} + 12q^{4} - 3q^{5} - 6q^{7} + 12q^{8} - 3q^{10} - 14q^{11} - 3q^{13} - 6q^{14} + 12q^{16} - 8q^{17} - 4q^{19} - 3q^{20} - 14q^{22} - 13q^{23} + 11q^{25} - 3q^{26} - 6q^{28} - 23q^{29} - 14q^{31} + 12q^{32} - 8q^{34} - 32q^{35} - 19q^{37} - 4q^{38} - 3q^{40} - 30q^{41} - 15q^{43} - 14q^{44} - 13q^{46} + q^{47} + 14q^{49} + 11q^{50} - 3q^{52} - 16q^{53} - 7q^{55} - 6q^{56} - 23q^{58} - 26q^{59} - 16q^{61} - 14q^{62} + 12q^{64} - 8q^{65} - 39q^{67} - 8q^{68} - 32q^{70} - 15q^{71} - 2q^{73} - 19q^{74} - 4q^{76} - 34q^{77} - 13q^{79} - 3q^{80} - 30q^{82} - 6q^{83} - 11q^{85} - 15q^{86} - 14q^{88} - 18q^{89} - 35q^{91} - 13q^{92} + q^{94} - 51q^{95} + 19q^{97} + 14q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 3 x^{11} - 31 x^{10} + 82 x^{9} + 334 x^{8} - 684 x^{7} - 1561 x^{6} + 1551 x^{5} + 3573 x^{4} + 345 x^{3} - 1607 x^{2} - 594 x - 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(24748733 \nu^{11} + 373555153 \nu^{10} - 2862765831 \nu^{9} - 8579109025 \nu^{8} + 62879646690 \nu^{7} + 50310595815 \nu^{6} - 465585432656 \nu^{5} - 38007995419 \nu^{4} + 994832840887 \nu^{3} + 294511464920 \nu^{2} - 403274368788 \nu - 135751602546\)\()/ 24466366011 \)
\(\beta_{3}\)\(=\)\((\)\(-142183724 \nu^{11} + 838314338 \nu^{10} + 2550850410 \nu^{9} - 22317146501 \nu^{8} + 3959704842 \nu^{7} + 175851726345 \nu^{6} - 208981623823 \nu^{5} - 377751239522 \nu^{4} + 495900208592 \nu^{3} + 403759068040 \nu^{2} - 184654740606 \nu - 100388062134\)\()/ 24466366011 \)
\(\beta_{4}\)\(=\)\((\)\(-743676686 \nu^{11} + 3071881331 \nu^{10} + 19932549972 \nu^{9} - 83789564729 \nu^{8} - 164381452443 \nu^{7} + 697617356013 \nu^{6} + 479743706735 \nu^{5} - 1672929464267 \nu^{4} - 1150914398347 \nu^{3} + 745303063093 \nu^{2} + 651960038514 \nu + 65302848768\)\()/ 24466366011 \)
\(\beta_{5}\)\(=\)\((\)\(260555422 \nu^{11} - 1008357826 \nu^{10} - 7145245871 \nu^{9} + 27363214940 \nu^{8} + 61692870961 \nu^{7} - 226011640501 \nu^{6} - 196533632656 \nu^{5} + 528630702028 \nu^{4} + 427946138875 \nu^{3} - 193999453699 \nu^{2} - 177941248499 \nu + 915421136\)\()/ 8155455337 \)
\(\beta_{6}\)\(=\)\((\)\(791457988 \nu^{11} - 2873219941 \nu^{10} - 22855402146 \nu^{9} + 79587640096 \nu^{8} + 218268434445 \nu^{7} - 686060598954 \nu^{6} - 843237849712 \nu^{5} + 1806674571358 \nu^{4} + 1819129337198 \nu^{3} - 899515205951 \nu^{2} - 830466305577 \nu - 29055232476\)\()/ 24466366011 \)
\(\beta_{7}\)\(=\)\((\)\(905897063 \nu^{11} - 3376570442 \nu^{10} - 25430053989 \nu^{9} + 92407418951 \nu^{8} + 230188534851 \nu^{7} - 779842057185 \nu^{6} - 806208285203 \nu^{5} + 1958545237151 \nu^{4} + 1735102707136 \nu^{3} - 954029651272 \nu^{2} - 857897373063 \nu - 7334114235\)\()/ 24466366011 \)
