# Properties

 Label 17.4.d Level 17 Weight 4 Character orbit d Rep. character $$\chi_{17}(2,\cdot)$$ Character field $$\Q(\zeta_{8})$$ Dimension 12 Newform subspaces 1 Sturm bound 6 Trace bound 0

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$17$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 17.d (of order $$8$$ and degree $$4$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$17$$ Character field: $$\Q(\zeta_{8})$$ Newform subspaces: $$1$$ Sturm bound: $$6$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(17, [\chi])$$.

Total New Old
Modular forms 20 20 0
Cusp forms 12 12 0
Eisenstein series 8 8 0

## Trace form

 $$12q - 4q^{2} - 4q^{3} - 20q^{5} + 20q^{6} - 4q^{7} + 28q^{8} - 64q^{9} + O(q^{10})$$ $$12q - 4q^{2} - 4q^{3} - 20q^{5} + 20q^{6} - 4q^{7} + 28q^{8} - 64q^{9} - 116q^{10} + 40q^{11} + 56q^{12} - 132q^{14} + 244q^{15} + 184q^{16} + 52q^{17} - 12q^{19} + 572q^{20} - 620q^{22} - 276q^{23} - 184q^{24} - 464q^{25} - 708q^{26} - 664q^{27} + 452q^{28} + 632q^{29} + 188q^{31} + 700q^{32} + 1400q^{33} + 764q^{34} - 632q^{35} + 524q^{36} + 940q^{37} - 1112q^{39} - 1864q^{40} + 176q^{41} + 48q^{42} - 1360q^{43} - 1364q^{44} - 32q^{45} + 452q^{46} - 540q^{48} + 1044q^{49} + 2856q^{50} + 340q^{51} + 792q^{52} - 360q^{53} - 244q^{54} - 1788q^{56} - 148q^{57} - 360q^{58} - 584q^{59} - 1792q^{60} - 1052q^{61} - 380q^{62} + 1752q^{63} + 404q^{65} + 1372q^{66} + 1080q^{67} + 2532q^{68} - 344q^{69} + 2072q^{70} + 28q^{71} + 824q^{73} - 2292q^{74} + 400q^{75} + 1328q^{76} - 1252q^{77} + 1128q^{78} - 196q^{79} - 904q^{80} - 1528q^{82} - 1008q^{83} - 4768q^{84} - 2824q^{85} - 1200q^{86} - 2516q^{87} - 56q^{88} - 860q^{90} + 2456q^{91} + 396q^{92} - 836q^{93} + 6360q^{94} + 2172q^{95} + 1668q^{96} - 904q^{97} + 3280q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(17, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
17.4.d.a $$12$$ $$1.003$$ $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ None $$-4$$ $$-4$$ $$-20$$ $$-4$$ $$q+(-\beta _{3}-\beta _{10})q^{2}+\beta _{8}q^{3}+(-\beta _{1}+\cdots)q^{4}+\cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 4 T + 8 T^{2} + 12 T^{3} - 62 T^{4} - 388 T^{5} - 984 T^{6} - 3700 T^{7} - 6783 T^{8} + 6520 T^{9} + 50208 T^{10} + 211072 T^{11} + 853440 T^{12} + 1688576 T^{13} + 3213312 T^{14} + 3338240 T^{15} - 27783168 T^{16} - 121241600 T^{17} - 257949696 T^{18} - 813694976 T^{19} - 1040187392 T^{20} + 1610612736 T^{21} + 8589934592 T^{22} + 34359738368 T^{23} + 68719476736 T^{24}$$
