# Properties

 Label 525.2.n Level 525 Weight 2 Character orbit n Rep. character $$\chi_{525}(106,\cdot)$$ Character field $$\Q(\zeta_{5})$$ Dimension 112 Newform subspaces 5 Sturm bound 160 Trace bound 3

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## Defining parameters

 Level: $$N$$ = $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 525.n (of order $$5$$ and degree $$4$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$25$$ Character field: $$\Q(\zeta_{5})$$ Newform subspaces: $$5$$ Sturm bound: $$160$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(525, [\chi])$$.

Total New Old
Modular forms 336 112 224
Cusp forms 304 112 192
Eisenstein series 32 0 32

## Trace form

 $$112q + 4q^{2} - 24q^{4} + 8q^{5} + 12q^{8} - 28q^{9} + O(q^{10})$$ $$112q + 4q^{2} - 24q^{4} + 8q^{5} + 12q^{8} - 28q^{9} + 12q^{10} + 16q^{12} + 24q^{13} + 4q^{15} - 40q^{16} + 32q^{17} - 16q^{18} - 12q^{19} - 60q^{20} + 4q^{21} - 56q^{22} - 12q^{23} + 8q^{25} + 64q^{26} + 20q^{29} - 8q^{30} - 72q^{32} - 12q^{33} - 36q^{34} + 4q^{35} - 24q^{36} - 16q^{37} + 56q^{38} + 16q^{39} - 96q^{40} + 48q^{41} + 48q^{43} - 84q^{44} + 8q^{45} + 4q^{46} - 48q^{47} + 32q^{48} + 112q^{49} - 72q^{50} - 64q^{51} + 88q^{52} - 12q^{53} - 68q^{55} + 24q^{57} - 112q^{58} + 24q^{59} + 28q^{60} + 8q^{61} - 28q^{62} - 24q^{64} - 88q^{65} + 16q^{66} + 24q^{67} + 128q^{68} + 8q^{69} + 16q^{70} - 32q^{71} + 12q^{72} + 64q^{73} + 24q^{74} + 32q^{75} + 72q^{76} - 24q^{77} + 24q^{78} + 24q^{79} + 152q^{80} - 28q^{81} + 56q^{82} + 68q^{83} + 12q^{84} - 8q^{85} - 120q^{86} - 48q^{87} + 36q^{88} - 96q^{89} + 12q^{90} + 16q^{91} - 48q^{92} - 80q^{93} + 52q^{94} - 180q^{95} + 20q^{96} + 12q^{97} + 4q^{98} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(525, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
525.2.n.a $$4$$ $$4.192$$ $$\Q(\zeta_{10})$$ None $$3$$ $$1$$ $$0$$ $$-4$$ $$q+(1-\zeta_{10}^{3})q^{2}+\zeta_{10}^{3}q^{3}+(1-\zeta_{10}+\cdots)q^{4}+\cdots$$
525.2.n.b $$20$$ $$4.192$$ $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ None $$-2$$ $$5$$ $$5$$ $$-20$$ $$q+\beta _{10}q^{2}+\beta _{4}q^{3}+(\beta _{1}-2\beta _{2}-\beta _{3}+\cdots)q^{4}+\cdots$$
525.2.n.c $$24$$ $$4.192$$ None $$1$$ $$-6$$ $$-1$$ $$24$$
525.2.n.d $$32$$ $$4.192$$ None $$1$$ $$-8$$ $$-3$$ $$-32$$
525.2.n.e $$32$$ $$4.192$$ None $$1$$ $$8$$ $$7$$ $$32$$

