# Properties

 Label 48.3.l Level 48 Weight 3 Character orbit l Rep. character $$\chi_{48}(19,\cdot)$$ Character field $$\Q(\zeta_{4})$$ Dimension 16 Newform subspaces 1 Sturm bound 24 Trace bound 0

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 48.l (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$16$$ Character field: $$\Q(i)$$ Newform subspaces: $$1$$ Sturm bound: $$24$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(48, [\chi])$$.

Total New Old
Modular forms 36 16 20
Cusp forms 28 16 12
Eisenstein series 8 0 8

## Trace form

 $$16q + 12q^{4} - 12q^{8} + O(q^{10})$$ $$16q + 12q^{4} - 12q^{8} - 56q^{10} + 32q^{11} - 24q^{12} - 44q^{14} + 32q^{16} + 12q^{18} - 32q^{19} + 80q^{20} + 32q^{22} - 128q^{23} + 36q^{24} - 100q^{26} - 120q^{28} + 32q^{29} + 72q^{30} + 160q^{32} + 96q^{34} + 96q^{35} + 12q^{36} - 96q^{37} + 168q^{38} + 48q^{40} - 60q^{42} + 160q^{43} + 88q^{44} + 136q^{46} - 144q^{48} + 112q^{49} - 236q^{50} - 96q^{51} - 48q^{52} - 160q^{53} - 36q^{54} - 256q^{55} - 224q^{56} + 144q^{58} - 128q^{59} - 72q^{60} - 32q^{61} - 276q^{62} - 408q^{64} - 32q^{65} + 72q^{66} + 320q^{67} - 448q^{68} + 96q^{69} - 384q^{70} + 512q^{71} + 60q^{72} + 348q^{74} + 192q^{75} + 72q^{76} + 224q^{77} + 396q^{78} + 552q^{80} - 144q^{81} - 40q^{82} - 160q^{83} + 72q^{84} + 160q^{85} + 528q^{86} + 480q^{88} - 24q^{90} - 480q^{91} + 496q^{92} + 312q^{94} - 480q^{96} - 440q^{98} + 96q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(48, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
48.3.l.a $$16$$ $$1.308$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{3}q^{2}-\beta _{5}q^{3}+(1-\beta _{2})q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{3}^{\mathrm{old}}(48, [\chi])$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(48, [\chi]) \cong$$ $$S_{3}^{\mathrm{new}}(16, [\chi])$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 6 T^{2} + 4 T^{3} + 10 T^{4} - 56 T^{5} + 88 T^{6} + 128 T^{7} - 496 T^{8} + 512 T^{9} + 1408 T^{10} - 3584 T^{11} + 2560 T^{12} + 4096 T^{13} - 24576 T^{14} + 65536 T^{16}$$
$3$ $$( 1 + 9 T^{4} )^{4}$$