\(\beta_{8}\)\(=\)\((\)\(-303077175 \nu^{11} + 1063333243 \nu^{10} + 8694135210 \nu^{9} - 28833755646 \nu^{8} - 82150526984 \nu^{7} + 238485257763 \nu^{6} + 313671881091 \nu^{5} - 556795852562 \nu^{4} - 689122894818 \nu^{3} + 140774506663 \nu^{2} + 293198015341 \nu + 44724727668\)\()/ 8155455337 \)
\(\beta_{9}\)\(=\)\((\)\(-329344029 \nu^{11} + 1140734420 \nu^{10} + 9678326681 \nu^{9} - 31578991963 \nu^{8} - 94978496783 \nu^{7} + 271887620614 \nu^{6} + 378855759913 \nu^{5} - 709008838650 \nu^{4} - 783228359697 \nu^{3} + 297129010822 \nu^{2} + 318441901306 \nu + 17837198367\)\()/ 8155455337 \)
\(\beta_{10}\)\(=\)\((\)\(-353735657 \nu^{11} + 1292153510 \nu^{10} + 10017640191 \nu^{9} - 35438802905 \nu^{8} - 91893899224 \nu^{7} + 299777725695 \nu^{6} + 328556990398 \nu^{5} - 756307087494 \nu^{4} - 702253132145 \nu^{3} + 389355266298 \nu^{2} + 322371833162 \nu - 27070193104\)\()/ 8155455337 \)
\(\beta_{11}\)\(=\)\((\)\(701406468 \nu^{11} - 2905422747 \nu^{10} - 18475689520 \nu^{9} + 78645513257 \nu^{8} + 146244224743 \nu^{7} - 647726374616 \nu^{6} - 374724551769 \nu^{5} + 1522325564529 \nu^{4} + 836181281764 \nu^{3} - 700613773249 \nu^{2} - 384310270718 \nu - 1717939244\)\()/ 8155455337 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{10} - \beta_{8} + \beta_{6} - \beta_{5} - \beta_{3} + \beta_{1} + 6\)
\(\nu^{3}\)\(=\)\(-\beta_{11} + 2 \beta_{10} + \beta_{9} + 3 \beta_{7} + \beta_{6} + 2 \beta_{5} - \beta_{3} + 10 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(-3 \beta_{11} + 16 \beta_{10} + 3 \beta_{9} - 14 \beta_{8} + \beta_{7} + 20 \beta_{6} - 6 \beta_{5} + \beta_{4} - 16 \beta_{3} - 2 \beta_{2} + 15 \beta_{1} + 68\)
\(\nu^{5}\)\(=\)\(-19 \beta_{11} + 43 \beta_{10} + 23 \beta_{9} - 3 \beta_{8} + 50 \beta_{7} + 35 \beta_{6} + 45 \beta_{5} + 8 \beta_{4} - 24 \beta_{3} - \beta_{2} + 127 \beta_{1} + 31\)
\(\nu^{6}\)\(=\)\(-62 \beta_{11} + 258 \beta_{10} + 85 \beta_{9} - 184 \beta_{8} + 55 \beta_{7} + 335 \beta_{6} + 4 \beta_{5} + 41 \beta_{4} - 239 \beta_{3} - 26 \beta_{2} + 238 \beta_{1} + 846\)
\(\nu^{7}\)\(=\)\(-313 \beta_{11} + 786 \beta_{10} + 465 \beta_{9} - 93 \beta_{8} + 795 \beta_{7} + 780 \beta_{6} + 809 \beta_{5} + 225 \beta_{4} - 459 \beta_{3} - 9 \beta_{2} + 1763 \beta_{1} + 576\)
\(\nu^{8}\)\(=\)\(-1051 \beta_{11} + 4141 \beta_{10} + 1808 \beta_{9} - 2429 \beta_{8} + 1441 \beta_{7} + 5432 \beta_{6} + 1061 \beta_{5} + 1011 \beta_{4} - 3568 \beta_{3} - 225 \beta_{2} + 3995 \beta_{1} + 11067\)
\(\nu^{9}\)\(=\)\(-4900 \beta_{11} + 13689 \beta_{10} + 8697 \beta_{9} - 2068 \beta_{8} + 12789 \beta_{7} + 14973 \beta_{6} + 13495 \beta_{5} + 4850 \beta_{4} - 8212 \beta_{3} + 193 \beta_{2} + 25829 \beta_{1} + 11256\)