## Decomposition of $$S_{2}^{\mathrm{old}}(525, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(525, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(25, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(75, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(175, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 - 3 T + 2 T^{2} + T^{4} + 8 T^{6} - 24 T^{7} + 16 T^{8}$$)($$1 + 2 T - 3 T^{2} - 8 T^{3} + 9 T^{4} + 24 T^{5} - 35 T^{6} - 76 T^{7} + 97 T^{8} + 174 T^{9} - 308 T^{10} - 375 T^{11} + 914 T^{12} + 753 T^{13} - 2368 T^{14} - 1381 T^{15} + 5578 T^{16} + 1957 T^{17} - 12263 T^{18} - 1170 T^{19} + 25773 T^{20} - 2340 T^{21} - 49052 T^{22} + 15656 T^{23} + 89248 T^{24} - 44192 T^{25} - 151552 T^{26} + 96384 T^{27} + 233984 T^{28} - 192000 T^{29} - 315392 T^{30} + 356352 T^{31} + 397312 T^{32} - 622592 T^{33} - 573440 T^{34} + 786432 T^{35} + 589824 T^{36} - 1048576 T^{37} - 786432 T^{38} + 1048576 T^{39} + 1048576 T^{40}$$)
$3$ ($$1 - T + T^{2} - T^{3} + T^{4}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{5}$$)
$5$ ($$( 1 - 5 T^{2} )^{2}$$)($$1 - 5 T + 20 T^{2} - 80 T^{3} + 205 T^{4} - 450 T^{5} + 850 T^{6} - 425 T^{7} - 1850 T^{8} + 9125 T^{9} - 28375 T^{10} + 45625 T^{11} - 46250 T^{12} - 53125 T^{13} + 531250 T^{14} - 1406250 T^{15} + 3203125 T^{16} - 6250000 T^{17} + 7812500 T^{18} - 9765625 T^{19} + 9765625 T^{20}$$)
$7$ ($$( 1 + T )^{4}$$)($$( 1 + T )^{20}$$)
$11$ ($$1 + 2 T + 13 T^{2} + 34 T^{3} + 225 T^{4} + 374 T^{5} + 1573 T^{6} + 2662 T^{7} + 14641 T^{8}$$)($$1 - 12 T + 5 T^{2} + 527 T^{3} - 2020 T^{4} - 7203 T^{5} + 56722 T^{6} - 4798 T^{7} - 711707 T^{8} + 1150473 T^{9} + 4220063 T^{10} - 12236295 T^{11} + 6043188 T^{12} - 37619910 T^{13} - 230334633 T^{14} + 2791361284 T^{15} - 3117216920 T^{16} - 35048914405 T^{17} + 113970259425 T^{18} + 155915532485 T^{19} - 1574551625415 T^{20} + 1715070857335 T^{21} + 13790401390425 T^{22} - 46650105073055 T^{23} - 45639172925720 T^{24} + 449551526149484 T^{25} - 408051852772113 T^{26} - 733105619174610 T^{27} + 1295411017352628 T^{28} - 28852543541644845 T^{29} + 109457565873969863 T^{30} + 328243373622849003 T^{31} - 2233641444710972747 T^{32} - 165639972866580938 T^{33} + 21540170060508596002 T^{34} - 30088718564300934153 T^{35} - 92818454324415765220 T^{36} +$$$$26\!\cdots\!17$$$$T^{37} + 27799586567461157405 T^{38} -$$$$73\!\cdots\!92$$$$T^{39} +$$$$67\!\cdots\!01$$$$T^{40}$$)
$13$ ($$1 - 9 T + 48 T^{2} - 235 T^{3} + 1011 T^{4} - 3055 T^{5} + 8112 T^{6} - 19773 T^{7} + 28561 T^{8}$$)($$1 + 17 T + 118 T^{2} + 511 T^{3} + 2352 T^{4} + 11040 T^{5} + 35384 T^{6} + 109963 T^{7} + 467215 T^{8} + 971364 T^{9} - 724062 T^{10} - 1470107 T^{11} - 9009942 T^{12} - 143708236 T^{13} - 89788267 T^{14} + 1601061003 T^{15} + 3142834670 T^{16} + 22309519952 T^{17} + 208993081204 T^{18} + 