$5$ $$1 + 32 T^{3} - 344 T^{4} + 5664 T^{5} + 512 T^{6} + 145600 T^{7} + 223452 T^{8} - 2255168 T^{9} + 20875776 T^{10} - 67282720 T^{11} + 753060504 T^{12} + 1881828576 T^{13} + 2740220928 T^{14} + 75471830656 T^{15} - 399298967994 T^{16} + 1886795766400 T^{17} + 1712638080000 T^{18} + 29403571500000 T^{19} + 294164259375000 T^{20} - 657057812500000 T^{21} + 5096625000000000 T^{22} - 13764453125000000 T^{23} + 34096069335937500 T^{24} + 555419921875000000 T^{25} + 48828125000000000 T^{26} + 13504028320312500000 T^{27} - 20503997802734375000 T^{28} + 47683715820312500000 T^{29} +$$$$23\!\cdots\!25$$$$T^{32}$$
$7$ $$( 1 + 168 T^{2} - 448 T^{3} + 15076 T^{4} - 56512 T^{5} + 1070392 T^{6} - 3649664 T^{7} + 60103046 T^{8} - 178833536 T^{9} + 2570011192 T^{10} - 6648580288 T^{11} + 86910139876 T^{12} - 126548911552 T^{13} + 2325336249768 T^{14} + 33232930569601 T^{16} )^{2}$$
$11$ $$1 - 32 T + 512 T^{2} - 8480 T^{3} + 137032 T^{4} - 1636576 T^{5} + 18165248 T^{6} - 219655136 T^{7} + 2263228700 T^{8} - 21867108000 T^{9} + 253620152832 T^{10} - 2916953293728 T^{11} + 33797606438392 T^{12} - 431768458252384 T^{13} + 5293166227138048 T^{14} - 61910274521995104 T^{15} + 703855220885889990 T^{16} - 7491143217161407584 T^{17} + 77497246731528160768 T^{18} -$$$$76\!\cdots\!24$$$$T^{19} +$$$$72\!\cdots\!52$$$$T^{20} -$$$$75\!\cdots\!28$$$$T^{21} +$$$$79\!\cdots\!72$$$$T^{22} -$$$$83\!\cdots\!00$$$$T^{23} +$$$$10\!\cdots\!00$$$$T^{24} -$$$$12\!\cdots\!16$$$$T^{25} +$$$$12\!\cdots\!48$$$$T^{26} -$$$$13\!\cdots\!96$$$$T^{27} +$$$$13\!\cdots\!12$$$$T^{28} -$$$$10\!\cdots\!80$$$$T^{29} +$$$$73\!\cdots\!72$$$$T^{30} -$$$$55\!\cdots\!32$$$$T^{31} +$$$$21\!\cdots\!21$$$$T^{32}$$
$13$ $$1 + 3200 T^{3} + 7608 T^{4} + 95360 T^{5} + 5120000 T^{6} + 68335872 T^{7} + 2004669468 T^{8} + 7270355200 T^{9} + 184268595200 T^{10} + 4889456013184 T^{11} + 5354592144136 T^{12} + 669839496880000 T^{13} + 7008632866619392 T^{14} + 70586941744778752 T^{15} + 2398056097119178950 T^{16} + 11929193154867609088 T^{17} +$$$$20\!\cdots\!12$$$$T^{18} +$$$$32\!\cdots\!00$$$$T^{19} +$$$$43\!\cdots\!56$$$$T^{20} +$$$$67\!\cdots\!16$$$$T^{21} +$$$$42\!\cdots\!00$$$$T^{22} +$$$$28\!\cdots\!00$$$$T^{23} +$$$$13\!\cdots\!88$$$$T^{24} +$$$$76\!\cdots\!88$$$$T^{25} +$$$$97\!\cdots\!00$$$$T^{26} +$$$$30\!\cdots\!40$$$$T^{27} +$$$$41\!\cdots\!88$$$$T^{28} +$$$$29\!\cdots\!00$$$$T^{29} +$$$$44\!\cdots\!81$$$$T^{32}$$
$17$ $$( 1 + 968 T^{2} - 2944 T^{3} + 516540 T^{4} - 3209600 T^{5} + 201700088 T^{6} - 1543904000 T^{7} + 63894476806 T^{8} - 446188256000 T^{9} + 16846193049848 T^{10} - 77471941462400 T^{11} + 3603257748574140 T^{12} - 5935086042921856 T^{13} + 563978325638408648 T^{14} + 48661191875666868481 T^{16} )^{2}$$