\(\nu^{10}\)\(=\)\(-16949 \beta_{11} + 66601 \beta_{10} + 34473 \beta_{9} - 32858 \beta_{8} + 30539 \beta_{7} + 87444 \beta_{6} + 27494 \beta_{5} + 20849 \beta_{4} - 54040 \beta_{3} - 667 \beta_{2} + 68339 \beta_{1} + 151126\)
\(\nu^{11}\)\(=\)\(-75864 \beta_{11} + 233383 \beta_{10} + 155287 \beta_{9} - 40971 \beta_{8} + 207800 \beta_{7} + 269383 \beta_{6} + 219090 \beta_{5} + 93810 \beta_{4} - 143006 \beta_{3} + 9222 \beta_{2} + 392614 \beta_{1} + 215351\)

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
4.06732
3.27568
2.95427
2.60206
0.761360
−0.00512154
−0.539392
−0.815716
−1.11280
−1.28984
−3.43402
−3.46379
1.00000 0 1.00000 −4.06732 0 1.57160 1.00000 0 −4.06732
1.2 1.00000 0 1.00000 −3.27568 0 4.39969 1.00000 0 −3.27568
1.3 1.00000 0 1.00000 −2.95427 0 0.794893 1.00000 0 −2.95427
1.4 1.00000 0 1.00000 −2.60206 0 −4.35416 1.00000 0 −2.60206
1.5 1.00000 0 1.00000 −0.761360 0 1.23946 1.00000 0 −0.761360
1.6 1.00000 0 1.00000 0.00512154 0 −2.98814 1.00000 0 0.00512154
1.7 1.00000 0 1.00000 0.539392 0 0.739124 1.00000 0 0.539392
1.8 1.00000 0 1.00000 0.815716 0 −4.13057 1.00000 0 0.815716
1.9 1.00000 0 1.00000 1.11280 0 −0.851837 1.00000 0 1.11280
1.10 1.00000 0 1.00000 1.28984 0 3.20787 1.00000 0 1.28984
1.11 1.00000 0 1.00000 3.43402 0 −1.90397 1.00000 0 3.43402
1.12 1.00000 0 1.00000 3.46379 0 −3.72396 1.00000 0 3.46379
\(n\): 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.n yes 12
3.b odd 2 1 8046.2.a.k 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8046.2.a.k 12 3.b odd 2 1
8046.2.a.n 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

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{12} \)
$3$ 1
$5$ \( 1 + 3 T + 29 T^{2} + 83 T^{3} + 434 T^{4} + 1119 T^{5} + 4424 T^{6} + 10464 T^{7} + 34918 T^{8} + 77730 T^{9} + 228728 T^{10} + 473919 T^{11} + 1254877 T^{12} + 2369595 T^{13} + 5718200 T^{14} + 9716250 T^{15} + 21823750 T^{16} + 32700000 T^{17} + 69125000 T^{18} + 87421875 T^{19} + 169531250 T^{20} + 162109375 T^{21} + 283203125 T^{22} + 146484375 T^{23} + 244140625 T^{24} \)
$7$ \( 1 + 6 T + 53 T^{2} + 242 T^{3} + 1308 T^{4} + 4846 T^{5} + 20325 T^{6} + 64568 T^{7} + 230705 T^{8} + 653407 T^{9} + 2089904 T^{10} + 5407980 T^{11} + 15862987 T^{12} + 37855860 T^{13} + 102405296 T^{14} + 224118601 T^{15} + 553922705 T^{16} + 1085194376 T^{17} + 2391215925 T^{18} + 3990889378 T^{19} + 7540359708 T^{20} + 9765572894 T^{21} + 14971188197 T^{22} + 11863960458 T^{23} + 13841287201 T^{24} \)
$11$ \( 1 + 14 T + 143 T^{2} + 1056 T^{3} + 6775 T^{4} + 37077 T^{5} + 185806 T^{6} + 838016 T^{7} + 3549961 T^{8} + 13913987 T^{9} + 52145040 T^{10} + 183959742 T^{11} + 626990511 T^{12} + 2023557162 T^{13} + 6309549840 T^{14} + 18519516697 T^{15} + 51974979001 T^{16} + 134963314816 T^{17} + 329166663166 T^{18} + 722525839167 T^{19} + 1452281418775 T^{20} + 2489992761696 T^{21} + 3709051717943 T^{22} + 3994363388554 T^{23} + 3138428376721 T^{24} \)