830991825843 T^{19} + 2569268002331 T^{20} + 10802893735959 T^{21} + 35319830723476 T^{22} + 49014015334544 T^{23} + 89762501009870 T^{24} + 594462742986879 T^{25} - 433390815250003 T^{26} - 9017478689686012 T^{27} - 7349686483828182 T^{28} - 15589748759742911 T^{29} - 99818095325170638 T^{30} + 1740840088993356468 T^{31} + 10885214840499960415 T^{32} + 33305055346203916639 T^{33} +$$$$13\!\cdots\!76$$$$T^{34} +$$$$56\!\cdots\!80$$$$T^{35} +$$$$15\!\cdots\!32$$$$T^{36} +$$$$44\!\cdots\!63$$$$T^{37} +$$$$13\!\cdots\!22$$$$T^{38} +$$$$24\!\cdots\!09$$$$T^{39} +$$$$19\!\cdots\!01$$$$T^{40}$$)
$17$ ($$1 - 13 T + 52 T^{2} + 25 T^{3} - 729 T^{4} + 425 T^{5} + 15028 T^{6} - 63869 T^{7} + 83521 T^{8}$$)($$1 + 9 T - 27 T^{2} - 620 T^{3} - 1290 T^{4} + 20557 T^{5} + 117068 T^{6} - 310247 T^{7} - 4476802 T^{8} - 4151366 T^{9} + 107746789 T^{10} + 390341611 T^{11} - 1579360503 T^{12} - 12891387238 T^{13} + 2870358378 T^{14} + 278972749333 T^{15} + 601233535747 T^{16} - 4051231959030 T^{17} - 20626010838649 T^{18} + 27189021791451 T^{19} + 423848770469272 T^{20} + 462213370454667 T^{21} - 5960917132369561 T^{22} - 19903702614714390 T^{23} + 50215626139125187 T^{24} + 396101410949705381 T^{25} + 69283473403703082 T^{26} - 5289834732370055174 T^{27} - 11017235780823752823 T^{28} + 46289782756908016667 T^{29} +$$$$21\!\cdots\!61$$$$T^{30} -$$$$14\!\cdots\!78$$$$T^{31} -$$$$26\!\cdots\!22$$$$T^{32} -$$$$30\!\cdots\!39$$$$T^{33} +$$$$19\!\cdots\!72$$$$T^{34} +$$$$58\!\cdots\!01$$$$T^{35} -$$$$62\!\cdots\!90$$$$T^{36} -$$$$51\!\cdots\!40$$$$T^{37} -$$$$37\!\cdots\!43$$$$T^{38} +$$$$21\!\cdots\!77$$$$T^{39} +$$$$40\!\cdots\!01$$$$T^{40}$$)
$19$ ($$1 - 9 T^{2} + 70 T^{3} + 291 T^{4} + 1330 T^{5} - 3249 T^{6} + 130321 T^{8}$$)($$1 + 9 T + 9 T^{2} + 10 T^{3} + 1521 T^{4} + 5182 T^{5} - 28341 T^{6} - 73578 T^{7} + 621886 T^{8} - 1041450 T^{9} - 21412968 T^{10} + 22324377 T^{11} + 309389811 T^{12} - 612021092 T^{13} + 809994731 T^{14} + 38047928694 T^{15} + 10122189378 T^{16} - 287816743238 T^{17} + 1931412339679 T^{18} + 1204750584038 T^{19} - 56973551738703 T^{20} + 22890261096722 T^{21} + 697239854624119 T^{22} - 1974135041869442 T^{23} + 1319133841930338 T^{24} + 94210438191284706 T^{25} + 38106915725253011 T^{26} - 547068357810718988 T^{27} + 5254541359361575251 T^{28} + 7203801818480458683 T^{29} -$$$$13\!\cdots\!68$$$$T^{30} -$$$$12\!\cdots\!50$$$$T^{31} +$$$$13\!\cdots\!46$$$$T^{32} -$$$$30\!\cdots\!02$$$$T^{33} -$$$$22\!\cdots\!61$$$$T^{34} +$$$$78\!\cdots\!18$$$$T^{35} +$$$$43\!\cdots\!01$$$$T^{36} +$$$$54\!\cdots\!90$$$$T^{37} +$$$$93\!\cdots\!69$$$$T^{38} +$$$$17\!\cdots\!11$$$$T^{39} +$$$$37\!\cdots\!01$$$$T^{40}$$)