$19$ $$1 + 32 T + 512 T^{2} - 2656 T^{3} - 523448 T^{4} - 8424608 T^{5} + 1945088 T^{6} + 4454446304 T^{7} + 107916937244 T^{8} - 703649376 T^{9} - 30601835632128 T^{10} - 698985761087712 T^{11} + 998616856187896 T^{12} + 253693358084547040 T^{13} + 3161998119961945600 T^{14} - 30474951661580761248 T^{15} -$$$$19\!\cdots\!42$$$$T^{16} -$$$$11\!\cdots\!28$$$$T^{17} +$$$$41\!\cdots\!00$$$$T^{18} +$$$$11\!\cdots\!40$$$$T^{19} +$$$$16\!\cdots\!36$$$$T^{20} -$$$$42\!\cdots\!12$$$$T^{21} -$$$$67\!\cdots\!08$$$$T^{22} -$$$$56\!\cdots\!96$$$$T^{23} +$$$$31\!\cdots\!64$$$$T^{24} +$$$$46\!\cdots\!64$$$$T^{25} +$$$$73\!\cdots\!88$$$$T^{26} -$$$$11\!\cdots\!88$$$$T^{27} -$$$$25\!\cdots\!08$$$$T^{28} -$$$$46\!\cdots\!36$$$$T^{29} +$$$$32\!\cdots\!92$$$$T^{30} +$$$$73\!\cdots\!32$$$$T^{31} +$$$$83\!\cdots\!61$$$$T^{32}$$
$23$ $$( 1 + 64 T + 3496 T^{2} + 127936 T^{3} + 4410332 T^{4} + 130001728 T^{5} + 3673719192 T^{6} + 94049622208 T^{7} + 2261818535238 T^{8} + 49752250148032 T^{9} + 1028057252408472 T^{10} + 19244921376016192 T^{11} + 345377444336323292 T^{12} + 5299942138629398464 T^{13} + 76613527014343042216 T^{14} +$$$$74\!\cdots\!76$$$$T^{15} +$$$$61\!\cdots\!61$$$$T^{16} )^{2}$$
$29$ $$1 - 32 T + 512 T^{2} + 18368 T^{3} - 1552984 T^{4} + 20596992 T^{5} + 304715776 T^{6} - 25469097376 T^{7} + 491466517980 T^{8} + 9791032230816 T^{9} - 347423504794624 T^{10} + 2649303176415616 T^{11} + 694517140133881240 T^{12} - 20658732330776531008 T^{13} +$$$$19\!\cdots\!28$$$$T^{14} +$$$$11\!\cdots\!40$$$$T^{15} -$$$$82\!\cdots\!10$$$$T^{16} +$$$$96\!\cdots\!40$$$$T^{17} +$$$$13\!\cdots\!68$$$$T^{18} -$$$$12\!\cdots\!68$$$$T^{19} +$$$$34\!\cdots\!40$$$$T^{20} +$$$$11\!\cdots\!16$$$$T^{21} -$$$$12\!\cdots\!84$$$$T^{22} +$$$$29\!\cdots\!96$$$$T^{23} +$$$$12\!\cdots\!80$$$$T^{24} -$$$$53\!\cdots\!36$$$$T^{25} +$$$$53\!\cdots\!76$$$$T^{26} +$$$$30\!\cdots\!72$$$$T^{27} -$$$$19\!\cdots\!04$$$$T^{28} +$$$$19\!\cdots\!28$$$$T^{29} +$$$$45\!\cdots\!32$$$$T^{30} -$$$$23\!\cdots\!32$$$$T^{31} +$$$$62\!\cdots\!41$$$$T^{32}$$