$3$ $$1 + 4 T + 40 T^{2} + 324 T^{3} + 1520 T^{4} + 17004 T^{5} + 93168 T^{6} + 623820 T^{7} + 3542815 T^{8} + 19106488 T^{9} + 116779528 T^{10} + 611518776 T^{11} + 3634702400 T^{12} + 16511006952 T^{13} + 85132275912 T^{14} + 376073003304 T^{15} + 1882797146415 T^{16} + 8951135164740 T^{17} + 36095192119152 T^{18} + 177867845863812 T^{19} + 429292895451120 T^{20} + 2470693585135788 T^{21} + 8235645283785960 T^{22} + 22236242266222092 T^{23} + 150094635296999121 T^{24}$$
$5$ $$1 + 20 T + 432 T^{2} + 5812 T^{3} + 76752 T^{4} + 671740 T^{5} + 5496800 T^{6} + 5471500 T^{7} - 426046641 T^{8} - 9209620720 T^{9} - 139560868176 T^{10} - 1662905986592 T^{11} - 19002322495072 T^{12} - 207863248324000 T^{13} - 2180638565250000 T^{14} - 17987540468750000 T^{15} - 104015293212890625 T^{16} + 166976928710937500 T^{17} + 20968627929687500000 T^{18} +$$$$32\!\cdots\!00$$$$T^{19} +$$$$45\!\cdots\!00$$$$T^{20} +$$$$43\!\cdots\!00$$$$T^{21} +$$$$40\!\cdots\!00$$$$T^{22} +$$$$23\!\cdots\!00$$$$T^{23} +$$$$14\!\cdots\!25$$$$T^{24}$$
$7$ $$1 + 4 T - 514 T^{2} - 6116 T^{3} + 115922 T^{4} + 1116940 T^{5} - 8154266 T^{6} + 477986852 T^{7} + 5461415235 T^{8} - 238144861440 T^{9} - 5762877749244 T^{10} + 45937354375984 T^{11} + 1745889417070628 T^{12} + 15756512550962512 T^{13} - 677996804320807356 T^{14} - 9610004147619214080 T^{15} + 75593016791551907235 T^{16} +$$$$22\!\cdots\!36$$$$T^{17} -$$$$13\!\cdots\!34$$$$T^{18} +$$$$62\!\cdots\!80$$$$T^{19} +$$$$22\!\cdots\!22$$$$T^{20} -$$$$40\!\cdots\!88$$$$T^{21} -$$$$11\!\cdots\!86$$$$T^{22} +$$$$30\!\cdots\!28$$$$T^{23} +$$$$26\!\cdots\!01$$$$T^{24}$$
$11$ $$1 - 40 T + 4148 T^{2} - 162224 T^{3} + 9095112 T^{4} - 332343560 T^{5} + 18025759684 T^{6} - 645155164752 T^{7} + 32705329538127 T^{8} - 1145801951121704 T^{9} + 50413798046517864 T^{10} - 1580384424194772536 T^{11} + 69762836759754864688 T^{12} -$$$$21\!\cdots\!16$$$$T^{13} +$$$$89\!\cdots\!04$$$$T^{14} -$$$$27\!\cdots\!64$$$$T^{15} +$$$$10\!\cdots\!67$$$$T^{16} -$$$$26\!\cdots\!52$$$$T^{17} +$$$$10\!\cdots\!04$$$$T^{18} -$$$$24\!\cdots\!60$$$$T^{19} +$$$$89\!\cdots\!92$$$$T^{20} -$$$$21\!\cdots\!04$$$$T^{21} +$$$$72\!\cdots\!48$$$$T^{22} -$$$$92\!\cdots\!40$$$$T^{23} +$$$$30\!\cdots\!61$$$$T^{24}$$