$23$ ($$1 + T - 17 T^{2} + 65 T^{3} + 576 T^{4} + 1495 T^{5} - 8993 T^{6} + 12167 T^{7} + 279841 T^{8}$$)($$1 - 7 T + 5 T^{2} + 232 T^{3} - 1052 T^{4} - 3365 T^{5} + 37947 T^{6} - 130501 T^{7} - 60947 T^{8} + 2537516 T^{9} - 10158836 T^{10} - 41777788 T^{11} + 620616929 T^{12} - 1197621959 T^{13} - 12815167944 T^{14} + 50019431341 T^{15} + 259769120270 T^{16} - 2481947544649 T^{17} + 5320406087173 T^{18} + 22264384484774 T^{19} - 176619524520221 T^{20} + 512080843149802 T^{21} + 2814494820114517 T^{22} - 30197855775744383 T^{23} + 72694050385477070 T^{24} + 321942216775625963 T^{25} - 1897104779274342216 T^{26} - 4077693721889190673 T^{27} + 48601123192058422049 T^{28} - 75248174046236983844 T^{29} -$$$$42\!\cdots\!64$$$$T^{30} +$$$$24\!\cdots\!32$$$$T^{31} -$$$$13\!\cdots\!87$$$$T^{32} -$$$$65\!\cdots\!83$$$$T^{33} +$$$$43\!\cdots\!23$$$$T^{34} -$$$$89\!\cdots\!55$$$$T^{35} -$$$$64\!\cdots\!72$$$$T^{36} +$$$$32\!\cdots\!96$$$$T^{37} +$$$$16\!\cdots\!45$$$$T^{38} -$$$$52\!\cdots\!09$$$$T^{39} +$$$$17\!\cdots\!01$$$$T^{40}$$)
$29$ ($$1 + 15 T + 106 T^{2} + 675 T^{3} + 4171 T^{4} + 19575 T^{5} + 89146 T^{6} + 365835 T^{7} + 707281 T^{8}$$)($$1 - 28 T + 292 T^{2} - 1111 T^{3} - 2637 T^{4} + 48056 T^{5} - 354977 T^{6} + 1751017 T^{7} + 1171058 T^{8} - 59724245 T^{9} + 265622419 T^{10} - 1367742638 T^{11} + 9549291711 T^{12} - 12436162284 T^{13} - 128894774032 T^{14} + 796946984854 T^{15} - 10162058212054 T^{16} + 58858081368164 T^{17} + 81427480890353 T^{18} - 1834790782280269 T^{19} + 9689391931459613 T^{20} - 53208932686127801 T^{21} + 68480511428786873 T^{22} + 1435489746488151796 T^{23} - 7187430694279765174 T^{24} + 16346298351441137246 T^{25} - 76669617549258800272 T^{26} -$$$$21\!\cdots\!56$$$$T^{27} +$$$$47\!\cdots\!71$$$$T^{28} -$$$$19\!\cdots\!22$$$$T^{29} +$$$$11\!\cdots\!19$$$$T^{30} -$$$$72\!\cdots\!05$$$$T^{31} +$$$$41\!\cdots\!78$$$$T^{32} +$$$$17\!\cdots\!13$$$$T^{33} -$$$$10\!\cdots\!37$$$$T^{34} +$$$$41\!\cdots\!44$$$$T^{35} -$$$$65\!\cdots\!77$$$$T^{36} -$$$$80\!\cdots\!99$$$$T^{37} +$$$$61\!\cdots\!12$$$$T^{38} -$$$$17\!\cdots\!32$$$$T^{39} +$$$$17\!\cdots\!01$$$$T^{40}$$)
