Properties

Label 108.5.g.a
Level 108
Weight 5
Character orbit 108.g
Analytic conductor 11.164
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 108 = 2^{2} \cdot 3^{3} \)
Weight: \( k \) = \( 5 \)
Character orbit: \([\chi]\) = 108.g (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.1639560131\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{10} \)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 + \beta_{2} + \beta_{4} + \beta_{7} ) q^{5} + ( 3 - \beta_{1} + 3 \beta_{2} - 2 \beta_{4} + \beta_{6} - \beta_{7} ) q^{7} +O(q^{10})\) \( q + ( 2 + \beta_{2} + \beta_{4} + \beta_{7} ) q^{5} + ( 3 - \beta_{1} + 3 \beta_{2} - 2 \beta_{4} + \beta_{6} - \beta_{7} ) q^{7} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{6} - 2 \beta_{7} ) q^{11} + ( -3 \beta_{1} + 5 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} - 3 \beta_{7} ) q^{13} + ( -53 - 3 \beta_{1} - 106 \beta_{2} - 2 \beta_{4} + 5 \beta_{5} - \beta_{6} ) q^{17} + ( 76 - 4 \beta_{1} + 8 \beta_{2} + 8 \beta_{3} + 5 \beta_{4} - 4 \beta_{5} + 10 \beta_{6} + 2 \beta_{7} ) q^{19} + ( 292 - 8 \beta_{1} + 141 \beta_{2} - 5 \beta_{3} - 10 \beta_{4} + 5 \beta_{5} + 16 \beta_{6} - 5 \beta_{7} ) q^{23} + ( 98 - 16 \beta_{1} + 100 \beta_{2} + 2 \beta_{3} + 6 \beta_{4} - 4 \beta_{5} + 18 \beta_{6} + \beta_{7} ) q^{25} + ( -172 + 181 \beta_{2} + 9 \beta_{3} + 9 \beta_{6} - \beta_{7} ) q^{29} + ( -14 \beta_{1} - 39 \beta_{2} + 7 \beta_{3} - 6 \beta_{4} + 7 \beta_{5} - \beta_{7} ) q^{31} + ( -533 + 12 \beta_{1} - 1066 \beta_{2} + 19 \beta_{4} + 10 \beta_{5} - 11 \beta_{6} ) q^{35} + ( -21 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 29 \beta_{4} - \beta_{5} - \beta_{6} - 60 \beta_{7} ) q^{37} + ( 1328 + 5 \beta_{1} + 675 \beta_{2} + 11 \beta_{3} + 37 \beta_{4} - 11 \beta_{5} - 10 \beta_{6} + 26 \beta_{7} ) q^{41} + ( -17 + 3 \beta_{1} - 23 \beta_{2} - 6 \beta_{3} + 12 \beta_{5} - 9 \beta_{6} + 6 \beta_{7} ) q^{43} + ( -1143 + 11 \beta_{1} + 1127 \beta_{2} - 16 \beta_{3} - 5 \beta_{6} + \beta_{7} ) q^{47} + ( 21 \beta_{1} + 42 \beta_{2} - 19 \beta_{3} - 20 \beta_{4} - 19 \beta_{5} + 39 \beta_{7} ) q^{49} + ( -961 - 45 \beta_{1} - 1922 \beta_{2} - 61 \beta_{4} - 45 \beta_{5} + 45 \beta_{6} ) q^{53} + ( -192 + 17 \beta_{1} - 34 \beta_{2} - 34 \beta_{3} + 42 \beta_{4} + 17 \beta_{5} - 18 \beta_{6} + 118 \beta_{7} ) q^{55} + ( 3286 + 41 \beta_{1} + 1627 \beta_{2} - 16 \beta_{3} - 46 \beta_{4} + 16 \beta_{5} - 82 \beta_{6} - 30 \beta_{7} ) q^{59} + ( -530 + 122 \beta_{1} - 519 \beta_{2} + 11 \beta_{3} + 48 \beta_{4} - 22 \beta_{5} - 111 \beta_{6} + 13 \beta_{7} ) q^{61} + ( -2210 - 56 \beta_{1} + 2181 \beta_{2} - 29 \beta_{3} - 85 \beta_{6} - 15 \beta_{7} ) q^{65} + ( 108 \beta_{1} - 25 \beta_{2} - 3 \beta_{3} + 93 \beta_{4} - 3 \beta_{5} - 90 \beta_{7} ) q^{67} + ( -2747 + 105 \beta_{1} - 5494 \beta_{2} + 31 \beta_{4} - 19 \beta_{5} - 43 \beta_{6} ) q^{71} + ( -1003 + 31 \beta_{1} - 62 \beta_{2} - 62 \beta_{3} + 42 \beta_{4} + 31 \beta_{5} - 165 \beta_{6} + 146 \beta_{7} ) q^{73} + ( 4222 - 22 \beta_{1} + 2185 \beta_{2} + 74 \beta_{3} - 73 \beta_{4} - 74 \beta_{5} + 44 \beta_{6} - 147 \beta_{7} ) q^{77} + ( -604 - 57 \beta_{1} - 635 \beta_{2} - 31 \beta_{3} - 244 \beta_{4} + 62 \beta_{5} + 26 \beta_{6} - 91 \beta_{7} ) q^{79} + ( -3015 + 47 \beta_{1} + 2999 \beta_{2} - 16 \beta_{3} + 31 \beta_{6} + 217 \beta_{7} ) q^{83} + ( -83 \beta_{1} + 724 \beta_{2} - 35 \beta_{3} - 69 \beta_{4} - 35 \beta_{5} + 104 \beta_{7} ) q^{85} + ( -1259 - 39 \beta_{1} - 2518 \beta_{2} + 241 \beta_{4} - 31 \beta_{5} + 35 \beta_{6} ) q^{89} + ( 1996 - 21 \beta_{1} + 42 \beta_{2} + 42 \beta_{3} - 120 \beta_{4} - 21 \beta_{5} + 246 \beta_{6} - 282 \beta_{7} ) q^{91} + ( 6376 + 8 \beta_{1} + 3094 \beta_{2} - 94 \beta_{3} + 302 \beta_{4} + 94 \beta_{5} - 16 \beta_{6} + 396 \beta_{7} ) q^{95} + ( 1874 - 34 \beta_{1} + 1911 \beta_{2} + 37 \beta_{3} + 122 \beta_{4} - 74 \beta_{5} + 71 \beta_{6} + 24 \beta_{7} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 9q^{5} + 13q^{7} + O(q^{10}) \) \( 8q + 9q^{5} + 13q^{7} + 18q^{11} - 5q^{13} + 562q^{19} + 1719q^{23} + 353q^{25} - 2115q^{29} + 187q^{31} + 16q^{37} + 7920q^{41} - 68q^{43} - 13689q^{47} - 327q^{49} - 1818q^{55} + 20052q^{59} - 1937q^{61} - 25965q^{65} + 154q^{67} - 7802q^{73} + 25641q^{77} - 2195q^{79} - 37017q^{83} - 3042q^{85} + 15830q^{91} + 37116q^{95} + 7282q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{7} + 6 x^{6} + 121 x^{5} + 1104 x^{4} - 1647 x^{3} + 6529 x^{2} + 85254 x + 440076\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 35 \nu^{7} - 3472 \nu^{6} - 34123 \nu^{5} + 527797 \nu^{4} - 988763 \nu^{3} - 3500708 \nu^{2} - 8283339 \nu + 307232328 \)\()/2814669\)
\(\beta_{2}\)\(=\)\((\)\( -115 \nu^{7} - 175 \nu^{6} + 4562 \nu^{5} - 26525 \nu^{4} - 65600 \nu^{3} + 13645 \nu^{2} + 1909485 \nu - 10751676 \)\()/5629338\)
