Properties

Label 7623.2.a.cy
Level 7623
Weight 2
Character orbit 7623.a
Self dual yes
Analytic conductor 60.870
Analytic rank 0
Dimension 10
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 7623.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.8699614608\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 2 + \beta_{2} ) q^{4} + \beta_{6} q^{5} + q^{7} + ( 1 + 2 \beta_{1} + \beta_{3} + \beta_{4} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 2 + \beta_{2} ) q^{4} + \beta_{6} q^{5} + q^{7} + ( 1 + 2 \beta_{1} + \beta_{3} + \beta_{4} ) q^{8} + ( -\beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{10} + ( -\beta_{3} + \beta_{5} - \beta_{9} ) q^{13} + \beta_{1} q^{14} + ( 4 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{5} - \beta_{7} + \beta_{8} ) q^{16} + ( -1 + \beta_{1} - \beta_{2} - \beta_{5} ) q^{17} + ( 1 + \beta_{3} + \beta_{4} - \beta_{7} ) q^{19} + ( 2 + 3 \beta_{3} - \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{9} ) q^{20} + ( \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} ) q^{23} + ( 3 - \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{9} ) q^{25} + ( -3 - \beta_{1} - 4 \beta_{3} + 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{26} + ( 2 + \beta_{2} ) q^{28} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{8} - \beta_{9} ) q^{29} + ( 2 + \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{31} + ( 7 + 3 \beta_{1} + 3 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{8} + 2 \beta_{9} ) q^{32} + ( 4 - 3 \beta_{1} + 3 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{9} ) q^{34} + \beta_{6} q^{35} + ( 3 + \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{37} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{5} + 2 \beta_{8} + \beta_{9} ) q^{38} + ( 1 + \beta_{1} - \beta_{2} + 3 \beta_{3} - 5 \beta_{5} + \beta_{8} + 2 \beta_{9} ) q^{40} + ( -2 + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{9} ) q^{41} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{8} - \beta_{9} ) q^{43} + ( 1 + \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{6} + 2 \beta_{7} + \beta_{9} ) q^{46} + ( -1 + \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{9} ) q^{47} + q^{49} + ( -4 + 3 \beta_{1} - \beta_{2} - 6 \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{50} + ( -7 - 3 \beta_{1} + 2 \beta_{2} - 7 \beta_{3} + 5 \beta_{5} - 3 \beta_{6} - \beta_{7} - 2 \beta_{8} - 2 \beta_{9} ) q^{52} + ( \beta_{2} - \beta_{3} + 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{53} + ( 1 + 2 \beta_{1} + \beta_{3} + \beta_{4} ) q^{56} + ( 2 + 2 \beta_{1} + 2 \beta_{4} + 3 \beta_{5} + \beta_{6} ) q^{58} + ( -1 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{6} + \beta_{7} + \beta_{9} ) q^{59} + ( 2 \beta_{1} + \beta_{3} + \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - \beta_{9} ) q^{61} + ( 2 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} + \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{9} ) q^{62} + ( 6 + 6 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - 4 \beta_{5} + 2 \beta_{6} + \beta_{8} + \beta_{9} ) q^{64} + ( 3 - 3 \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} + 3 \beta_{6} + \beta_{7} + 2 \beta_{9} ) q^{65} + ( 5 - \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{67} + ( -8 + 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - 3 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} ) q^{68} + ( -\beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{70} + ( -5 - \beta_{1} + \beta_{2} - 5 \beta_{3} - 2 \beta_{5} - \beta_{8} - \beta_{9} ) q^{71} + ( 1 + 3 \beta_{1} + 2 \beta_{3} - 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{9} ) q^{73} + ( 3 + 2 \beta_{1} + 3 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{9} ) q^{74} + ( 5 + 2 \beta_{1} + 5 \beta_{3} + \beta_{4} + 2 \beta_{5} + 4 \beta_{6} - \beta_{7} + 2 \beta_{9} ) q^{76} + ( 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{79} + ( 7 + \beta_{1} - 2 \beta_{2} + 12 \beta_{3} - 2 \beta_{5} + 4 \beta_{6} + \beta_{7} + 3 \beta_{9} ) q^{80} + ( -3 - 3 \beta_{1} + \beta_{2} - 5 \beta_{3} + \beta_{5} - 2 \beta_{6} - \beta_{8} - 2 \beta_{9} ) q^{82} + ( -2 - 4 \beta_{3} - 2 \beta_{5} - 2 \beta_{7} ) q^{83} + ( -2 + \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} + 3 \beta_{7} + \beta_{8} ) q^{85} + ( 4 + 2 \beta_{3} + \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{86} + ( 2 - \beta_{2} + 2 \beta_{3} + 3 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} ) q^{89} + ( -\beta_{3} + \beta_{5} - \beta_{9} ) q^{91} + ( 5 - \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} - 3 \beta_{5} - \beta_{6} - 3 \beta_{7} - \beta_{9} ) q^{92} + ( 2 - 2 \beta_{2} + 5 \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{7} + \beta_{9} ) q^{94} + ( 6 - 2 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} - \beta_{5} - 2 \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{95} + ( 2 - 2 \beta_{1} - \beta_{3} - \beta_{5} + \beta_{9} ) q^{97} + \beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 18q^{4} - 5q^{5} + 10q^{7} + 3q^{8} + O(q^{10}) \) \( 10q + 18q^{4} - 5q^{5} + 10q^{7} + 3q^{8} - 6q^{10} + 6q^{13} + 38q^{16} - 8q^{17} - 7q^{20} + 31q^{25} - q^{26} + 18q^{28} + 14q^{29} + 26q^{31} + 41q^{32} + 21q^{34} - 5q^{35} + 24q^{37} - 8q^{38} - 5q^{40} - 19q^{41} - 6q^{43} - q^{46} - 15q^{47} + 10q^{49} + q^{50} - 25q^{52} + q^{53} + 3q^{56} + 11q^{58} - 23q^{59} - 11q^{62} + 53q^{64} + 29q^{65} + 38q^{67} - 87q^{68} - 6q^{70} - 26q^{71} - q^{73} + 39q^{74} - 2q^{76} + 5q^{79} - 6q^{80} + 5q^{82} - 6q^{83} - q^{85} + 41q^{86} + 9q^{89} + 6q^{91} + 48q^{92} + 42q^{95} + 24q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 19 x^{8} - x^{7} + 124 x^{6} + 6 x^{5} - 316 x^{4} + 17 x^{3} + 253 x^{2} - 70 x - 11\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\((\)\( -25 \nu^{9} + 31 \nu^{8} + 558 \nu^{7} - 505 \nu^{6} - 4093 \nu^{5} + 2537 \nu^{4} + 11069 \nu^{3} - 4152 \nu^{2} - 8301 \nu + 385 \)\()/2024\)