$31$ ($$1 + 7 T + 3 T^{2} + 119 T^{3} + 1640 T^{4} + 3689 T^{5} + 2883 T^{6} + 208537 T^{7} + 923521 T^{8}$$)($$1 - 6 T - 140 T^{2} + 841 T^{3} + 10547 T^{4} - 68492 T^{5} - 515260 T^{6} + 4069562 T^{7} + 17326096 T^{8} - 190777850 T^{9} - 393262670 T^{10} + 7140215105 T^{11} + 5355496331 T^{12} - 210301021603 T^{13} - 32554607310 T^{14} + 4743770235228 T^{15} + 963101506732 T^{16} - 77868139958804 T^{17} - 93076859779940 T^{18} + 691653607625279 T^{19} + 3973675961861677 T^{20} + 21441261836383649 T^{21} - 89446862248522340 T^{22} - 2319769757512729964 T^{23} + 889444466598643372 T^{24} +$$$$13\!\cdots\!28$$$$T^{25} - 28892333821134508110 T^{26} -$$$$57\!\cdots\!33$$$$T^{27} +$$$$45\!\cdots\!71$$$$T^{28} +$$$$18\!\cdots\!55$$$$T^{29} -$$$$32\!\cdots\!70$$$$T^{30} -$$$$48\!\cdots\!50$$$$T^{31} +$$$$13\!\cdots\!56$$$$T^{32} +$$$$99\!\cdots\!42$$$$T^{33} -$$$$39\!\cdots\!60$$$$T^{34} -$$$$16\!\cdots\!92$$$$T^{35} +$$$$76\!\cdots\!07$$$$T^{36} +$$$$18\!\cdots\!51$$$$T^{37} -$$$$97\!\cdots\!40$$$$T^{38} -$$$$13\!\cdots\!26$$$$T^{39} +$$$$67\!\cdots\!01$$$$T^{40}$$)
$37$ ($$1 - 3 T + 17 T^{2} - 225 T^{3} + 2116 T^{4} - 8325 T^{5} + 23273 T^{6} - 151959 T^{7} + 1874161 T^{8}$$)($$1 + 5 T - 81 T^{2} - 679 T^{3} + 3744 T^{4} + 60925 T^{5} + 60905 T^{6} - 3234461 T^{7} - 14856815 T^{8} + 110399361 T^{9} + 1143959304 T^{10} - 675160546 T^{11} - 47596373540 T^{12} - 138591309660 T^{13} + 1351597549496 T^{14} + 10023063834413 T^{15} - 4790109168240 T^{16} - 343820155009758 T^{17} - 1189329408355585 T^{18} + 5609372386700900 T^{19} + 73074032491390783 T^{20} + 207546778307933300 T^{21} - 1628191960038795865 T^{22} - 17415522311709271974 T^{23} - 8977435788857846640 T^{24} +$$$$69\!\cdots\!41$$$$T^{25} +$$$$34\!\cdots\!64$$$$T^{26} -$$$$13\!\cdots\!80$$$$T^{27} -$$$$16\!\cdots\!40$$$$T^{28} -$$$$87\!\cdots\!42$$$$T^{29} +$$$$55\!\cdots\!96$$$$T^{30} +$$$$19\!\cdots\!93$$$$T^{31} -$$$$97\!\cdots\!15$$$$T^{32} -$$$$78\!\cdots\!17$$$$T^{33} +$$$$54\!\cdots\!45$$$$T^{34} +$$$$20\!\cdots\!25$$$$T^{35} +$$$$46\!\cdots\!04$$$$T^{36} -$$$$30\!\cdots\!43$$$$T^{37} -$$$$13\!\cdots\!49$$$$T^{38} +$$$$31\!\cdots\!65$$$$T^{39} +$$$$23\!\cdots\!01$$$$T^{40}$$)
$41$ ($$1 + 12 T + 53 T^{2} + 444 T^{3} + 4405 T^{4} + 18204 T^{5} + 89093 T^{6} + 827052 T^{7} + 2825761 T^{8}$$)($$1 - 11 T - 74 T^{2} + 1250 T^{3} + 74 T^{4} - 40033 T^{5} - 23006 T^{6} + 264250 T^{7} + 14104671 T^{8} - 64818221 T^{9} - 407915891 T^{10} + 4491189267 T^{11} - 20807200322 T^{12} + 22786129000 T^{13} + 615571236857 T^{14} - 4246961124649 T^{15} + 23630290658722 T^{16} - 244694380689250 T^{17} + 940209666208138 T^{18} + 10616650231934837 T^{19} - 133412201560405859 T^{20} + 435282659509328317 T^{21} + 1580492448895879978 T^{22} - 16864581411483799250 T^{23} + 66773553762080937442 T^{24} -$$$$49\!\cdots\!49$$$$T^{25} +$$$$29\!\cdots\!37$$$$T^{26} +$$$$44\!\cdots\!00$$$$T^{27} -$$$$16\!\cdots\!62$$$$T^{28} +$$$$14\!\cdots\!87$$$$T^{29} -$$$$54\!\cdots\!91$$$$T^{30} -$$$$35\!\cdots\!61$$$$T^{31} +$$$$31\!\cdots\!51$$$$T^{32} +$$$$24\!\cdots\!50$$$$T^{33} -$$$$87\!\cdots\!66$$$$T^{34} -$$$$62\!\cdots\!33$$$$T^{35} +$$$$47\!\cdots\!34$$$$T^{36} +$$$$32\!\cdots\!50$$$$T^{37} -$$$$79\!\cdots\!54$$$$T^{38} -$$$$48\!\cdots\!71$$$$T^{39} +$$$$18\!\cdots\!01$$$$T^{40}$$)