$13$ $$1 - 14448 T^{2} + 107666638 T^{4} - 546851229616 T^{6} + 2094008234358095 T^{8} - 6327905324283989088 T^{10} +$$$$15\!\cdots\!00$$$$T^{12} -$$$$30\!\cdots\!92$$$$T^{14} +$$$$48\!\cdots\!95$$$$T^{16} -$$$$61\!\cdots\!64$$$$T^{18} +$$$$58\!\cdots\!18$$$$T^{20} -$$$$37\!\cdots\!52$$$$T^{22} +$$$$12\!\cdots\!41$$$$T^{24}$$
$17$ $$1 - 52 T - 1224 T^{2} + 491300 T^{3} - 28461009 T^{4} - 1278873552 T^{5} + 118745480624 T^{6} - 6283105760976 T^{7} - 686979568547121 T^{8} + 58262223722976100 T^{9} - 713129618369227464 T^{10} -$$$$14\!\cdots\!36$$$$T^{11} +$$$$14\!\cdots\!09$$$$T^{12}$$
$19$ $$1 + 12 T + 72 T^{2} + 1356220 T^{3} + 80627506 T^{4} + 1256173316 T^{5} + 928935243560 T^{6} + 122232243190932 T^{7} + 4866957296223951 T^{8} + 356984531389493592 T^{9} + 95598460087921452240 T^{10} +$$$$61\!\cdots\!04$$$$T^{11} +$$$$12\!\cdots\!48$$$$T^{12} +$$$$42\!\cdots\!36$$$$T^{13} +$$$$44\!\cdots\!40$$$$T^{14} +$$$$11\!\cdots\!68$$$$T^{15} +$$$$10\!\cdots\!11$$$$T^{16} +$$$$18\!\cdots\!68$$$$T^{17} +$$$$96\!\cdots\!60$$$$T^{18} +$$$$89\!\cdots\!04$$$$T^{19} +$$$$39\!\cdots\!26$$$$T^{20} +$$$$45\!\cdots\!80$$$$T^{21} +$$$$16\!\cdots\!72$$$$T^{22} +$$$$18\!\cdots\!08$$$$T^{23} +$$$$10\!\cdots\!81$$$$T^{24}$$
$23$ $$1 + 276 T + 17214 T^{2} - 1142692 T^{3} - 27820014 T^{4} + 28497242460 T^{5} + 1982704026150 T^{6} - 65861887634364 T^{7} + 2296978221239043 T^{8} - 875617922313069328 T^{9} -$$$$39\!\cdots\!16$$$$T^{10} +$$$$14\!\cdots\!48$$$$T^{11} +$$$$80\!\cdots\!56$$$$T^{12} +$$$$17\!\cdots\!16$$$$T^{13} -$$$$58\!\cdots\!24$$$$T^{14} -$$$$15\!\cdots\!64$$$$T^{15} +$$$$50\!\cdots\!03$$$$T^{16} -$$$$17\!\cdots\!48$$$$T^{17} +$$$$64\!\cdots\!50$$$$T^{18} +$$$$11\!\cdots\!80$$$$T^{19} -$$$$13\!\cdots\!74$$$$T^{20} -$$$$66\!\cdots\!24$$$$T^{21} +$$$$12\!\cdots\!86$$$$T^{22} +$$$$23\!\cdots\!08$$$$T^{23} +$$$$10\!\cdots\!61$$$$T^{24}$$
$29$ $$1 - 632 T + 193012 T^{2} - 31241744 T^{3} + 1162856136 T^{4} + 678887482016 T^{5} - 143704921685972 T^{6} + 1869446104826552 T^{7} + 4704974888482939551 T^{8} -$$$$10\!\cdots\!44$$$$T^{9} +$$$$72\!\cdots\!24$$$$T^{10} +$$$$12\!\cdots\!52$$$$T^{11} -$$$$36\!\cdots\!96$$$$T^{12} +$$$$29\!\cdots\!28$$$$T^{13} +$$$$42\!\cdots\!04$$$$T^{14} -$$$$14\!\cdots\!36$$$$T^{15} +$$$$16\!\cdots\!91$$$$T^{16} +$$$$16\!\cdots\!48$$$$T^{17} -$$$$30\!\cdots\!92$$$$T^{18} +$$$$34\!\cdots\!64$$$$T^{19} +$$$$14\!\cdots\!16$$$$T^{20} -$$$$95\!\cdots\!96$$$$T^{21} +$$$$14\!\cdots\!12$$$$T^{22} -$$$$11\!\cdots\!48$$$$T^{23} +$$$$44\!\cdots\!21$$$$T^{24}$$