\(\beta_{4}\)\(=\)\((\)\( 25 \nu^{9} - 31 \nu^{8} - 558 \nu^{7} + 505 \nu^{6} + 4093 \nu^{5} - 2537 \nu^{4} - 9045 \nu^{3} + 4152 \nu^{2} - 3843 \nu - 2409 \)\()/2024\)
\(\beta_{5}\)\(=\)\((\)\( -31 \nu^{9} - 83 \nu^{8} + 530 \nu^{7} + 993 \nu^{6} - 2687 \nu^{5} - 3169 \nu^{4} + 3727 \nu^{3} + 1976 \nu^{2} + 1365 \nu + 275 \)\()/2024\)
\(\beta_{6}\)\(=\)\((\)\( -97 \nu^{9} - 325 \nu^{8} + 2246 \nu^{7} + 5327 \nu^{6} - 16569 \nu^{5} - 27479 \nu^{4} + 43393 \nu^{3} + 43072 \nu^{2} - 29941 \nu - 5995 \)\()/2024\)
\(\beta_{7}\)\(=\)\((\)\( -229 \nu^{9} + 203 \nu^{8} + 3654 \nu^{7} - 2197 \nu^{6} - 19033 \nu^{5} + 4861 \nu^{4} + 34681 \nu^{3} + 5848 \nu^{2} - 13617 \nu - 7403 \)\()/2024\)
\(\beta_{8}\)\(=\)\((\)\( -285 \nu^{9} + 151 \nu^{8} + 4742 \nu^{7} - 1709 \nu^{6} - 25813 \nu^{5} + 6253 \nu^{4} + 49477 \nu^{3} - 12520 \nu^{2} - 24601 \nu + 9449 \)\()/2024\)
\(\beta_{9}\)\(=\)\((\)\( 70 \nu^{9} + 65 \nu^{8} - 1360 \nu^{7} - 1116 \nu^{6} + 8981 \nu^{5} + 5850 \nu^{4} - 22543 \nu^{3} - 8412 \nu^{2} + 16260 \nu - 1331 \)\()/506\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{4} + \beta_{3} + 6 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{8} - \beta_{7} - \beta_{5} - \beta_{3} + 8 \beta_{2} + 2 \beta_{1} + 24\)
\(\nu^{5}\)\(=\)\(2 \beta_{9} + \beta_{8} + 2 \beta_{6} + \beta_{5} + 9 \beta_{4} + 11 \beta_{3} + 39 \beta_{1} + 15\)
\(\nu^{6}\)\(=\)\(\beta_{9} + 11 \beta_{8} - 10 \beta_{7} + 2 \beta_{6} - 14 \beta_{5} + \beta_{4} - 12 \beta_{3} + 58 \beta_{2} + 26 \beta_{1} + 158\)
\(\nu^{7}\)\(=\)\(26 \beta_{9} + 13 \beta_{8} - \beta_{7} + 27 \beta_{6} + 10 \beta_{5} + 69 \beta_{4} + 104 \beta_{3} + 3 \beta_{2} + 266 \beta_{1} + 157\)
\(\nu^{8}\)\(=\)\(18 \beta_{9} + 97 \beta_{8} - 80 \beta_{7} + 30 \beta_{6} - 145 \beta_{5} + 16 \beta_{4} - 92 \beta_{3} + 412 \beta_{2} + 257 \beta_{1} + 1095\)
\(\nu^{9}\)\(=\)\(255 \beta_{9} + 126 \beta_{8} - 21 \beta_{7} + 272 \beta_{6} + 61 \beta_{5} + 509 \beta_{4} + 909 \beta_{3} + 52 \beta_{2} + 1873 \beta_{1} + 1444\)

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
−2.65195
−2.39396
−2.09767
−1.33330
−0.112481
0.473713
0.871604
1.80545
2.63994
2.79866
−2.65195 0 5.03286 −1.71311 0 1.00000 −8.04301 0 4.54308
1.2 −2.39396 0 3.73106 3.93829 0 1.00000 −4.14409 0 −9.42812
1.3 −2.09767 0 2.40021 −3.15947 0 1.00000 −0.839503 0 6.62751
1.4 −1.33330 0 −0.222305 0.873210 0 1.00000 2.96300 0 −1.16425
1.5 −0.112481 0 −1.98735 −1.06131 0 1.00000 0.448501 0 0.119378
1.6 0.473713 0 −1.77560 −3.75881 0 1.00000 −1.78855 0 −1.78060
1.7 0.871604 0 −1.24031 4.06436 0 1.00000 −2.82426 0 3.54252
1.8 1.80545 0 1.25966 −2.77715 0 1.00000 −1.33666 0 −5.01400
1.9 2.63994 0 4.96928 −3.08369 0 1.00000 7.83871 0 −8.14075
1.10 2.79866 0 5.83249 1.67767 0 1.00000 10.7258 0 4.69523
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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 7623.2.a.cy 10
3.b odd 2 1 2541.2.a.br 10
11.b odd 2 1 7623.2.a.cx 10
11.d odd 10 2 693.2.m.j 20
33.d even 2 1 2541.2.a.bq 10
33.f even 10 2 231.2.j.g 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.j.g 20 33.f even 10 2