\(\beta_{3}\)\(=\)\((\)\( -95 \nu^{7} + 21007 \nu^{6} - 8318 \nu^{5} + 612635 \nu^{4} + 681581 \nu^{3} + 17133464 \nu^{2} + 49979736 \nu + 416903721 \)\()/2814669\)
\(\beta_{4}\)\(=\)\((\)\( 574 \nu^{7} - 3659 \nu^{6} + 12583 \nu^{5} - 3580 \nu^{4} + 549521 \nu^{3} - 4310506 \nu^{2} + 10634175 \nu - 2489799 \)\()/2814669\)
\(\beta_{5}\)\(=\)\((\)\( 28 \nu^{7} - 164 \nu^{6} - 509 \nu^{5} + 16298 \nu^{4} + 11840 \nu^{3} - 102856 \nu^{2} + 736731 \nu + 9836232 \)\()/72171\)
\(\beta_{6}\)\(=\)\((\)\( 4 \nu^{7} - 35 \nu^{6} + 43 \nu^{5} + 419 \nu^{4} + 1460 \nu^{3} - 18628 \nu^{2} - 20592 \nu + 201354 \)\()/6561\)
\(\beta_{7}\)\(=\)\((\)\( 5003 \nu^{7} - 30661 \nu^{6} + 137138 \nu^{5} + 631207 \nu^{4} + 837436 \nu^{3} - 4541909 \nu^{2} + 80817471 \nu + 323067342 \)\()/5629338\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} - 3 \beta_{6} + 2 \beta_{5} - \beta_{3} - 7 \beta_{2} - \beta_{1} + 6\)\()/27\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{7} + 3 \beta_{6} + 8 \beta_{5} - 27 \beta_{4} - \beta_{3} + 11 \beta_{2} - 10 \beta_{1} - 3\)\()/27\)
\(\nu^{3}\)\(=\)\((\)\(-50 \beta_{7} + 84 \beta_{6} + 32 \beta_{5} - 27 \beta_{4} + 23 \beta_{3} + 602 \beta_{2} - 49 \beta_{1} - 1002\)\()/27\)
\(\nu^{4}\)\(=\)\((\)\(-89 \beta_{7} + 228 \beta_{6} - 64 \beta_{5} + 54 \beta_{4} + 116 \beta_{3} + 2135 \beta_{2} + 116 \beta_{1} - 18885\)\()/27\)
\(\nu^{5}\)\(=\)\((\)\(-335 \beta_{7} + 1875 \beta_{6} - 2095 \beta_{5} + 2160 \beta_{4} + 1145 \beta_{3} + 22073 \beta_{2} + 1325 \beta_{1} - 22944\)\()/27\)
\(\nu^{6}\)\(=\)\((\)\(1120 \beta_{7} - 4254 \beta_{6} - 11443 \beta_{5} + 22410 \beta_{4} + 3065 \beta_{3} - 71200 \beta_{2} + 9365 \beta_{1} + 33477\)\()/27\)
\(\nu^{7}\)\(=\)\((\)\(50779 \beta_{7} - 69108 \beta_{6} - 35029 \beta_{5} + 51327 \beta_{4} - 15841 \beta_{3} - 1288465 \beta_{2} + 21716 \beta_{1} + 1541283\)\()/27\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/108\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(55\)
\(\chi(n)\) \(-\beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
−3.05006 3.25531i
4.23522 + 4.06612i
3.72537 4.42407i
−3.41053 + 2.74723i
−3.05006 + 3.25531i
4.23522 4.06612i
3.72537 + 4.42407i
−3.41053 2.74723i
0 0 0 −27.4152 15.8282i 0 37.6830 + 65.2688i 0 0 0
17.2 0 0 0 −10.6364 6.14094i 0 7.14202 + 12.3703i 0 0 0
17.3 0 0 0 7.67992 + 4.43400i 0 −30.9381 53.5864i 0 0 0
17.4 0 0 0 34.8718 + 20.1332i 0 −7.38688 12.7945i 0 0 0
89.1 0 0 0 −27.4152 + 15.8282i 0 37.6830 65.2688i 0 0 0