$31$ $$1 - 7312 T^{2} + 29025544 T^{4} - 80335806576 T^{6} + 171125889681052 T^{8} - 295006946315669072 T^{10} +$$$$42\!\cdots\!64$$$$T^{12} -$$$$51\!\cdots\!00$$$$T^{14} +$$$$53\!\cdots\!38$$$$T^{16} -$$$$47\!\cdots\!00$$$$T^{18} +$$$$36\!\cdots\!24$$$$T^{20} -$$$$23\!\cdots\!92$$$$T^{22} +$$$$12\!\cdots\!12$$$$T^{24} -$$$$53\!\cdots\!76$$$$T^{26} +$$$$18\!\cdots\!24$$$$T^{28} -$$$$41\!\cdots\!92$$$$T^{30} +$$$$52\!\cdots\!61$$$$T^{32}$$
$37$ $$1 + 96 T + 4608 T^{2} + 145952 T^{3} + 4040888 T^{4} + 217733344 T^{5} + 12932982272 T^{6} + 602883756192 T^{7} + 21839639792924 T^{8} + 655265530977504 T^{9} + 21703692469355008 T^{10} + 815191556064282016 T^{11} + 35433653736114978312 T^{12} +$$$$14\!\cdots\!40$$$$T^{13} +$$$$50\!\cdots\!52$$$$T^{14} +$$$$14\!\cdots\!84$$$$T^{15} +$$$$43\!\cdots\!90$$$$T^{16} +$$$$20\!\cdots\!96$$$$T^{17} +$$$$95\!\cdots\!72$$$$T^{18} +$$$$37\!\cdots\!60$$$$T^{19} +$$$$12\!\cdots\!52$$$$T^{20} +$$$$39\!\cdots\!84$$$$T^{21} +$$$$14\!\cdots\!48$$$$T^{22} +$$$$59\!\cdots\!56$$$$T^{23} +$$$$26\!\cdots\!84$$$$T^{24} +$$$$10\!\cdots\!68$$$$T^{25} +$$$$29\!\cdots\!72$$$$T^{26} +$$$$68\!\cdots\!36$$$$T^{27} +$$$$17\!\cdots\!68$$$$T^{28} +$$$$86\!\cdots\!68$$$$T^{29} +$$$$37\!\cdots\!68$$$$T^{30} +$$$$10\!\cdots\!04$$$$T^{31} +$$$$15\!\cdots\!81$$$$T^{32}$$
$41$ $$1 - 13840 T^{2} + 102706104 T^{4} - 524939980080 T^{6} + 2044068651261084 T^{8} - 6376104819902485008 T^{10} +$$$$16\!\cdots\!68$$$$T^{12} -$$$$35\!\cdots\!72$$$$T^{14} +$$$$64\!\cdots\!06$$$$T^{16} -$$$$99\!\cdots\!92$$$$T^{18} +$$$$13\!\cdots\!28$$$$T^{20} -$$$$14\!\cdots\!48$$$$T^{22} +$$$$13\!\cdots\!44$$$$T^{24} -$$$$94\!\cdots\!80$$$$T^{26} +$$$$52\!\cdots\!44$$$$T^{28} -$$$$19\!\cdots\!40$$$$T^{30} +$$$$40\!\cdots\!81$$$$T^{32}$$
$43$ $$1 - 160 T + 12800 T^{2} - 978464 T^{3} + 71106632 T^{4} - 3813053664 T^{5} + 178619596288 T^{6} - 8719368905312 T^{7} + 336417491247900 T^{8} - 9339737479444512 T^{9} + 288453906337733120 T^{10} - 7137460469658328480 T^{11} -$$$$12\!\cdots\!76$$$$T^{12} +$$$$13\!\cdots\!84$$$$T^{13} -$$$$44\!\cdots\!20$$$$T^{14} +$$$$28\!\cdots\!76$$$$T^{15} -$$$$17\!\cdots\!30$$$$T^{16} +$$$$53\!\cdots\!24$$$$T^{17} -$$$$15\!\cdots\!20$$$$T^{18} +$$$$83\!\cdots\!16$$$$T^{19} -$$$$15\!\cdots\!76$$$$T^{20} -$$$$15\!\cdots\!20$$$$T^{21} +$$$$11\!\cdots\!20$$$$T^{22} -$$$$69\!\cdots\!88$$$$T^{23} +$$$$45\!\cdots\!00$$$$T^{24} -$$$$22\!\cdots\!88$$$$T^{25} +$$$$83\!\cdots\!88$$$$T^{26} -$$$$32\!\cdots\!36$$$$T^{27} +$$$$11\!\cdots\!32$$$$T^{28} -$$$$28\!\cdots\!36$$$$T^{29} +$$$$69\!\cdots\!00$$$$T^{30} -$$$$16\!\cdots\!40$$$$T^{31} +$$$$18\!\cdots\!01$$$$T^{32}$$