$31$ $$1 - 188 T + 22934 T^{2} + 18386644 T^{3} - 3691984606 T^{4} + 776536429588 T^{5} + 99645628308078 T^{6} - 25120109404874988 T^{7} + 8682201249054683203 T^{8} -$$$$15\!\cdots\!80$$$$T^{9} +$$$$20\!\cdots\!44$$$$T^{10} +$$$$40\!\cdots\!64$$$$T^{11} -$$$$34\!\cdots\!20$$$$T^{12} +$$$$11\!\cdots\!24$$$$T^{13} +$$$$17\!\cdots\!64$$$$T^{14} -$$$$41\!\cdots\!80$$$$T^{15} +$$$$68\!\cdots\!83$$$$T^{16} -$$$$58\!\cdots\!88$$$$T^{17} +$$$$69\!\cdots\!98$$$$T^{18} +$$$$16\!\cdots\!28$$$$T^{19} -$$$$22\!\cdots\!26$$$$T^{20} +$$$$33\!\cdots\!84$$$$T^{21} +$$$$12\!\cdots\!34$$$$T^{22} -$$$$30\!\cdots\!08$$$$T^{23} +$$$$48\!\cdots\!81$$$$T^{24}$$
$37$ $$1 - 940 T + 400776 T^{2} - 69989748 T^{3} - 12837369392 T^{4} + 10952154351924 T^{5} - 2855667555891144 T^{6} + 200153198828812748 T^{7} +$$$$10\!\cdots\!87$$$$T^{8} -$$$$40\!\cdots\!08$$$$T^{9} +$$$$48\!\cdots\!68$$$$T^{10} +$$$$89\!\cdots\!52$$$$T^{11} -$$$$42\!\cdots\!88$$$$T^{12} +$$$$45\!\cdots\!56$$$$T^{13} +$$$$12\!\cdots\!12$$$$T^{14} -$$$$53\!\cdots\!16$$$$T^{15} +$$$$71\!\cdots\!47$$$$T^{16} +$$$$66\!\cdots\!64$$$$T^{17} -$$$$48\!\cdots\!76$$$$T^{18} +$$$$93\!\cdots\!88$$$$T^{19} -$$$$55\!\cdots\!12$$$$T^{20} -$$$$15\!\cdots\!84$$$$T^{21} +$$$$44\!\cdots\!24$$$$T^{22} -$$$$52\!\cdots\!80$$$$T^{23} +$$$$28\!\cdots\!41$$$$T^{24}$$
$41$ $$1 - 176 T - 88034 T^{2} + 34173648 T^{3} + 827249858 T^{4} - 2621758340656 T^{5} + 280349260544702 T^{6} + 147031456973046352 T^{7} - 22298918693016177249 T^{8} -$$$$82\!\cdots\!96$$$$T^{9} +$$$$33\!\cdots\!96$$$$T^{10} +$$$$28\!\cdots\!24$$$$T^{11} -$$$$30\!\cdots\!88$$$$T^{12} +$$$$19\!\cdots\!04$$$$T^{13} +$$$$15\!\cdots\!36$$$$T^{14} -$$$$26\!\cdots\!56$$$$T^{15} -$$$$50\!\cdots\!69$$$$T^{16} +$$$$22\!\cdots\!52$$$$T^{17} +$$$$30\!\cdots\!42$$$$T^{18} -$$$$19\!\cdots\!96$$$$T^{19} +$$$$42\!\cdots\!38$$$$T^{20} +$$$$11\!\cdots\!88$$$$T^{21} -$$$$21\!\cdots\!34$$$$T^{22} -$$$$29\!\cdots\!96$$$$T^{23} +$$$$11\!\cdots\!41$$$$T^{24}$$