89.2 0 0 0 −10.6364 + 6.14094i 0 7.14202 12.3703i 0 0 0
89.3 0 0 0 7.67992 4.43400i 0 −30.9381 + 53.5864i 0 0 0
89.4 0 0 0 34.8718 20.1332i 0 −7.38688 + 12.7945i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 89.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.5.g.a 8
3.b odd 2 1 36.5.g.a 8
4.b odd 2 1 432.5.q.c 8
9.c even 3 1 36.5.g.a 8
9.c even 3 1 324.5.c.a 8
9.d odd 6 1 inner 108.5.g.a 8
9.d odd 6 1 324.5.c.a 8
12.b even 2 1 144.5.q.c 8
36.f odd 6 1 144.5.q.c 8
36.f odd 6 1 1296.5.e.g 8
36.h even 6 1 432.5.q.c 8
36.h even 6 1 1296.5.e.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.5.g.a 8 3.b odd 2 1
36.5.g.a 8 9.c even 3 1
108.5.g.a 8 1.a even 1 1 trivial
108.5.g.a 8 9.d odd 6 1 inner
144.5.q.c 8 12.b even 2 1
144.5.q.c 8 36.f odd 6 1
324.5.c.a 8 9.c even 3 1
324.5.c.a 8 9.d odd 6 1
432.5.q.c 8 4.b odd 2 1
432.5.q.c 8 36.h even 6 1
1296.5.e.g 8 36.f odd 6 1
1296.5.e.g 8 36.h even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(108, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - 9 T + 1114 T^{2} - 9783 T^{3} + 533599 T^{4} - 10528056 T^{5} + 59806456 T^{6} - 10305069192 T^{7} - 6001445444 T^{8} - 6440668245000 T^{9} + 23361896875000 T^{10} - 2570326171875000 T^{11} + 81420745849609375 T^{12} - 932979583740234375 T^{13} + 66399574279785156250 T^{14} - \)\(33\!\cdots\!25\)\( T^{15} + \)\(23\!\cdots\!25\)\( T^{16} \)
$7$ \( 1 - 13 T - 4554 T^{2} + 124753 T^{3} + 8962391 T^{4} - 347580324 T^{5} + 2901735784 T^{6} + 447642835484 T^{7} - 30706182623268 T^{8} + 1074790447997084 T^{9} + 16727929349338984 T^{10} - 4810959089900633124 T^{11} + \)\(29\!\cdots\!91\)\( T^{12} + \)\(99\!\cdots\!53\)\( T^{13} - \)\(87\!\cdots\!54\)\( T^{14} - \)\(59\!\cdots\!13\)\( T^{15} + \)\(11\!\cdots\!01\)\( T^{16} \)
$11$ \( 1 - 18 T + 33850 T^{2} - 607356 T^{3} + 452208721 T^{4} - 17443950048 T^{5} + 9074477968558 T^{6} - 445095012639474 T^{7} + 191968264536523468 T^{8} - 6516636080054538834 T^{9} + \)\(19\!\cdots\!98\)\( T^{10} - \)\(54\!\cdots\!08\)\( T^{11} + \)\(20\!\cdots\!81\)\( T^{12} - \)\(40\!\cdots\!56\)\( T^{13} + \)\(33\!\cdots\!50\)\( T^{14} - \)\(25\!\cdots\!58\)\( T^{15} + \)\(21\!\cdots\!21\)\( T^{16} \)
$13$ \( 1 + 5 T - 42054 T^{2} - 7266665 T^{3} + 352176575 T^{4} + 250092029040 T^{5} + 23191943061904 T^{6} - 3464858176098460 T^{7} - 548440340475170196 T^{8} - 98959814367548116060 T^{9} + \)\(18\!\cdots\!84\)\( T^{10} + \)\(58\!\cdots\!40\)\( T^{11} + \)\(23\!\cdots\!75\)\( T^{12} - \)\(13\!\cdots\!65\)\( T^{13} - \)\(22\!\cdots\!94\)\( T^{14} + \)\(77\!\cdots\!05\)\( T^{15} + \)\(44\!\cdots\!81\)\( T^{16} \)
