Properties

Label 8036.2.a.p
Level $8036$
Weight $2$
Character orbit 8036.a
Self dual yes
Analytic conductor $64.168$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8036.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1677830643\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 2 x^{9} - 18 x^{8} + 32 x^{7} + 110 x^{6} - 154 x^{5} - 282 x^{4} + 256 x^{3} + 253 x^{2} - 126 x - 49\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + \beta_{2} q^{5} + ( 1 + \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + \beta_{2} q^{5} + ( 1 + \beta_{2} + \beta_{3} ) q^{9} + \beta_{6} q^{11} + ( 1 + \beta_{7} ) q^{13} + ( -1 + 2 \beta_{1} - \beta_{3} - \beta_{5} - \beta_{8} + \beta_{9} ) q^{15} + ( 1 - \beta_{4} + \beta_{9} ) q^{17} + ( 1 - \beta_{1} + \beta_{3} + \beta_{5} ) q^{19} + ( -\beta_{4} + \beta_{5} ) q^{23} + ( 2 - 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{25} + ( \beta_{1} - \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{27} + ( 1 + \beta_{3} - \beta_{4} + \beta_{7} + \beta_{8} ) q^{29} + ( 1 + \beta_{1} + \beta_{2} + \beta_{4} - 2 \beta_{5} - \beta_{8} ) q^{31} + ( \beta_{3} + \beta_{6} - \beta_{9} ) q^{33} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{37} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} ) q^{39} + q^{41} + ( 1 + \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{43} + ( 5 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{9} ) q^{45} + ( -\beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{47} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{7} + 2 \beta_{9} ) q^{51} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{53} + ( 1 - 2 \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} ) q^{55} + ( -3 + 3 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} ) q^{57} + ( 1 + \beta_{1} - \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} ) q^{59} + ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{8} ) q^{61} + ( -2 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{65} + ( -2 + \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{9} ) q^{67} + ( 2 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{69} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{8} - \beta_{9} ) q^{71} + ( 4 - \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{73} + ( -9 + 6 \beta_{1} - 6 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} - 3 \beta_{8} + 2 \beta_{9} ) q^{75} + ( 2 + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} - \beta_{9} ) q^{79} + ( -1 - \beta_{1} - \beta_{2} - \beta_{4} + 2 \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{81} + ( 2 - \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} ) q^{83} + ( \beta_{1} + \beta_{2} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{9} ) q^{85} + ( 1 + 2 \beta_{1} - \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{9} ) q^{87} + ( 1 - \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{89} + ( 2 + \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - \beta_{6} + 3 \beta_{7} - 3 \beta_{9} ) q^{93} + ( 1 - 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{95} + ( 6 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{9} ) q^{97} + ( 2 + \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 2q^{3} + 4q^{5} + 10q^{9} + O(q^{10}) \) \( 10q + 2q^{3} + 4q^{5} + 10q^{9} + 6q^{13} - 4q^{15} + 12q^{17} + 8q^{19} + 4q^{23} + 16q^{25} + 8q^{27} + 2q^{29} + 8q^{31} - 6q^{33} - 2q^{37} - 2q^{39} + 10q^{41} - 2q^{43} + 44q^{45} - 14q^{47} + 14q^{51} + 8q^{53} + 8q^{55} - 10q^{57} + 24q^{59} + 14q^{61} + 2q^{65} - 8q^{67} + 16q^{69} + 10q^{71} + 44q^{73} - 50q^{75} + 10q^{79} - 14q^{81} + 20q^{83} + 8q^{85} + 20q^{87} + 6q^{89} + 8q^{93} + 4q^{95} + 46q^{97} + 2q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 2 x^{9} - 18 x^{8} + 32 x^{7} + 110 x^{6} - 154 x^{5} - 282 x^{4} + 256 x^{3} + 253 x^{2} - 126 x - 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -2 \nu^{9} - 3 \nu^{8} + 36 \nu^{7} + 41 \nu^{6} - 213 \nu^{5} - 154 \nu^{4} + 445 \nu^{3} + 132 \nu^{2} - 275 \nu - 28 \)\()/21\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{9} + 3 \nu^{8} - 36 \nu^{7} - 41 \nu^{6} + 213 \nu^{5} + 154 \nu^{4} - 445 \nu^{3} - 111 \nu^{2} + 275 \nu - 56 \)\()/21\)
\(\beta_{4}\)\(=\)\((\)\( -3 \nu^{9} - \nu^{8} + 47 \nu^{7} + 16 \nu^{6} - 218 \nu^{5} - 84 \nu^{4} + 293 \nu^{3} + 72 \nu^{2} - 73 \nu - 7 \)\()/7\)
\(\beta_{5}\)\(=\)\((\)\( 13 \nu^{9} + 9 \nu^{8} - 213 \nu^{7} - 151 \nu^{6} + 1080 \nu^{5} + 812 \nu^{4} - 1811 \nu^{3} - 1173 \nu^{2} + 895 \nu + 371 \)\()/21\)
\(\beta_{6}\)\(=\)\((\)\( -22 \nu^{9} - 12 \nu^{8} + 354 \nu^{7} + 199 \nu^{6} - 1734 \nu^{5} - 1043 \nu^{4} + 2711 \nu^{3} + 1221 \nu^{2} - 1240 \nu - 266 \)\()/21\)
\(\beta_{7}\)\(=\)\((\)\( -29 \nu^{9} - 12 \nu^{8} + 459 \nu^{7} + 206 \nu^{6} - 2196 \nu^{5} - 1162 \nu^{4} + 3334 \nu^{3} + 1452 \nu^{2} - 1415 \nu - 343 \)\()/21\)
\(\beta_{8}\)\(=\)\((\)\( 4 \nu^{9} + 3 \nu^{8} - 63 \nu^{7} - 49 \nu^{6} + 297 \nu^{5} + 251 \nu^{4} - 425 \nu^{3} - 309 \nu^{2} + 175 \nu + 74 \)\()/3\)
\(\beta_{9}\)\(=\)\((\)\( 12 \nu^{9} + 11 \nu^{8} - 195 \nu^{7} - 176 \nu^{6} + 970 \nu^{5} + 868 \nu^{4} - 1529 \nu^{3} - 1072 \nu^{2} + 691 \nu + 252 \)\()/7\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} + 7 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{9} + \beta_{8} + \beta_{7} + 2 \beta_{5} - \beta_{4} + 9 \beta_{3} + 8 \beta_{2} - \beta_{1} + 26\)
\(\nu^{5}\)\(=\)\(10 \beta_{9} - 10 \beta_{8} - 11 \beta_{7} + 10 \beta_{6} - 11 \beta_{5} + 2 \beta_{4} - 11 \beta_{3} - 2 \beta_{2} + 53 \beta_{1} - 2\)
\(\nu^{6}\)\(=\)\(-14 \beta_{9} + 14 \beta_{8} + 14 \beta_{7} - 2 \beta_{6} + 25 \beta_{5} - 13 \beta_{4} + 77 \beta_{3} + 61 \beta_{2} - 20 \beta_{1} + 194\)
\(\nu^{7}\)\(=\)\(87 \beta_{9} - 86 \beta_{8} - 101 \beta_{7} + 88 \beta_{6} - 102 \beta_{5} + 27 \beta_{4} - 108 \beta_{3} - 34 \beta_{2} + 417 \beta_{1} - 50\)
\(\nu^{8}\)\(=\)\(-149 \beta_{9} + 151 \beta_{8} + 150 \beta_{7} - 33 \beta_{6} + 249 \beta_{5} - 129 \beta_{4} + 655 \beta_{3} + 474 \beta_{2} - 269 \beta_{1} + 1522\)
\(\nu^{9}\)\(=\)\(737 \beta_{9} - 722 \beta_{8} - 884 \beta_{7} + 750 \beta_{6} - 902 \beta_{5} + 277 \beta_{4} - 1026 \beta_{3} - 420 \beta_{2} + 3352 \beta_{1} - 745\)

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
−2.96176
−1.91097
−1.73395
−1.05099
−0.284914
0.729469
1.20067
2.58799
2.70664
2.71781