$13$ \( 1 + 3 T + 55 T^{2} + 141 T^{3} + 1951 T^{4} + 4476 T^{5} + 50093 T^{6} + 103950 T^{7} + 1024442 T^{8} + 1954957 T^{9} + 17370533 T^{10} + 30386550 T^{11} + 245221481 T^{12} + 395025150 T^{13} + 2935620077 T^{14} + 4295040529 T^{15} + 29259087962 T^{16} + 38595907350 T^{17} + 241789343237 T^{18} + 280862362092 T^{19} + 1591490636671 T^{20} + 1495234411593 T^{21} + 7582217051695 T^{22} + 5376481182111 T^{23} + 23298085122481 T^{24} \)
$17$ \( 1 + 8 T + 129 T^{2} + 717 T^{3} + 6947 T^{4} + 28060 T^{5} + 211253 T^{6} + 567037 T^{7} + 3951482 T^{8} + 4417300 T^{9} + 48836470 T^{10} - 44174689 T^{11} + 607256915 T^{12} - 750969713 T^{13} + 14113739830 T^{14} + 21702194900 T^{15} + 330031728122 T^{16} + 805111453709 T^{17} + 5099133863957 T^{18} + 11514103164380 T^{19} + 48460586942627 T^{20} + 85027507448349 T^{21} + 260063213157921 T^{22} + 274175170461064 T^{23} + 582622237229761 T^{24} \)
$19$ \( 1 + 4 T + 81 T^{2} + 258 T^{3} + 3843 T^{4} + 10693 T^{5} + 125306 T^{6} + 298422 T^{7} + 3198555 T^{8} + 6708439 T^{9} + 68590826 T^{10} + 131253174 T^{11} + 1346523227 T^{12} + 2493810306 T^{13} + 24761288186 T^{14} + 46013183101 T^{15} + 416838886155 T^{16} + 738922415778 T^{17} + 5895131164586 T^{18} + 9558170505127 T^{19} + 65267832766563 T^{20} + 83253426026982 T^{21} + 496616366881881 T^{22} + 465961035592876 T^{23} + 2213314919066161 T^{24} \)
$23$ \( 1 + 13 T + 142 T^{2} + 1143 T^{3} + 8673 T^{4} + 59528 T^{5} + 394597 T^{6} + 2406983 T^{7} + 14176685 T^{8} + 77968405 T^{9} + 418881201 T^{10} + 2133677593 T^{11} + 10521531035 T^{12} + 49074584639 T^{13} + 221588155329 T^{14} + 948641583635 T^{15} + 3967217707085 T^{16} + 15492168183169 T^{17} + 58414517691733 T^{18} + 202682449209016 T^{19} + 679191175342113 T^{20} + 2058717492052209 T^{21} + 5882564592338158 T^{22} + 12386526852881051 T^{23} + 21914624432020321 T^{24} \)
$29$ \( 1 + 23 T + 395 T^{2} + 4877 T^{3} + 51758 T^{4} + 469685 T^{5} + 3858600 T^{6} + 28750460 T^{7} + 199074860 T^{8} + 1286588538 T^{9} + 7851838688 T^{10} + 45444927621 T^{11} + 250591717289 T^{12} + 1317902901009 T^{13} + 6603396336608 T^{14} + 31378607853282 T^{15} + 140801866055660 T^{16} + 589704968878540 T^{17} + 2295185266410600 T^{18} + 8102008154192665 T^{19} + 25891753842035438 T^{20} + 70751350924313113 T^{21} + 166179357153579395 T^{22} + 280611724611234067 T^{23} + 353814783205469041 T^{24} \)