$17$ \( 1 - 288125 T^{2} + 52320681154 T^{4} - 6759733382202755 T^{6} + \)\(63\!\cdots\!86\)\( T^{8} - \)\(47\!\cdots\!55\)\( T^{10} + \)\(25\!\cdots\!74\)\( T^{12} - \)\(97\!\cdots\!25\)\( T^{14} + \)\(23\!\cdots\!61\)\( T^{16} \)
$19$ \( ( 1 - 281 T + 110170 T^{2} + 68843041 T^{3} - 21846246566 T^{4} + 8971693946161 T^{5} + 1871079140226970 T^{6} - 621941492257591241 T^{7} + \)\(28\!\cdots\!81\)\( T^{8} )^{2} \)
$23$ \( 1 - 1719 T + 1529458 T^{2} - 935945649 T^{3} + 342642958747 T^{4} + 16333135609500 T^{5} - 118516645434237428 T^{6} + \)\(10\!\cdots\!68\)\( T^{7} - \)\(65\!\cdots\!00\)\( T^{8} + \)\(29\!\cdots\!88\)\( T^{9} - \)\(92\!\cdots\!68\)\( T^{10} + \)\(35\!\cdots\!00\)\( T^{11} + \)\(21\!\cdots\!67\)\( T^{12} - \)\(16\!\cdots\!49\)\( T^{13} + \)\(73\!\cdots\!78\)\( T^{14} - \)\(23\!\cdots\!39\)\( T^{15} + \)\(37\!\cdots\!21\)\( T^{16} \)
$29$ \( 1 + 2115 T + 4091014 T^{2} + 5498870985 T^{3} + 7048695081595 T^{4} + 8024737206821040 T^{5} + 8297140249856169556 T^{6} + \)\(79\!\cdots\!80\)\( T^{7} + \)\(68\!\cdots\!24\)\( T^{8} + \)\(56\!\cdots\!80\)\( T^{9} + \)\(41\!\cdots\!16\)\( T^{10} + \)\(28\!\cdots\!40\)\( T^{11} + \)\(17\!\cdots\!95\)\( T^{12} + \)\(97\!\cdots\!85\)\( T^{13} + \)\(51\!\cdots\!34\)\( T^{14} + \)\(18\!\cdots\!15\)\( T^{15} + \)\(62\!\cdots\!41\)\( T^{16} \)
$31$ \( 1 - 187 T - 2516004 T^{2} + 186847537 T^{3} + 3362431719041 T^{4} - 296059350096 T^{5} - 3460282108678916846 T^{6} - 13487262715981377034 T^{7} + \)\(31\!\cdots\!92\)\( T^{8} - \)\(12\!\cdots\!14\)\( T^{9} - \)\(29\!\cdots\!86\)\( T^{10} - \)\(23\!\cdots\!56\)\( T^{11} + \)\(24\!\cdots\!21\)\( T^{12} + \)\(12\!\cdots\!37\)\( T^{13} - \)\(15\!\cdots\!84\)\( T^{14} - \)\(10\!\cdots\!67\)\( T^{15} + \)\(52\!\cdots\!61\)\( T^{16} \)
$37$ \( ( 1 - 8 T + 3611368 T^{2} + 1256575624 T^{3} + 6911619203950 T^{4} + 2355025028051464 T^{5} + 12684855900547773928 T^{6} - 52663616046720282248 T^{7} + \)\(12\!\cdots\!41\)\( T^{8} )^{2} \)
$41$ \( 1 - 7920 T + 37687894 T^{2} - 132890424480 T^{3} + 385083705354505 T^{4} - 963185727644706960 T^{5} + \)\(21\!\cdots\!66\)\( T^{6} - \)\(41\!\cdots\!20\)\( T^{7} + \)\(74\!\cdots\!64\)\( T^{8} - \)\(11\!\cdots\!20\)\( T^{9} + \)\(16\!\cdots\!86\)\( T^{10} - \)\(21\!\cdots\!60\)\( T^{11} + \)\(24\!\cdots\!05\)\( T^{12} - \)\(23\!\cdots\!80\)\( T^{13} + \)\(19\!\cdots\!34\)\( T^{14} - \)\(11\!\cdots\!20\)\( T^{15} + \)\(40\!\cdots\!81\)\( T^{16} \)