0 −2.96176 0 4.34462 0 0 0 5.77201 0
1.2 0 −1.91097 0 1.95825 0 0 0 0.651806 0
1.3 0 −1.73395 0 −3.41343 0 0 0 0.00658743 0
1.4 0 −1.05099 0 −1.02904 0 0 0 −1.89542 0
1.5 0 −0.284914 0 2.38934 0 0 0 −2.91882 0
1.6 0 0.729469 0 −3.02160 0 0 0 −2.46788 0
1.7 0 1.20067 0 −0.960855 0 0 0 −1.55839 0
1.8 0 2.58799 0 3.78481 0 0 0 3.69770 0
1.9 0 2.70664 0 0.469317 0 0 0 4.32591 0
1.10 0 2.71781 0 −0.521406 0 0 0 4.38650 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)
\(41\) \(-1\)

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 8036.2.a.p yes 10
7.b odd 2 1 8036.2.a.o 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8036.2.a.o 10 7.b odd 2 1
8036.2.a.p yes 10 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8036))\):

\(T_{3}^{10} - \cdots\)
\(T_{5}^{10} - \cdots\)
\(T_{11}^{10} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 2 T + 12 T^{2} - 22 T^{3} + 83 T^{4} - 130 T^{5} + 402 T^{6} - 542 T^{7} + 1513 T^{8} - 1854 T^{9} + 4817 T^{10} - 5562 T^{11} + 13617 T^{12} - 14634 T^{13} + 32562 T^{14} - 31590 T^{15} + 60507 T^{16} - 48114 T^{17} + 78732 T^{18} - 39366 T^{19} + 59049 T^{20} \)
$5$ \( 1 - 4 T + 25 T^{2} - 88 T^{3} + 326 T^{4} - 994 T^{5} + 2938 T^{6} - 7978 T^{7} + 20745 T^{8} - 50112 T^{9} + 116858 T^{10} - 250560 T^{11} + 518625 T^{12} - 997250 T^{13} + 1836250 T^{14} - 3106250 T^{15} + 5093750 T^{16} - 6875000 T^{17} + 9765625 T^{18} - 7812500 T^{19} + 9765625 T^{20} \)
$7$ 1
$11$ \( 1 + 60 T^{2} - 48 T^{3} + 1750 T^{4} - 2512 T^{5} + 35046 T^{6} - 60128 T^{7} + 548345 T^{8} - 910432 T^{9} + 6806204 T^{10} - 10014752 T^{11} + 66349745 T^{12} - 80030368 T^{13} + 513108486 T^{14} - 404560112 T^{15} + 3100231750 T^{16} - 935384208 T^{17} + 12861532860 T^{18} + 25937424601 T^{20} \)
$13$ \( 1 - 6 T + 77 T^{2} - 378 T^{3} + 2756 T^{4} - 11378 T^{5} + 63254 T^{6} - 226122 T^{7} + 1083085 T^{8} - 3478468 T^{9} + 15235973 T^{10} - 45220084 T^{11} + 183041365 T^{12} - 496790034 T^{13} + 1806597494 T^{14} - 4224571754 T^{15} + 13302685604 T^{16} - 23718939426 T^{17} + 62811265517 T^{18} - 63626996238 T^{19} + 137858491849 T^{20} \)
$17$ \( 1 - 12 T + 153 T^{2} - 1250 T^{3} + 9866 T^{4} - 64080 T^{5} + 390146 T^{6} - 2122674 T^{7} + 10664977 T^{8} - 49516210 T^{9} + 211311741 T^{10} - 841775570 T^{11} + 3082178353 T^{12} - 10428697362 T^{13} + 32585384066 T^{14} - 90984436560 T^{15} + 238141255754 T^{16} - 512923341250 T^{17} + 1067290888473 T^{18} - 1423054517964 T^{19} + 2015993900449 T^{20} \)
$19$ \( 1 - 8 T + 142 T^{2} - 854 T^{3} + 8779 T^{4} - 41722 T^{5} + 324726 T^{6} - 1272034 T^{7} + 8468743 T^{8} - 28827300 T^{9} + 175736485 T^{10} - 547718700 T^{11} + 3057216223 T^{12} - 8724881206 T^{13} + 42318617046 T^{14} - 103307802478 T^{15} + 413015789299 T^{16} - 763366465106 T^{17} + 2411665951822 T^{18} - 2581501582232 T^{19} + 6131066257801 T^{20} \)
$23$ \( 1 - 4 T + 150 T^{2} - 472 T^{3} + 10995 T^{4} - 29014 T^{5} + 530388 T^{6} - 1210146 T^{7} + 18532865 T^{8} - 36844898 T^{9} + 487837739 T^{10} - 847432654 T^{11} + 9803885585 T^{12} - 14723846382 T^{13} + 148424308308 T^{14} - 186744055802 T^{15} + 1627654599555 T^{16} - 1607077610984 T^{17} + 11746647792150 T^{18} - 7204610645852 T^{19} + 41426511213649 T^{20} \)