$43$ $$1 + 1360 T + 924800 T^{2} + 445088144 T^{3} + 187950043870 T^{4} + 76345837363280 T^{5} + 29065866207767168 T^{6} + 10133858719746215760 T^{7} +$$$$33\!\cdots\!95$$$$T^{8} +$$$$10\!\cdots\!56$$$$T^{9} +$$$$34\!\cdots\!40$$$$T^{10} +$$$$10\!\cdots\!60$$$$T^{11} +$$$$29\!\cdots\!16$$$$T^{12} +$$$$81\!\cdots\!20$$$$T^{13} +$$$$21\!\cdots\!60$$$$T^{14} +$$$$54\!\cdots\!08$$$$T^{15} +$$$$13\!\cdots\!95$$$$T^{16} +$$$$32\!\cdots\!20$$$$T^{17} +$$$$73\!\cdots\!32$$$$T^{18} +$$$$15\!\cdots\!40$$$$T^{19} +$$$$30\!\cdots\!70$$$$T^{20} +$$$$56\!\cdots\!08$$$$T^{21} +$$$$93\!\cdots\!00$$$$T^{22} +$$$$10\!\cdots\!80$$$$T^{23} +$$$$63\!\cdots\!01$$$$T^{24}$$
$47$ $$1 - 580412 T^{2} + 182778426498 T^{4} - 40216713491295692 T^{6} +$$$$68\!\cdots\!79$$$$T^{8} -$$$$94\!\cdots\!00$$$$T^{10} +$$$$10\!\cdots\!68$$$$T^{12} -$$$$10\!\cdots\!00$$$$T^{14} +$$$$79\!\cdots\!39$$$$T^{16} -$$$$50\!\cdots\!88$$$$T^{18} +$$$$24\!\cdots\!38$$$$T^{20} -$$$$84\!\cdots\!88$$$$T^{22} +$$$$15\!\cdots\!21$$$$T^{24}$$
$53$ $$1 + 360 T + 64800 T^{2} + 135249864 T^{3} + 60322424078 T^{4} + 11373697219528 T^{5} + 9331900774784928 T^{6} + 6170118859401097768 T^{7} +$$$$17\!\cdots\!63$$$$T^{8} +$$$$52\!\cdots\!24$$$$T^{9} +$$$$41\!\cdots\!00$$$$T^{10} +$$$$14\!\cdots\!20$$$$T^{11} +$$$$24\!\cdots\!08$$$$T^{12} +$$$$21\!\cdots\!40$$$$T^{13} +$$$$91\!\cdots\!00$$$$T^{14} +$$$$17\!\cdots\!92$$$$T^{15} +$$$$84\!\cdots\!83$$$$T^{16} +$$$$45\!\cdots\!76$$$$T^{17} +$$$$10\!\cdots\!92$$$$T^{18} +$$$$18\!\cdots\!84$$$$T^{19} +$$$$14\!\cdots\!18$$$$T^{20} +$$$$48\!\cdots\!68$$$$T^{21} +$$$$34\!\cdots\!00$$$$T^{22} +$$$$28\!\cdots\!80$$$$T^{23} +$$$$11\!\cdots\!21$$$$T^{24}$$
$59$ $$1 + 584 T + 170528 T^{2} - 175550056 T^{3} - 94146703170 T^{4} + 18580823027848 T^{5} + 42314760727238560 T^{6} + 18131962503470660184 T^{7} -$$$$19\!\cdots\!73$$$$T^{8} -$$$$38\!\cdots\!88$$$$T^{9} -$$$$96\!\cdots\!00$$$$T^{10} +$$$$60\!\cdots\!68$$$$T^{11} +$$$$52\!\cdots\!08$$$$T^{12} +$$$$12\!\cdots\!72$$$$T^{13} -$$$$40\!\cdots\!00$$$$T^{14} -$$$$33\!\cdots\!32$$$$T^{15} -$$$$34\!\cdots\!13$$$$T^{16} +$$$$66\!\cdots\!16$$$$T^{17} +$$$$31\!\cdots\!60$$$$T^{18} +$$$$28\!\cdots\!32$$$$T^{19} -$$$$29\!\cdots\!70$$$$T^{20} -$$$$11\!\cdots\!64$$$$T^{21} +$$$$22\!\cdots\!28$$$$T^{22} +$$$$16\!\cdots\!36$$$$T^{23} +$$$$56\!\cdots\!41$$$$T^{24}$$