$43$ \( 1 + 68 T - 12950604 T^{2} - 209786648 T^{3} + 102574547445791 T^{4} + 59145034173804 T^{5} - \)\(54\!\cdots\!36\)\( T^{6} + \)\(27\!\cdots\!16\)\( T^{7} + \)\(21\!\cdots\!12\)\( T^{8} + \)\(94\!\cdots\!16\)\( T^{9} - \)\(63\!\cdots\!36\)\( T^{10} + \)\(23\!\cdots\!04\)\( T^{11} + \)\(14\!\cdots\!91\)\( T^{12} - \)\(97\!\cdots\!48\)\( T^{13} - \)\(20\!\cdots\!04\)\( T^{14} + \)\(37\!\cdots\!68\)\( T^{15} + \)\(18\!\cdots\!01\)\( T^{16} \)
$47$ \( 1 + 13689 T + 103685338 T^{2} + 564293857959 T^{3} + 2435028217967227 T^{4} + 8736748337842042500 T^{5} + \)\(26\!\cdots\!72\)\( T^{6} + \)\(71\!\cdots\!32\)\( T^{7} + \)\(16\!\cdots\!20\)\( T^{8} + \)\(35\!\cdots\!92\)\( T^{9} + \)\(63\!\cdots\!92\)\( T^{10} + \)\(10\!\cdots\!00\)\( T^{11} + \)\(13\!\cdots\!67\)\( T^{12} + \)\(15\!\cdots\!59\)\( T^{13} + \)\(13\!\cdots\!78\)\( T^{14} + \)\(90\!\cdots\!29\)\( T^{15} + \)\(32\!\cdots\!41\)\( T^{16} \)
$53$ \( 1 - 5145920 T^{2} + 115452291970684 T^{4} - 84051566001475463360 T^{6} + \)\(67\!\cdots\!26\)\( T^{8} - \)\(52\!\cdots\!60\)\( T^{10} + \)\(44\!\cdots\!64\)\( T^{12} - \)\(12\!\cdots\!20\)\( T^{14} + \)\(15\!\cdots\!41\)\( T^{16} \)
$59$ \( 1 - 20052 T + 216711700 T^{2} - 1657982214864 T^{3} + 9931594296358591 T^{4} - 49579528565018409012 T^{5} + \)\(21\!\cdots\!48\)\( T^{6} - \)\(84\!\cdots\!76\)\( T^{7} + \)\(30\!\cdots\!08\)\( T^{8} - \)\(10\!\cdots\!36\)\( T^{9} + \)\(31\!\cdots\!08\)\( T^{10} - \)\(88\!\cdots\!72\)\( T^{11} + \)\(21\!\cdots\!31\)\( T^{12} - \)\(43\!\cdots\!64\)\( T^{13} + \)\(68\!\cdots\!00\)\( T^{14} - \)\(76\!\cdots\!92\)\( T^{15} + \)\(46\!\cdots\!81\)\( T^{16} \)
$61$ \( 1 + 1937 T - 10529634 T^{2} + 149647181023 T^{3} + 416288373490931 T^{4} - 1350680282380662864 T^{5} + \)\(12\!\cdots\!24\)\( T^{6} + \)\(40\!\cdots\!44\)\( T^{7} - \)\(98\!\cdots\!68\)\( T^{8} + \)\(55\!\cdots\!04\)\( T^{9} + \)\(23\!\cdots\!44\)\( T^{10} - \)\(35\!\cdots\!44\)\( T^{11} + \)\(15\!\cdots\!91\)\( T^{12} + \)\(76\!\cdots\!23\)\( T^{13} - \)\(74\!\cdots\!94\)\( T^{14} + \)\(18\!\cdots\!97\)\( T^{15} + \)\(13\!\cdots\!21\)\( T^{16} \)