$29$ \( 1 - 2 T + 169 T^{2} - 200 T^{3} + 13298 T^{4} - 8180 T^{5} + 689398 T^{6} - 273872 T^{7} + 27573885 T^{8} - 10844658 T^{9} + 890329250 T^{10} - 314495082 T^{11} + 23189637285 T^{12} - 6679464208 T^{13} + 487598106838 T^{14} - 167781198820 T^{15} + 7909960522658 T^{16} - 3449975261800 T^{17} + 84541643790409 T^{18} - 29014291951738 T^{19} + 420707233300201 T^{20} \)
$31$ \( 1 - 8 T + 173 T^{2} - 1172 T^{3} + 15518 T^{4} - 93450 T^{5} + 940146 T^{6} - 5050870 T^{7} + 42210833 T^{8} - 203386776 T^{9} + 1475895842 T^{10} - 6304990056 T^{11} + 40564610513 T^{12} - 150470468170 T^{13} + 868244574066 T^{14} - 2675394160950 T^{15} + 13772282121758 T^{16} - 32244783738092 T^{17} + 147550149477293 T^{18} - 211516977285368 T^{19} + 819628286980801 T^{20} \)
$37$ \( 1 + 2 T + 207 T^{2} + 440 T^{3} + 23279 T^{4} + 49382 T^{5} + 1758529 T^{6} + 3562350 T^{7} + 97635520 T^{8} + 181110266 T^{9} + 4119614704 T^{10} + 6701079842 T^{11} + 133663026880 T^{12} + 180443714550 T^{13} + 3295766469169 T^{14} + 3424343284574 T^{15} + 59727545075111 T^{16} + 41770025938520 T^{17} + 727083246961647 T^{18} + 259923479590154 T^{19} + 4808584372417849 T^{20} \)
$41$ \( ( 1 - T )^{10} \)
$43$ \( 1 + 2 T + 221 T^{2} + 468 T^{3} + 25826 T^{4} + 55060 T^{5} + 2056468 T^{6} + 4339622 T^{7} + 123599993 T^{8} + 247838392 T^{9} + 5913403015 T^{10} + 10657050856 T^{11} + 228536387057 T^{12} + 345030326354 T^{13} + 7030654854868 T^{14} + 8094284871580 T^{15} + 163255522103474 T^{16} + 127211109998076 T^{17} + 2583092261349821 T^{18} + 1005185223873686 T^{19} + 21611482313284249 T^{20} \)
$47$ \( 1 + 14 T + 347 T^{2} + 4286 T^{3} + 61213 T^{4} + 634688 T^{5} + 6842191 T^{6} + 59787324 T^{7} + 525899550 T^{8} + 3920374812 T^{9} + 28988297588 T^{10} + 184257616164 T^{11} + 1161712105950 T^{12} + 6207299339652 T^{13} + 33387709421071 T^{14} + 145562523802816 T^{15} + 659828107934077 T^{16} + 2171386694304418 T^{17} + 8262516471631067 T^{18} + 15667826623438738 T^{19} + 52599132235830049 T^{20} \)
$53$ \( 1 - 8 T + 245 T^{2} - 1196 T^{3} + 23954 T^{4} - 58632 T^{5} + 1591642 T^{6} - 2732816 T^{7} + 116936509 T^{8} - 293998276 T^{9} + 7553425474 T^{10} - 15581908628 T^{11} + 328474653781 T^{12} - 406853447632 T^{13} + 12558820959802 T^{14} - 24519638145576 T^{15} + 530925106484066 T^{16} - 1404954523245052 T^{17} + 15253624150783445 T^{18} - 26398108734417064 T^{19} + 174887470365513049 T^{20} \)
$59$ \( 1 - 24 T + 629 T^{2} - 8776 T^{3} + 126918 T^{4} - 1161538 T^{5} + 11387310 T^{6} - 65476022 T^{7} + 482347017 T^{8} - 1374681036 T^{9} + 15949341018 T^{10} - 81106181124 T^{11} + 1679049966177 T^{12} - 13447399922338 T^{13} + 137984146088910 T^{14} - 830411740411862 T^{15} + 5353468968648438 T^{16} - 21840405430771544 T^{17} + 92356345253117909 T^{18} - 207911899647718536 T^{19} + 511116753300641401 T^{20} \)
$61$ \( 1 - 14 T + 528 T^{2} - 6362 T^{3} + 130430 T^{4} - 1349402 T^{5} + 19744598 T^{6} - 175069242 T^{7} + 2021257345 T^{8} - 15296907412 T^{9} + 145932869524 T^{10} - 933111352132 T^{11} + 7521098580745 T^{12} - 39737391618402 T^{13} + 273380564516918 T^{14} - 1139699937762002 T^{15} + 6719802427905230 T^{16} - 19994129922765602 T^{17} + 101221461262564368 T^{18} - 163718045299677974 T^{19} + 713342911662882601 T^{20} \)