693.2.m.j 20 11.d odd 10 2
2541.2.a.bq 10 33.d even 2 1
2541.2.a.br 10 3.b odd 2 1
7623.2.a.cx 10 11.b odd 2 1
7623.2.a.cy 10 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(-1\)
\(11\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

\(T_{2}^{10} - \cdots\)
\(T_{5}^{10} + \cdots\)
\(T_{13}^{10} - \cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} - T^{3} - 8 T^{5} + 4 T^{6} - 7 T^{7} + 13 T^{8} - 8 T^{9} + 41 T^{10} - 16 T^{11} + 52 T^{12} - 56 T^{13} + 64 T^{14} - 256 T^{15} - 128 T^{17} + 256 T^{18} + 1024 T^{20} \)
$3$ \( \)
$5$ \( 1 + 5 T + 22 T^{2} + 46 T^{3} + 113 T^{4} + 108 T^{5} + 391 T^{6} + 538 T^{7} + 4002 T^{8} + 9087 T^{9} + 31806 T^{10} + 45435 T^{11} + 100050 T^{12} + 67250 T^{13} + 244375 T^{14} + 337500 T^{15} + 1765625 T^{16} + 3593750 T^{17} + 8593750 T^{18} + 9765625 T^{19} + 9765625 T^{20} \)
$7$ \( ( 1 - T )^{10} \)
$11$ \( \)
$13$ \( 1 - 6 T + 65 T^{2} - 368 T^{3} + 2385 T^{4} - 10918 T^{5} + 56756 T^{6} - 221994 T^{7} + 984630 T^{8} - 3474482 T^{9} + 13929014 T^{10} - 45168266 T^{11} + 166402470 T^{12} - 487720818 T^{13} + 1621008116 T^{14} - 4053776974 T^{15} + 11511939465 T^{16} - 23091454256 T^{17} + 53022496865 T^{18} - 63626996238 T^{19} + 137858491849 T^{20} \)
$17$ \( 1 + 8 T + 136 T^{2} + 954 T^{3} + 9093 T^{4} + 54096 T^{5} + 378365 T^{6} + 1910556 T^{7} + 10713786 T^{8} + 46048194 T^{9} + 215035238 T^{10} + 782819298 T^{11} + 3096284154 T^{12} + 9386561628 T^{13} + 31601423165 T^{14} + 76808584272 T^{15} + 219482914917 T^{16} + 391463094042 T^{17} + 948703011976 T^{18} + 948703011976 T^{19} + 2015993900449 T^{20} \)
$19$ \( 1 + 69 T^{2} + 136 T^{3} + 2758 T^{4} + 8362 T^{5} + 89118 T^{6} + 289638 T^{7} + 2274617 T^{8} + 7545660 T^{9} + 46843450 T^{10} + 143367540 T^{11} + 821136737 T^{12} + 1986627042 T^{13} + 11613946878 T^{14} + 20705139838 T^{15} + 129752539798 T^{16} + 121566556504 T^{17} + 1171865849829 T^{18} + 6131066257801 T^{20} \)
$23$ \( 1 + 124 T^{2} - 38 T^{3} + 8330 T^{4} - 3634 T^{5} + 381506 T^{6} - 180342 T^{7} + 12965000 T^{8} - 5936040 T^{9} + 338330610 T^{10} - 136528920 T^{11} + 6858485000 T^{12} - 2194221114 T^{13} + 106761020546 T^{14} - 23389670462 T^{15} + 1233138955370 T^{16} - 129383366986 T^{17} + 9710562174844 T^{18} + 41426511213649 T^{20} \)
$29$ \( 1 - 14 T + 200 T^{2} - 1928 T^{3} + 18575 T^{4} - 144068 T^{5} + 1110481 T^{6} - 7337936 T^{7} + 48191364 T^{8} - 277661302 T^{9} + 1587859174 T^{10} - 8052177758 T^{11} + 40528937124 T^{12} - 178964921104 T^{13} + 785422112161 T^{14} - 2955000214132 T^{15} + 11048843187575 T^{16} - 33257761523752 T^{17} + 100049282592200 T^{18} - 203100043662166 T^{19} + 420707233300201 T^{20} \)
$31$ \( 1 - 26 T + 482 T^{2} - 6400 T^{3} + 71453 T^{4} - 670846 T^{5} + 5624329 T^{6} - 41976980 T^{7} + 287918626 T^{8} - 1800633596 T^{9} + 10465823578 T^{10} - 55819641476 T^{11} + 276689799586 T^{12} - 1250536211180 T^{13} + 5194185942409 T^{14} - 19205751431746 T^{15} + 63414800518493 T^{16} - 176080730310400 T^{17} + 411093480046562 T^{18} - 687430176177446 T^{19} + 819628286980801 T^{20} \)