$67$ \( 1 - 154 T - 33835854 T^{2} - 25606229228 T^{3} + 539365905411977 T^{4} + 738160924156362336 T^{5} + \)\(70\!\cdots\!02\)\( T^{6} - \)\(11\!\cdots\!78\)\( T^{7} - \)\(26\!\cdots\!64\)\( T^{8} - \)\(22\!\cdots\!38\)\( T^{9} + \)\(28\!\cdots\!82\)\( T^{10} + \)\(60\!\cdots\!96\)\( T^{11} + \)\(88\!\cdots\!37\)\( T^{12} - \)\(85\!\cdots\!28\)\( T^{13} - \)\(22\!\cdots\!34\)\( T^{14} - \)\(20\!\cdots\!14\)\( T^{15} + \)\(27\!\cdots\!61\)\( T^{16} \)
$71$ \( 1 - 68871716 T^{2} + 3244147638477940 T^{4} - \)\(11\!\cdots\!24\)\( T^{6} + \)\(30\!\cdots\!74\)\( T^{8} - \)\(73\!\cdots\!64\)\( T^{10} + \)\(13\!\cdots\!40\)\( T^{12} - \)\(18\!\cdots\!96\)\( T^{14} + \)\(17\!\cdots\!41\)\( T^{16} \)
$73$ \( ( 1 + 3901 T + 59309470 T^{2} + 292589317519 T^{3} + 2279602007321194 T^{4} + 8309021952930084079 T^{5} + \)\(47\!\cdots\!70\)\( T^{6} + \)\(89\!\cdots\!21\)\( T^{7} + \)\(65\!\cdots\!61\)\( T^{8} )^{2} \)
$79$ \( 1 + 2195 T - 87724914 T^{2} - 187644610415 T^{3} + 3128319215246375 T^{4} + 3711455091635884260 T^{5} - \)\(16\!\cdots\!36\)\( T^{6} + \)\(76\!\cdots\!80\)\( T^{7} + \)\(88\!\cdots\!64\)\( T^{8} + \)\(29\!\cdots\!80\)\( T^{9} - \)\(24\!\cdots\!96\)\( T^{10} + \)\(21\!\cdots\!60\)\( T^{11} + \)\(72\!\cdots\!75\)\( T^{12} - \)\(16\!\cdots\!15\)\( T^{13} - \)\(30\!\cdots\!34\)\( T^{14} + \)\(29\!\cdots\!95\)\( T^{15} + \)\(52\!\cdots\!41\)\( T^{16} \)
$83$ \( 1 + 37017 T + 725723290 T^{2} + 9956481997959 T^{3} + 104510585134438411 T^{4} + \)\(87\!\cdots\!72\)\( T^{5} + \)\(62\!\cdots\!28\)\( T^{6} + \)\(40\!\cdots\!76\)\( T^{7} + \)\(26\!\cdots\!48\)\( T^{8} + \)\(19\!\cdots\!96\)\( T^{9} + \)\(14\!\cdots\!48\)\( T^{10} + \)\(93\!\cdots\!92\)\( T^{11} + \)\(53\!\cdots\!91\)\( T^{12} + \)\(23\!\cdots\!59\)\( T^{13} + \)\(82\!\cdots\!90\)\( T^{14} + \)\(20\!\cdots\!97\)\( T^{15} + \)\(25\!\cdots\!61\)\( T^{16} \)
$89$ \( 1 - 294759296 T^{2} + 46567064448316540 T^{4} - \)\(48\!\cdots\!04\)\( T^{6} + \)\(35\!\cdots\!14\)\( T^{8} - \)\(19\!\cdots\!24\)\( T^{10} + \)\(72\!\cdots\!40\)\( T^{12} - \)\(17\!\cdots\!36\)\( T^{14} + \)\(24\!\cdots\!21\)\( T^{16} \)
$97$ \( 1 - 7282 T - 283226964 T^{2} + 993163976152 T^{3} + 57034963146137471 T^{4} - 97460573991801682656 T^{5} - \)\(75\!\cdots\!16\)\( T^{6} + \)\(33\!\cdots\!26\)\( T^{7} + \)\(75\!\cdots\!52\)\( T^{8} + \)\(29\!\cdots\!06\)\( T^{9} - \)\(58\!\cdots\!76\)\( T^{10} - \)\(67\!\cdots\!96\)\( T^{11} + \)\(35\!\cdots\!91\)\( T^{12} + \)\(54\!\cdots\!52\)\( T^{13} - \)\(13\!\cdots\!84\)\( T^{14} - \)\(31\!\cdots\!02\)\( T^{15} + \)\(37\!\cdots\!41\)\( T^{16} \)
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