$47$ $$1 - 24144 T^{2} + 280869112 T^{4} - 2097883923184 T^{6} + 11327375509374492 T^{8} - 47271044690493269328 T^{10} +$$$$15\!\cdots\!16$$$$T^{12} -$$$$44\!\cdots\!04$$$$T^{14} +$$$$10\!\cdots\!58$$$$T^{16} -$$$$21\!\cdots\!24$$$$T^{18} +$$$$37\!\cdots\!76$$$$T^{20} -$$$$54\!\cdots\!48$$$$T^{22} +$$$$64\!\cdots\!32$$$$T^{24} -$$$$58\!\cdots\!84$$$$T^{26} +$$$$37\!\cdots\!72$$$$T^{28} -$$$$15\!\cdots\!84$$$$T^{30} +$$$$32\!\cdots\!41$$$$T^{32}$$
$53$ $$1 + 160 T + 12800 T^{2} + 602944 T^{3} + 3948712 T^{4} - 1481707264 T^{5} - 105845942272 T^{6} - 3791430241760 T^{7} + 34861972067036 T^{8} + 14471440004155872 T^{9} + 1098860393015073792 T^{10} + 54880211634179791488 T^{11} +$$$$13\!\cdots\!12$$$$T^{12} -$$$$17\!\cdots\!00$$$$T^{13} -$$$$18\!\cdots\!48$$$$T^{14} +$$$$16\!\cdots\!08$$$$T^{15} +$$$$51\!\cdots\!54$$$$T^{16} +$$$$46\!\cdots\!72$$$$T^{17} -$$$$14\!\cdots\!88$$$$T^{18} -$$$$38\!\cdots\!00$$$$T^{19} +$$$$84\!\cdots\!32$$$$T^{20} +$$$$95\!\cdots\!12$$$$T^{21} +$$$$53\!\cdots\!72$$$$T^{22} +$$$$19\!\cdots\!68$$$$T^{23} +$$$$13\!\cdots\!56$$$$T^{24} -$$$$41\!\cdots\!40$$$$T^{25} -$$$$32\!\cdots\!72$$$$T^{26} -$$$$12\!\cdots\!76$$$$T^{27} +$$$$95\!\cdots\!72$$$$T^{28} +$$$$40\!\cdots\!76$$$$T^{29} +$$$$24\!\cdots\!00$$$$T^{30} +$$$$85\!\cdots\!40$$$$T^{31} +$$$$15\!\cdots\!41$$$$T^{32}$$
$59$ $$1 + 128 T + 8192 T^{2} + 1121408 T^{3} + 136226184 T^{4} + 9279937408 T^{5} + 700645040128 T^{6} + 71627082366848 T^{7} + 5234572115355804 T^{8} + 316007889653226112 T^{9} + 25502997282495045632 T^{10} +$$$$19\!\cdots\!80$$$$T^{11} +$$$$10\!\cdots\!40$$$$T^{12} +$$$$69\!\cdots\!16$$$$T^{13} +$$$$51\!\cdots\!56$$$$T^{14} +$$$$29\!\cdots\!24$$$$T^{15} +$$$$15\!\cdots\!38$$$$T^{16} +$$$$10\!\cdots\!44$$$$T^{17} +$$$$62\!\cdots\!16$$$$T^{18} +$$$$29\!\cdots\!56$$$$T^{19} +$$$$16\!\cdots\!40$$$$T^{20} +$$$$98\!\cdots\!80$$$$T^{21} +$$$$45\!\cdots\!92$$$$T^{22} +$$$$19\!\cdots\!32$$$$T^{23} +$$$$11\!\cdots\!64$$$$T^{24} +$$$$53\!\cdots\!08$$$$T^{25} +$$$$18\!\cdots\!28$$$$T^{26} +$$$$84\!\cdots\!48$$$$T^{27} +$$$$43\!\cdots\!24$$$$T^{28} +$$$$12\!\cdots\!28$$$$T^{29} +$$$$31\!\cdots\!32$$$$T^{30} +$$$$17\!\cdots\!28$$$$T^{31} +$$$$46\!\cdots\!81$$$$T^{32}$$