$37$ \( 1 - 24 T + 440 T^{2} - 5840 T^{3} + 65456 T^{4} - 634664 T^{5} + 5522032 T^{6} - 43933344 T^{7} + 323265704 T^{8} - 2198445368 T^{9} + 13921174566 T^{10} - 81342478616 T^{11} + 442550748776 T^{12} - 2225355673632 T^{13} + 10349177015152 T^{14} - 44010113125448 T^{15} + 167942187827504 T^{16} - 554402162456720 T^{17} + 1545490959725240 T^{18} - 3119081755081848 T^{19} + 4808584372417849 T^{20} \)
$41$ \( 1 + 19 T + 484 T^{2} + 6388 T^{3} + 94475 T^{4} + 963532 T^{5} + 10457893 T^{6} + 86623352 T^{7} + 748989936 T^{8} + 5154844493 T^{9} + 36776727094 T^{10} + 211348624213 T^{11} + 1259052082416 T^{12} + 5970168043192 T^{13} + 29551506181573 T^{14} + 111631157061932 T^{15} + 448766098168475 T^{16} + 1244090301551828 T^{17} + 3864703810894564 T^{18} + 6220256753485259 T^{19} + 13422659310152401 T^{20} \)
$43$ \( 1 + 6 T + 312 T^{2} + 1828 T^{3} + 47319 T^{4} + 259736 T^{5} + 4576717 T^{6} + 22838020 T^{7} + 311402464 T^{8} + 1377461130 T^{9} + 15540959670 T^{10} + 59230828590 T^{11} + 575783155936 T^{12} + 1815782456140 T^{13} + 15646884656317 T^{14} + 38183384951048 T^{15} + 299120578115631 T^{16} + 496884421103596 T^{17} + 3646718486611512 T^{18} + 3015555671621058 T^{19} + 21611482313284249 T^{20} \)
$47$ \( 1 + 15 T + 333 T^{2} + 4161 T^{3} + 56109 T^{4} + 576028 T^{5} + 5980092 T^{6} + 51937604 T^{7} + 445599026 T^{8} + 3323250890 T^{9} + 24323199470 T^{10} + 156192791830 T^{11} + 984328248434 T^{12} + 5392317860092 T^{13} + 29180941310652 T^{14} + 132109145692196 T^{15} + 604810992894861 T^{16} + 2108058804246543 T^{17} + 7929158458366413 T^{18} + 16786957096541505 T^{19} + 52599132235830049 T^{20} \)
$53$ \( 1 - T + 254 T^{2} + 378 T^{3} + 28817 T^{4} + 120372 T^{5} + 2050339 T^{6} + 14991722 T^{7} + 109654718 T^{8} + 1160429241 T^{9} + 5512758766 T^{10} + 61502749773 T^{11} + 308020102862 T^{12} + 2231922596194 T^{13} + 16178160923059 T^{14} + 50339027883396 T^{15} + 638710394654393 T^{16} + 444040810858386 T^{17} + 15813961364485694 T^{18} - 3299763591802133 T^{19} + 174887470365513049 T^{20} \)
$59$ \( 1 + 23 T + 495 T^{2} + 6037 T^{3} + 71081 T^{4} + 556772 T^{5} + 4700924 T^{6} + 26168124 T^{7} + 209482574 T^{8} + 1016636482 T^{9} + 10462754634 T^{10} + 59981552438 T^{11} + 729208840094 T^{12} + 5374383138996 T^{13} + 56962793141564 T^{14} + 398049831802828 T^{15} + 2998234511735921 T^{16} + 15023989013852303 T^{17} + 72681066614138895 T^{18} + 199248903829063597 T^{19} + 511116753300641401 T^{20} \)
$61$ \( 1 + 215 T^{2} - 490 T^{3} + 23725 T^{4} - 88654 T^{5} + 1906100 T^{6} - 8267970 T^{7} + 125871770 T^{8} - 598496950 T^{9} + 7588984906 T^{10} - 36508313950 T^{11} + 468368856170 T^{12} - 1876672098570 T^{13} + 26391557530100 T^{14} - 74876840468854 T^{15} + 1222320881714725 T^{16} - 1539943989650290 T^{17} + 41217072294415415 T^{18} + 713342911662882601 T^{20} \)
