# 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

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

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