$61$ $$1 + 32 T + 512 T^{2} - 38048 T^{3} - 60439624 T^{4} - 1520787552 T^{5} - 16996289024 T^{6} + 2981900088544 T^{7} + 2018049968078364 T^{8} + 40394929489472928 T^{9} + 275088896591278592 T^{10} -$$$$10\!\cdots\!56$$$$T^{11} -$$$$45\!\cdots\!20$$$$T^{12} -$$$$71\!\cdots\!16$$$$T^{13} -$$$$18\!\cdots\!28$$$$T^{14} +$$$$23\!\cdots\!64$$$$T^{15} +$$$$72\!\cdots\!42$$$$T^{16} +$$$$88\!\cdots\!44$$$$T^{17} -$$$$26\!\cdots\!48$$$$T^{18} -$$$$36\!\cdots\!76$$$$T^{19} -$$$$86\!\cdots\!20$$$$T^{20} -$$$$75\!\cdots\!56$$$$T^{21} +$$$$73\!\cdots\!32$$$$T^{22} +$$$$39\!\cdots\!48$$$$T^{23} +$$$$74\!\cdots\!04$$$$T^{24} +$$$$40\!\cdots\!64$$$$T^{25} -$$$$86\!\cdots\!24$$$$T^{26} -$$$$28\!\cdots\!92$$$$T^{27} -$$$$42\!\cdots\!84$$$$T^{28} -$$$$99\!\cdots\!28$$$$T^{29} +$$$$49\!\cdots\!72$$$$T^{30} +$$$$11\!\cdots\!32$$$$T^{31} +$$$$13\!\cdots\!21$$$$T^{32}$$
$67$ $$1 - 320 T + 51200 T^{2} - 6047552 T^{3} + 641735304 T^{4} - 64228593856 T^{5} + 5982745065472 T^{6} - 525110406070976 T^{7} + 43992629224199580 T^{8} - 3502096836597496384 T^{9} +$$$$26\!\cdots\!12$$$$T^{10} -$$$$19\!\cdots\!80$$$$T^{11} +$$$$14\!\cdots\!20$$$$T^{12} -$$$$10\!\cdots\!88$$$$T^{13} +$$$$71\!\cdots\!04$$$$T^{14} -$$$$48\!\cdots\!52$$$$T^{15} +$$$$32\!\cdots\!74$$$$T^{16} -$$$$21\!\cdots\!28$$$$T^{17} +$$$$14\!\cdots\!84$$$$T^{18} -$$$$93\!\cdots\!72$$$$T^{19} +$$$$58\!\cdots\!20$$$$T^{20} -$$$$36\!\cdots\!20$$$$T^{21} +$$$$21\!\cdots\!32$$$$T^{22} -$$$$12\!\cdots\!36$$$$T^{23} +$$$$72\!\cdots\!80$$$$T^{24} -$$$$38\!\cdots\!84$$$$T^{25} +$$$$19\!\cdots\!72$$$$T^{26} -$$$$95\!\cdots\!84$$$$T^{27} +$$$$42\!\cdots\!84$$$$T^{28} -$$$$18\!\cdots\!88$$$$T^{29} +$$$$69\!\cdots\!00$$$$T^{30} -$$$$19\!\cdots\!80$$$$T^{31} +$$$$27\!\cdots\!61$$$$T^{32}$$
$71$ $$( 1 - 256 T + 68104 T^{2} - 10692864 T^{3} + 1610923548 T^{4} - 179723087616 T^{5} + 18972832358712 T^{6} - 1588998739085056 T^{7} + 125568612540426694 T^{8} - 8010142643727767296 T^{9} +$$$$48\!\cdots\!72$$$$T^{10} -$$$$23\!\cdots\!36$$$$T^{11} +$$$$10\!\cdots\!28$$$$T^{12} -$$$$34\!\cdots\!64$$$$T^{13} +$$$$11\!\cdots\!64$$$$T^{14} -$$$$21\!\cdots\!36$$$$T^{15} +$$$$41\!\cdots\!21$$$$T^{16} )^{2}$$
$73$ $$1 - 42768 T^{2} + 946714744 T^{4} - 14391245893936 T^{6} + 167549428359087132 T^{8} -$$$$15\!\cdots\!24$$$$T^{10} +$$$$12\!\cdots\!76$$$$T^{12} -$$$$83\!\cdots\!96$$$$T^{14} +$$$$47\!\cdots\!22$$$$T^{16} -$$$$23\!\cdots\!36$$$$T^{18} +$$$$10\!\cdots\!56$$$$T^{20} -$$$$36\!\cdots\!04$$$$T^{22} +$$$$10\!\cdots\!52$$$$T^{24} -$$$$26\!\cdots\!36$$$$T^{26} +$$$$49\!\cdots\!04$$$$T^{28} -$$$$63\!\cdots\!08$$$$T^{30} +$$$$42\!\cdots\!21$$$$T^{32}$$