$31$ \( 1 + 14 T + 160 T^{2} + 1419 T^{3} + 12651 T^{4} + 93889 T^{5} + 681265 T^{6} + 4370044 T^{7} + 28168003 T^{8} + 165872107 T^{9} + 1001880791 T^{10} + 5534240029 T^{11} + 31712932293 T^{12} + 171561440899 T^{13} + 962807440151 T^{14} + 4941495939637 T^{15} + 26013742298563 T^{16} + 125110649552644 T^{17} + 604625195236465 T^{18} + 2583131826267679 T^{19} + 10789924514666091 T^{20} + 37517823845992149 T^{21} + 131140525916928160 T^{22} + 355718676549667634 T^{23} + 787662783788549761 T^{24} \)
$37$ \( 1 + 19 T + 435 T^{2} + 6117 T^{3} + 83873 T^{4} + 934343 T^{5} + 9736417 T^{6} + 89608258 T^{7} + 769486826 T^{8} + 6007577417 T^{9} + 44012372877 T^{10} + 296078381605 T^{11} + 1877232834877 T^{12} + 10954900119385 T^{13} + 60252938468613 T^{14} + 304301818903301 T^{15} + 1442142199302986 T^{16} + 6213791189596906 T^{17} + 24980982225936553 T^{18} + 88698934876078619 T^{19} + 294602189238716033 T^{20} + 794975962326486009 T^{21} + 2091734202001764315 T^{22} + 3380434813809747847 T^{23} + 6582952005840035281 T^{24} \)
$41$ \( 1 + 30 T + 630 T^{2} + 9922 T^{3} + 132934 T^{4} + 1536253 T^{5} + 15942916 T^{6} + 149623872 T^{7} + 1294936134 T^{8} + 10367083477 T^{9} + 77655276302 T^{10} + 544482255134 T^{11} + 3596802825854 T^{12} + 22323772460494 T^{13} + 130538519463662 T^{14} + 714509760318317 T^{15} + 3659180024947974 T^{16} + 17334853388830272 T^{17} + 75730512905506756 T^{18} + 299191837512507893 T^{19} + 1061468050407971014 T^{20} + 3248283553056881042 T^{21} + 8456275365396012630 T^{22} + 16509870951487453230 T^{23} + 22563490300366186081 T^{24} \)
$43$ \( 1 + 15 T + 246 T^{2} + 2619 T^{3} + 30033 T^{4} + 285144 T^{5} + 2655040 T^{6} + 21909881 T^{7} + 177479979 T^{8} + 1346416517 T^{9} + 9873603109 T^{10} + 68255483876 T^{11} + 456026671185 T^{12} + 2934985806668 T^{13} + 18256292148541 T^{14} + 107049538017119 T^{15} + 606768729685179 T^{16} + 3220937492125283 T^{17} + 16783471749616960 T^{18} + 77507446045494408 T^{19} + 351031718937190833 T^{20} + 1316290050662591817 T^{21} + 5316424649067925254 T^{22} + 13939406092068340605 T^{23} + 39959630797262576401 T^{24} \)
$47$ \( 1 - T + 258 T^{2} + 686 T^{3} + 32746 T^{4} + 173833 T^{5} + 3159377 T^{6} + 19666499 T^{7} + 250177506 T^{8} + 1522093066 T^{9} + 16078008221 T^{10} + 89911618542 T^{11} + 839979747933 T^{12} + 4225846071474 T^{13} + 35516320160189 T^{14} + 158028268391318 T^{15} + 1220786422655586 T^{16} + 4510413350820493 T^{17} + 34055604988490033 T^{18} + 88067816899444679 T^{19} + 779724393026025706 T^{20} + 767723504548498162 T^{21} + 13570576116844152642 T^{22} - 2472159215084012303 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \)
$53$ \( 1 + 16 T + 395 T^{2} + 4441 T^{3} + 67581 T^{4} + 623297 T^{5} + 7577923 T^{6} + 61778296 T^{7} + 651876542 T^{8} + 4808080523 T^{9} + 45326804659 T^{10} + 304835394648 T^{11} + 2617377641701 T^{12} + 16156275916344 T^{13} + 127322994287131 T^{14} + 715812604022671 T^{15} + 5143619468996702 T^{16} + 25835404952419928 T^{17} + 167959821979755067 T^{18} + 732193929326982589 T^{19} + 4207572137690187741 T^{20} + 14654250111193272653 T^{21} + 69080550794377654355 T^{22} + \)\(14\!\cdots\!52\)\( T^{23} + \)\(49\!\cdots\!41\)\( T^{24} \)