$43$ ($$( 1 - T + 43 T^{2} )^{4}$$)($$( 1 - 14 T + 271 T^{2} - 2568 T^{3} + 26841 T^{4} - 162823 T^{5} + 953702 T^{6} - 904287 T^{7} - 22940467 T^{8} + 388741980 T^{9} - 2790795889 T^{10} + 16715905140 T^{11} - 42416923483 T^{12} - 71897146509 T^{13} + 3260517351302 T^{14} - 23936355714589 T^{15} + 169671705598209 T^{16} - 698030193322776 T^{17} + 3167502275229871 T^{18} - 7036296567115802 T^{19} + 21611482313284249 T^{20} )^{2}$$)
$47$ ($$1 + 12 T + 97 T^{2} + 600 T^{3} + 2641 T^{4} + 28200 T^{5} + 214273 T^{6} + 1245876 T^{7} + 4879681 T^{8}$$)($$1 + 24 T + 214 T^{2} + 1028 T^{3} + 12897 T^{4} + 148620 T^{5} + 168624 T^{6} - 6740131 T^{7} - 13459903 T^{8} + 17951408 T^{9} - 4846961466 T^{10} - 43154146793 T^{11} - 4688333873 T^{12} + 105775993364 T^{13} - 13620645345307 T^{14} - 46895778039065 T^{15} + 646122293050745 T^{16} + 2673706913658631 T^{17} - 13814120092137086 T^{18} + 122417875161538166 T^{19} + 2510353529002371099 T^{20} + 5753640132592293802 T^{21} - 30515391283530822974 T^{22} +$$$$27\!\cdots\!13$$$$T^{23} +$$$$31\!\cdots\!45$$$$T^{24} -$$$$10\!\cdots\!55$$$$T^{25} -$$$$14\!\cdots\!03$$$$T^{26} +$$$$53\!\cdots\!32$$$$T^{27} -$$$$11\!\cdots\!53$$$$T^{28} -$$$$48\!\cdots\!31$$$$T^{29} -$$$$25\!\cdots\!34$$$$T^{30} +$$$$44\!\cdots\!24$$$$T^{31} -$$$$15\!\cdots\!23$$$$T^{32} -$$$$36\!\cdots\!37$$$$T^{33} +$$$$43\!\cdots\!56$$$$T^{34} +$$$$17\!\cdots\!60$$$$T^{35} +$$$$73\!\cdots\!37$$$$T^{36} +$$$$27\!\cdots\!36$$$$T^{37} +$$$$26\!\cdots\!46$$$$T^{38} +$$$$14\!\cdots\!92$$$$T^{39} +$$$$27\!\cdots\!01$$$$T^{40}$$)
$53$ ($$1 - 14 T + 43 T^{2} - 160 T^{3} + 2961 T^{4} - 8480 T^{5} + 120787 T^{6} - 2084278 T^{7} + 7890481 T^{8}$$)($$1 + 26 T + 189 T^{2} - 687 T^{3} - 12003 T^{4} + 33403 T^{5} + 552417 T^{6} - 5988920 T^{7} - 70924623 T^{8} + 151543065 T^{9} + 3746438244 T^{10} - 1341982202 T^{11} - 133505239607 T^{12} + 94806090571 T^{13} + 3402269873522 T^{14} - 28374866549812 T^{15} - 304279860832395 T^{16} - 79730970585402 T^{17} + 9927808060918525 T^{18} + 92743126415480970 T^{19} + 733823132261482620 T^{20} + 4915385700020491410 T^{21} + 27887212843120136725 T^{22} - 11870107707842893554 T^{23} -$$$$24\!\cdots\!95$$$$T^{24} -$$$$11\!\cdots\!16$$$$T^{25} +$$$$75\!\cdots\!38$$$$T^{26} +$$$$11\!\cdots\!27$$$$T^{27} -$$$$83\!