$61$ $$1 + 1052 T + 1006376 T^{2} + 412979636 T^{3} + 133288994960 T^{4} - 72852168865956 T^{5} - 77682489128187272 T^{6} - 64250295850085738220 T^{7} -$$$$24\!\cdots\!17$$$$T^{8} -$$$$69\!\cdots\!76$$$$T^{9} +$$$$27\!\cdots\!44$$$$T^{10} +$$$$28\!\cdots\!40$$$$T^{11} +$$$$20\!\cdots\!96$$$$T^{12} +$$$$65\!\cdots\!40$$$$T^{13} +$$$$13\!\cdots\!84$$$$T^{14} -$$$$81\!\cdots\!16$$$$T^{15} -$$$$65\!\cdots\!57$$$$T^{16} -$$$$38\!\cdots\!20$$$$T^{17} -$$$$10\!\cdots\!32$$$$T^{18} -$$$$22\!\cdots\!16$$$$T^{19} +$$$$93\!\cdots\!60$$$$T^{20} +$$$$66\!\cdots\!56$$$$T^{21} +$$$$36\!\cdots\!76$$$$T^{22} +$$$$86\!\cdots\!12$$$$T^{23} +$$$$18\!\cdots\!61$$$$T^{24}$$
$67$ $$( 1 - 540 T + 1237196 T^{2} - 437516948 T^{3} + 687431155711 T^{4} - 191267075797288 T^{5} + 250196846973981368 T^{6} - 57526059518019730744 T^{7} +$$$$62\!\cdots\!59$$$$T^{8} -$$$$11\!\cdots\!56$$$$T^{9} +$$$$10\!\cdots\!56$$$$T^{10} -$$$$13\!\cdots\!20$$$$T^{11} +$$$$74\!\cdots\!09$$$$T^{12} )^{2}$$
$71$ $$1 - 28 T - 195882 T^{2} + 194007508 T^{3} + 13906393890 T^{4} - 24274230169564 T^{5} + 39419381006302414 T^{6} + 24734369575425164292 T^{7} -$$$$95\!\cdots\!37$$$$T^{8} -$$$$41\!\cdots\!16$$$$T^{9} +$$$$42\!\cdots\!20$$$$T^{10} +$$$$13\!\cdots\!48$$$$T^{11} -$$$$63\!\cdots\!80$$$$T^{12} +$$$$49\!\cdots\!28$$$$T^{13} +$$$$54\!\cdots\!20$$$$T^{14} -$$$$19\!\cdots\!96$$$$T^{15} -$$$$15\!\cdots\!17$$$$T^{16} +$$$$14\!\cdots\!92$$$$T^{17} +$$$$82\!\cdots\!54$$$$T^{18} -$$$$18\!\cdots\!44$$$$T^{19} +$$$$37\!\cdots\!90$$$$T^{20} +$$$$18\!\cdots\!28$$$$T^{21} -$$$$67\!\cdots\!82$$$$T^{22} -$$$$34\!\cdots\!08$$$$T^{23} +$$$$44\!\cdots\!21$$$$T^{24}$$
$73$ $$1 - 824 T + 588222 T^{2} + 9262472 T^{3} - 118766234814 T^{4} + 230361433132360 T^{5} - 186489223297230882 T^{6} +$$$$15\!\cdots\!92$$$$T^{7} -$$$$70\!\cdots\!77$$$$T^{8} +$$$$10\!\cdots\!48$$$$T^{9} +$$$$24\!\cdots\!96$$$$T^{10} -$$$$26\!\cdots\!64$$$$T^{11} +$$$$16\!\cdots\!68$$$$T^{12} -$$$$10\!\cdots\!88$$$$T^{13} +$$$$37\!\cdots\!44$$$$T^{14} +$$$$61\!\cdots\!24$$$$T^{15} -$$$$16\!\cdots\!17$$$$T^{16} +$$$$13\!\cdots\!44$$$$T^{17} -$$$$64\!\cdots\!58$$$$T^{18} +$$$$31\!\cdots\!80$$$$T^{19} -$$$$62\!\cdots\!74$$$$T^{20} +$$$$18\!\cdots\!84$$$$T^{21} +$$$$46\!\cdots\!78$$$$T^{22} -$$$$25\!\cdots\!92$$$$T^{23} +$$$$12\!\cdots\!61$$$$T^{24}$$