$67$ \( 1 + 8 T + 449 T^{2} + 2320 T^{3} + 87078 T^{4} + 265904 T^{5} + 10436462 T^{6} + 16581400 T^{7} + 935678857 T^{8} + 796918512 T^{9} + 68666864514 T^{10} + 53393540304 T^{11} + 4200262389073 T^{12} + 4987071608200 T^{13} + 210306408573902 T^{14} + 359003666451728 T^{15} + 7876935002512182 T^{16} + 14060850924349360 T^{17} + 182324387222931809 T^{18} + 217652275170359576 T^{19} + 1822837804551761449 T^{20} \)
$71$ \( 1 - 10 T + 444 T^{2} - 4262 T^{3} + 86730 T^{4} - 810086 T^{5} + 9918442 T^{6} - 94008490 T^{7} + 789911765 T^{8} - 7996076064 T^{9} + 55295925748 T^{10} - 567721400544 T^{11} + 3981945207365 T^{12} - 33646672664390 T^{13} + 252044284121002 T^{14} - 1461580938034186 T^{15} + 11110137624468330 T^{16} - 38763402115062442 T^{17} + 286714567873117884 T^{18} - 458485007184490310 T^{19} + 3255243551009881201 T^{20} \)
$73$ \( 1 - 44 T + 1220 T^{2} - 24508 T^{3} + 403542 T^{4} - 5667300 T^{5} + 70965058 T^{6} - 804708756 T^{7} + 8403572217 T^{8} - 80839769728 T^{9} + 719421847316 T^{10} - 5901303190144 T^{11} + 44782636344393 T^{12} - 313045386132852 T^{13} + 2015282819662978 T^{14} - 11748718639008900 T^{15} + 61069716345115638 T^{16} - 270749642906029276 T^{17} + 983881312110778820 T^{18} - 2590349815163788172 T^{19} + 4297625829703557649 T^{20} \)
$79$ \( 1 - 10 T + 614 T^{2} - 5902 T^{3} + 179069 T^{4} - 1630352 T^{5} + 32727400 T^{6} - 275265280 T^{7} + 4153988066 T^{8} - 31232651988 T^{9} + 382732416868 T^{10} - 2467379507052 T^{11} + 25925039519906 T^{12} - 135716518385920 T^{13} + 1274734880919400 T^{14} - 5016685054222448 T^{15} + 43529427572689949 T^{16} - 113341470836310418 T^{17} + 931504809282628454 T^{18} - 1198515959826183190 T^{19} + 9468276082626847201 T^{20} \)
$83$ \( 1 - 20 T + 528 T^{2} - 7496 T^{3} + 121918 T^{4} - 1442276 T^{5} + 18880926 T^{6} - 198399444 T^{7} + 2251070337 T^{8} - 21080323924 T^{9} + 210699872260 T^{10} - 1749666885692 T^{11} + 15507623551593 T^{12} - 113442222886428 T^{13} + 896057046885246 T^{14} - 5681183782423468 T^{15} + 39859916440401742 T^{16} - 203411838218243992 T^{17} + 1189210298569413648 T^{18} - 3738805105350808060 T^{19} + 15516041187205853449 T^{20} \)
$89$ \( 1 - 6 T + 178 T^{2} - 40 T^{3} + 19025 T^{4} + 13312 T^{5} + 2644880 T^{6} - 1951728 T^{7} + 298984707 T^{8} + 249612878 T^{9} + 23616419385 T^{10} + 22215546142 T^{11} + 2368257864147 T^{12} - 1375907736432 T^{13} + 165945698376080 T^{14} + 74334999385088 T^{15} + 9455069060533025 T^{16} - 1769253395821160 T^{17} + 700712807414970418 T^{18} - 2102138422244911254 T^{19} + 31181719929966183601 T^{20} \)
$97$ \( 1 - 46 T + 1292 T^{2} - 26530 T^{3} + 430839 T^{4} - 5783946 T^{5} + 66062922 T^{6} - 659196450 T^{7} + 5979674797 T^{8} - 52593327006 T^{9} + 491620931417 T^{10} - 5101552719582 T^{11} + 56262760164973 T^{12} - 601630801610850 T^{13} + 5848502985419082 T^{14} - 49668712330114122 T^{15} + 358876825631605431 T^{16} - 2143578487204337890 T^{17} + 10125964203935033612 T^{18} - 34970628698109999982 T^{19} + 73742412689492826049 T^{20} \)
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