$59$ \( 1 + 26 T + 739 T^{2} + 13160 T^{3} + 227786 T^{4} + 3166215 T^{5} + 41699950 T^{6} + 479752210 T^{7} + 5194514199 T^{8} + 50976622899 T^{9} + 470037330734 T^{10} + 3994115219949 T^{11} + 31880627164561 T^{12} + 235652797976991 T^{13} + 1636199948285054 T^{14} + 10469527834373721 T^{15} + 62943803768908839 T^{16} + 342986512427950790 T^{17} + 1758926143803017950 T^{18} + 7879605661006190085 T^{19} + 33445918060137863306 T^{20} + \)\(11\!\cdots\!40\)\( T^{21} + \)\(37\!\cdots\!39\)\( T^{22} + \)\(78\!\cdots\!34\)\( T^{23} + \)\(17\!\cdots\!81\)\( T^{24} \)
$61$ \( 1 + 16 T + 458 T^{2} + 6342 T^{3} + 102854 T^{4} + 1253623 T^{5} + 15283299 T^{6} + 164257682 T^{7} + 1687820982 T^{8} + 15974386386 T^{9} + 145384440141 T^{10} + 1216682940343 T^{11} + 9937155937675 T^{12} + 74217659360923 T^{13} + 540975501764661 T^{14} + 3625882196280666 T^{15} + 23369300953235862 T^{16} + 138731430628034282 T^{17} + 787401285951096939 T^{18} + 3939814702321154083 T^{19} + 19717863971022339974 T^{20} + 74164274520754122222 T^{21} + \)\(32\!\cdots\!58\)\( T^{22} + \)\(69\!\cdots\!76\)\( T^{23} + \)\(26\!\cdots\!21\)\( T^{24} \)
$67$ \( 1 + 39 T + 1249 T^{2} + 27317 T^{3} + 524434 T^{4} + 8280215 T^{5} + 118933121 T^{6} + 1498064992 T^{7} + 17521520034 T^{8} + 185053888185 T^{9} + 1834186332603 T^{10} + 16629487163152 T^{11} + 142069839742145 T^{12} + 1114175639931184 T^{13} + 8233662447054867 T^{14} + 55657362572185155 T^{15} + 353078270309058114 T^{16} + 2022575157616954144 T^{17} + 10758497711969919449 T^{18} + 50183995145069584445 T^{19} + \)\(21\!\cdots\!94\)\( T^{20} + \)\(74\!\cdots\!99\)\( T^{21} + \)\(22\!\cdots\!01\)\( T^{22} + \)\(47\!\cdots\!37\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \)
$71$ \( 1 + 15 T + 575 T^{2} + 6821 T^{3} + 150303 T^{4} + 1537962 T^{5} + 25378542 T^{6} + 235061520 T^{7} + 3179437558 T^{8} + 26943753841 T^{9} + 311687255347 T^{10} + 2405044514488 T^{11} + 24550756948835 T^{12} + 170758160528648 T^{13} + 1571215454204227 T^{14} + 9643465880986151 T^{15} + 80794852983314998 T^{16} + 424104893674673520 T^{17} + 3250998435701023182 T^{18} + 13987949189039339142 T^{19} + 97058693006831615583 T^{20} + \)\(31\!\cdots\!51\)\( T^{21} + \)\(18\!\cdots\!75\)\( T^{22} + \)\(34\!\cdots\!65\)\( T^{23} + \)\(16\!\cdots\!41\)\( T^{24} \)