$79$ $$1 - 62928 T^{2} + 1905826568 T^{4} - 37296559235888 T^{6} + 534425714020543644 T^{8} -$$$$60\!\cdots\!08$$$$T^{10} +$$$$55\!\cdots\!96$$$$T^{12} -$$$$43\!\cdots\!36$$$$T^{14} +$$$$29\!\cdots\!62$$$$T^{16} -$$$$16\!\cdots\!16$$$$T^{18} +$$$$84\!\cdots\!56$$$$T^{20} -$$$$35\!\cdots\!28$$$$T^{22} +$$$$12\!\cdots\!24$$$$T^{24} -$$$$33\!\cdots\!88$$$$T^{26} +$$$$66\!\cdots\!08$$$$T^{28} -$$$$85\!\cdots\!08$$$$T^{30} +$$$$52\!\cdots\!41$$$$T^{32}$$
$83$ $$1 + 160 T + 12800 T^{2} + 895904 T^{3} + 107479624 T^{4} + 16432771168 T^{5} + 1654826188288 T^{6} + 174484645067104 T^{7} + 18280323695716892 T^{8} + 1483531366054758688 T^{9} +$$$$11\!\cdots\!96$$$$T^{10} +$$$$11\!\cdots\!36$$$$T^{11} +$$$$13\!\cdots\!00$$$$T^{12} +$$$$12\!\cdots\!44$$$$T^{13} +$$$$89\!\cdots\!76$$$$T^{14} +$$$$74\!\cdots\!76$$$$T^{15} +$$$$61\!\cdots\!66$$$$T^{16} +$$$$51\!\cdots\!64$$$$T^{17} +$$$$42\!\cdots\!96$$$$T^{18} +$$$$39\!\cdots\!36$$$$T^{19} +$$$$30\!\cdots\!00$$$$T^{20} +$$$$17\!\cdots\!64$$$$T^{21} +$$$$12\!\cdots\!56$$$$T^{22} +$$$$10\!\cdots\!52$$$$T^{23} +$$$$92\!\cdots\!52$$$$T^{24} +$$$$60\!\cdots\!36$$$$T^{25} +$$$$39\!\cdots\!88$$$$T^{26} +$$$$27\!\cdots\!52$$$$T^{27} +$$$$12\!\cdots\!04$$$$T^{28} +$$$$70\!\cdots\!76$$$$T^{29} +$$$$69\!\cdots\!00$$$$T^{30} +$$$$59\!\cdots\!40$$$$T^{31} +$$$$25\!\cdots\!61$$$$T^{32}$$
$89$ $$1 - 81008 T^{2} + 3201135736 T^{4} - 82544801381712 T^{6} + 1567286911309649436 T^{8} -$$$$23\!\cdots\!04$$$$T^{10} +$$$$28\!\cdots\!72$$$$T^{12} -$$$$29\!\cdots\!36$$$$T^{14} +$$$$25\!\cdots\!10$$$$T^{16} -$$$$18\!\cdots\!76$$$$T^{18} +$$$$11\!\cdots\!32$$$$T^{20} -$$$$57\!\cdots\!84$$$$T^{22} +$$$$24\!\cdots\!96$$$$T^{24} -$$$$80\!\cdots\!12$$$$T^{26} +$$$$19\!\cdots\!76$$$$T^{28} -$$$$31\!\cdots\!48$$$$T^{30} +$$$$24\!\cdots\!21$$$$T^{32}$$
$97$ $$( 1 + 38216 T^{2} + 116224 T^{3} + 770481564 T^{4} + 3485408768 T^{5} + 10857255215864 T^{6} + 49274039499776 T^{7} + 116292098553803590 T^{8} + 463619437653392384 T^{9} +$$$$96\!\cdots\!84$$$$T^{10} +$$$$29\!\cdots\!72$$$$T^{11} +$$$$60\!\cdots\!04$$$$T^{12} +$$$$85\!\cdots\!76$$$$T^{13} +$$$$26\!\cdots\!56$$$$T^{14} +$$$$61\!\cdots\!21$$$$T^{16} )^{2}$$