$79$ $$1 + 196 T + 788046 T^{2} + 351355900 T^{3} + 349469377586 T^{4} + 376015608150652 T^{5} + 21556020940992630 T^{6} +$$$$18\!\cdots\!52$$$$T^{7} +$$$$20\!\cdots\!43$$$$T^{8} +$$$$54\!\cdots\!52$$$$T^{9} +$$$$53\!\cdots\!64$$$$T^{10} -$$$$49\!\cdots\!12$$$$T^{11} +$$$$41\!\cdots\!72$$$$T^{12} -$$$$24\!\cdots\!68$$$$T^{13} +$$$$13\!\cdots\!44$$$$T^{14} +$$$$64\!\cdots\!88$$$$T^{15} +$$$$12\!\cdots\!63$$$$T^{16} +$$$$54\!\cdots\!48$$$$T^{17} +$$$$30\!\cdots\!30$$$$T^{18} +$$$$26\!\cdots\!08$$$$T^{19} +$$$$12\!\cdots\!66$$$$T^{20} +$$$$60\!\cdots\!00$$$$T^{21} +$$$$66\!\cdots\!46$$$$T^{22} +$$$$82\!\cdots\!44$$$$T^{23} +$$$$20\!\cdots\!21$$$$T^{24}$$
$83$ $$1 + 1008 T + 508032 T^{2} + 608566144 T^{3} + 1311140396574 T^{4} + 925961362981888 T^{5} + 452444151744975104 T^{6} +$$$$51\!\cdots\!40$$$$T^{7} +$$$$71\!\cdots\!79$$$$T^{8} +$$$$40\!\cdots\!08$$$$T^{9} +$$$$20\!\cdots\!72$$$$T^{10} +$$$$22\!\cdots\!60$$$$T^{11} +$$$$25\!\cdots\!56$$$$T^{12} +$$$$12\!\cdots\!20$$$$T^{13} +$$$$65\!\cdots\!68$$$$T^{14} +$$$$76\!\cdots\!24$$$$T^{15} +$$$$76\!\cdots\!19$$$$T^{16} +$$$$31\!\cdots\!80$$$$T^{17} +$$$$15\!\cdots\!36$$$$T^{18} +$$$$18\!\cdots\!04$$$$T^{19} +$$$$14\!\cdots\!54$$$$T^{20} +$$$$39\!\cdots\!88$$$$T^{21} +$$$$18\!\cdots\!68$$$$T^{22} +$$$$21\!\cdots\!04$$$$T^{23} +$$$$12\!\cdots\!81$$$$T^{24}$$
$89$ $$1 - 5719348 T^{2} + 16180825465546 T^{4} - 29845545554168255940 T^{6} +$$$$39\!\cdots\!99$$$$T^{8} -$$$$40\!\cdots\!04$$$$T^{10} +$$$$32\!\cdots\!60$$$$T^{12} -$$$$20\!\cdots\!44$$$$T^{14} +$$$$98\!\cdots\!79$$$$T^{16} -$$$$36\!\cdots\!40$$$$T^{18} +$$$$98\!\cdots\!86$$$$T^{20} -$$$$17\!\cdots\!48$$$$T^{22} +$$$$15\!\cdots\!61$$$$T^{24}$$
$97$ $$1 + 904 T + 307166 T^{2} - 2157443544 T^{3} - 1987213960190 T^{4} - 448060583775704 T^{5} + 3336323390652175454 T^{6} +$$$$24\!\cdots\!08$$$$T^{7} +$$$$10\!\cdots\!15$$$$T^{8} -$$$$37\!\cdots\!80$$$$T^{9} -$$$$19\!\cdots\!56$$$$T^{10} +$$$$70\!\cdots\!36$$$$T^{11} +$$$$35\!\cdots\!84$$$$T^{12} +$$$$64\!\cdots\!28$$$$T^{13} -$$$$15\!\cdots\!24$$$$T^{14} -$$$$28\!\cdots\!60$$$$T^{15} +$$$$74\!\cdots\!15$$$$T^{16} +$$$$15\!\cdots\!44$$$$T^{17} +$$$$19\!\cdots\!06$$$$T^{18} -$$$$23\!\cdots\!88$$$$T^{19} -$$$$95\!\cdots\!90$$$$T^{20} -$$$$94\!\cdots\!72$$$$T^{21} +$$$$12\!\cdots\!34$$$$T^{22} +$$$$33\!\cdots\!08$$$$T^{23} +$$$$33\!\cdots\!21$$$$T^{24}$$