$73$ \( 1 + 2 T + 430 T^{2} + 1781 T^{3} + 97769 T^{4} + 544940 T^{5} + 15643484 T^{6} + 97962235 T^{7} + 1933240123 T^{8} + 12369386893 T^{9} + 190995008857 T^{10} + 1175619536159 T^{11} + 15381529083285 T^{12} + 85820226139607 T^{13} + 1017812402198953 T^{14} + 4811901780954181 T^{15} + 54900618923823643 T^{16} + 203082726565290355 T^{17} + 2367394547604350876 T^{18} + 6020169348996719180 T^{19} + 78846796724392405289 T^{20} + \)\(10\!\cdots\!53\)\( T^{21} + \)\(18\!\cdots\!70\)\( T^{22} + \)\(62\!\cdots\!54\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \)
$79$ \( 1 + 13 T + 642 T^{2} + 7931 T^{3} + 208102 T^{4} + 2350248 T^{5} + 43881217 T^{6} + 448689818 T^{7} + 6656264326 T^{8} + 61306315608 T^{9} + 764644126911 T^{10} + 6302701751558 T^{11} + 68251618197003 T^{12} + 497913438373082 T^{13} + 4772143996051551 T^{14} + 30226404541052712 T^{15} + 259262034655110406 T^{16} + 1380643875643045382 T^{17} + 10666973385694849057 T^{18} + 45133948686902217432 T^{19} + \)\(31\!\cdots\!22\)\( T^{20} + \)\(95\!\cdots\!89\)\( T^{21} + \)\(60\!\cdots\!42\)\( T^{22} + \)\(97\!\cdots\!27\)\( T^{23} + \)\(59\!\cdots\!41\)\( T^{24} \)
$83$ \( 1 + 6 T + 499 T^{2} + 1996 T^{3} + 124008 T^{4} + 380973 T^{5} + 21623828 T^{6} + 57244490 T^{7} + 2928997939 T^{8} + 6884893093 T^{9} + 321051017318 T^{10} + 680931897969 T^{11} + 29175527008207 T^{12} + 56517347531427 T^{13} + 2211720458303702 T^{14} + 3936692366967191 T^{15} + 139005324397400419 T^{16} + 225488372697807070 T^{17} + 7069702399987036532 T^{18} + 10338102753671167071 T^{19} + \)\(27\!\cdots\!28\)\( T^{20} + \)\(37\!\cdots\!88\)\( T^{21} + \)\(77\!\cdots\!51\)\( T^{22} + \)\(77\!\cdots\!02\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \)
$89$ \( 1 + 18 T + 746 T^{2} + 10931 T^{3} + 258147 T^{4} + 3211901 T^{5} + 56563606 T^{6} + 617030340 T^{7} + 8993867452 T^{8} + 87725667134 T^{9} + 1111985478215 T^{10} + 9767955242077 T^{11} + 110148098602543 T^{12} + 869348016544853 T^{13} + 8808036972941015 T^{14} + 61843875833788846 T^{15} + 564295399195439932 T^{16} + 3445534100396682660 T^{17} + 28111053931289365366 T^{18} + \)\(14\!\cdots\!29\)\( T^{19} + \)\(10\!\cdots\!07\)\( T^{20} + \)\(38\!\cdots\!79\)\( T^{21} + \)\(23\!\cdots\!46\)\( T^{22} + \)\(49\!\cdots\!02\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} \)
$97$ \( 1 - 19 T + 267 T^{2} - 559 T^{3} - 14104 T^{4} + 276882 T^{5} + 496472 T^{6} - 28917735 T^{7} + 500203798 T^{8} - 2112028751 T^{9} + 11279268340 T^{10} + 94022750560 T^{11} + 142599766015 T^{12} + 9120206804320 T^{13} + 106126635811060 T^{14} - 1927591616261423 T^{15} + 44282682590409238 T^{16} - 248326429906757895 T^{17} + 413547277231110488 T^{18} + 22371590602868883666 T^{19} - \)\(11\!\cdots\!44\)\( T^{20} - \)\(42\!\cdots\!03\)\( T^{21} + \)\(19\!\cdots\!83\)\( T^{22} - \)\(13\!\cdots\!07\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \)
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