$67$ \( 1 - 38 T + 939 T^{2} - 16920 T^{3} + 253022 T^{4} - 3231648 T^{5} + 37034650 T^{6} - 384823180 T^{7} + 3720320353 T^{8} - 33446868814 T^{9} + 283134556406 T^{10} - 2240940210538 T^{11} + 16700518064617 T^{12} - 115740574086340 T^{13} + 746289713342650 T^{14} - 4363129101786336 T^{15} + 22887960773164718 T^{16} - 102547240362065160 T^{17} + 381297549225685899 T^{18} - 1033848307059207986 T^{19} + 1822837804551761449 T^{20} \)
$71$ \( 1 + 26 T + 632 T^{2} + 9130 T^{3} + 126663 T^{4} + 1295526 T^{5} + 13933979 T^{6} + 123894450 T^{7} + 1257395276 T^{8} + 10721089596 T^{9} + 100241082738 T^{10} + 761197361316 T^{11} + 6338529586316 T^{12} + 44343186493950 T^{13} + 354085829408699 T^{14} + 2337426034183626 T^{15} + 16225566262285623 T^{16} + 83038447046109830 T^{17} + 408116231747320952 T^{18} + 1192061018679674806 T^{19} + 3255243551009881201 T^{20} \)
$73$ \( 1 + T + 393 T^{2} + 817 T^{3} + 74513 T^{4} + 250396 T^{5} + 9185572 T^{6} + 43089004 T^{7} + 854063166 T^{8} + 4726643438 T^{9} + 66584553318 T^{10} + 345044970974 T^{11} + 4551302611614 T^{12} + 16762355069068 T^{13} + 260854087378852 T^{14} + 519088834600828 T^{15} + 11276367203472257 T^{16} + 9025724590102249 T^{17} + 316938816114373833 T^{18} + 58871586708267913 T^{19} + 4297625829703557649 T^{20} \)
$79$ \( 1 - 5 T + 552 T^{2} - 2604 T^{3} + 151027 T^{4} - 658718 T^{5} + 26569829 T^{6} - 105054020 T^{7} + 3304178864 T^{8} - 11590016269 T^{9} + 302495065710 T^{10} - 915611285251 T^{11} + 20621380290224 T^{12} - 51795728966780 T^{13} + 1034896991706149 T^{14} - 2026912437036482 T^{15} + 36712769144970067 T^{16} - 50006978999958036 T^{17} + 837444063068421672 T^{18} - 599257979913091595 T^{19} + 9468276082626847201 T^{20} \)
$83$ \( 1 + 6 T + 490 T^{2} + 3578 T^{3} + 118805 T^{4} + 976848 T^{5} + 19084536 T^{6} + 164833504 T^{7} + 2271305170 T^{8} + 19148739052 T^{9} + 211320377724 T^{10} + 1589345341316 T^{11} + 15647021316130 T^{12} + 94249654751648 T^{13} + 905720035624056 T^{14} + 3847843974033264 T^{15} + 38842151058104045 T^{16} + 97092790440885406 T^{17} + 1103623193748130090 T^{18} + 1121641531605242418 T^{19} + 15516041187205853449 T^{20} \)
$89$ \( 1 - 9 T + 640 T^{2} - 4050 T^{3} + 178355 T^{4} - 698876 T^{5} + 29313265 T^{6} - 51732294 T^{7} + 3369419368 T^{8} - 1036172651 T^{9} + 317703552166 T^{10} - 92219365939 T^{11} + 26689170813928 T^{12} - 36469663568886 T^{13} + 1839179937126865 T^{14} - 3902565131479324 T^{15} + 88639098149349155 T^{16} - 179136906326892450 T^{17} + 2519416835649331840 T^{18} - 3153207633367366881 T^{19} + 31181719929966183601 T^{20} \)
$97$ \( 1 - 24 T + 1069 T^{2} - 19604 T^{3} + 490777 T^{4} - 7233428 T^{5} + 130302052 T^{6} - 1582958620 T^{7} + 22393814246 T^{8} - 226465922980 T^{9} + 2616113341790 T^{10} - 21967194529060 T^{11} + 210703398240614 T^{12} - 1444723592591260 T^{13} + 11535546976384612 T^{14} - 62115907460510996 T^{15} + 408803501663039833 T^{16} - 1583969568908927252 T^{17} + 8378216512388971309 T^{18} - 18245545407709565208 T^{19} + 73742412689492826049 T^{20} \)
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