\cdots\!27$$$$T^{28} -$$$$44\!\cdots\!66$$$$T^{29} +$$$$65\!\cdots\!56$$$$T^{30} +$$$$14\!\cdots\!05$$$$T^{31} -$$$$34\!\cdots\!43$$$$T^{32} -$$$$15\!\cdots\!60$$$$T^{33} +$$$$76\!\cdots\!73$$$$T^{34} +$$$$24\!\cdots\!71$$$$T^{35} -$$$$46\!\cdots\!63$$$$T^{36} -$$$$14\!\cdots\!31$$$$T^{37} +$$$$20\!\cdots\!21$$$$T^{38} +$$$$15\!\cdots\!42$$$$T^{39} +$$$$30\!\cdots\!01$$$$T^{40}$$)
$59$ ($$1 + 20 T + 131 T^{2} + 530 T^{3} + 3851 T^{4} + 31270 T^{5} + 456011 T^{6} + 4107580 T^{7} + 12117361 T^{8}$$)($$1 - 64 T + 1835 T^{2} - 30211 T^{3} + 294923 T^{4} - 1319681 T^{5} - 6472851 T^{6} + 158363147 T^{7} - 1301770421 T^{8} + 5307853640 T^{9} + 6106474336 T^{10} - 292966186104 T^{11} + 2789886458001 T^{12} - 14757440708664 T^{13} - 27120694739231 T^{14} + 1672303716112849 T^{15} - 19763683840572248 T^{16} + 114997595957857866 T^{17} + 62338242339194563 T^{18} - 8477577091481604596 T^{19} + 91524505213402944705 T^{20} -$$$$50\!\cdots\!64$$$$T^{21} +$$$$21\!\cdots\!03$$$$T^{22} +$$$$23\!\cdots\!14$$$$T^{23} -$$$$23\!\cdots\!28$$$$T^{24} +$$$$11\!\cdots\!51$$$$T^{25} -$$$$11\!\cdots\!71$$$$T^{26} -$$$$36\!\cdots\!16$$$$T^{27} +$$$$40\!\cdots\!21$$$$T^{28} -$$$$25\!\cdots\!56$$$$T^{29} +$$$$31\!\cdots\!36$$$$T^{30} +$$$$16\!\cdots\!60$$$$T^{31} -$$$$23\!\cdots\!01$$$$T^{32} +$$$$16\!\cdots\!13$$$$T^{33} -$$$$40\!\cdots\!11$$$$T^{34} -$$$$48\!\cdots\!19$$$$T^{35} +$$$$63\!\cdots\!43$$$$T^{36} -$$$$38\!\cdots\!09$$$$T^{37} +$$$$13\!\cdots\!35$$$$T^{38} -$$$$28\!\cdots\!96$$$$T^{39} +$$$$26\!\cdots\!01$$$$T^{40}$$)
$61$ ($$1 - 8 T + 123 T^{2} - 766 T^{3} + 10325 T^{4} - 46726 T^{5} + 457683 T^{6} - 1815848 T^{7} + 13845841 T^{8}$$)($$1 - 8 T - 250 T^{2} + 2547 T^{3} + 23448 T^{4} - 409872 T^{5} + 30251 T^{6} + 40503616 T^{7} - 251606971 T^{8} - 2220692740 T^{9} + 32366729049 T^{10} - 15508614507 T^{11} - 2274901784809 T^{12} + 15563607606792 T^{13} + 79216530591596 T^{14} - 1576254516302698 T^{15} + 2970947510282232 T^{16} + 87783662074531477 T^{17} - 687319481699616148 T^{18} - 2134990650502795753 T^{19} + 55437519869949000215 T^{20} -$$$$13\!\cdots\!33$$$$T^{21} -$$$$25\!\cdots\!08$$$$T^{22} +$$$$19\!\cdots\!37$$$$T^{23} +$$$$41\!\cdots\!12$$$$T^{24} -$$$$13\!\cdots\!98$$$$T^{25} +$$$$40\!\cdots\!56$$$$T^{26} +$$$$48\!\cdots\!32$$$$T^{27} -$$$$43\!\cdots\!29$$$$T^{28} -$$$$18\!\cdots\!87$$$$T^{29} +$$$$23\!\cdots\!49$$$$T^{30} -$$$$96\!\cdots\!40$$$$T^{31} -$$$$66\!\cdots\!91$$$$T^{32} +$$$$65\!\cdots\!96$$$$T^{33} +$$$$29\!\cdots\!91$$$$T^{34} -$$$$24\!\cdots\!72$$$$T^{35} +$$$$86\!\cdots\!28$$$$T^{36} +$$$$57\!\cdots\!87$$$$T^{37} -$$$$34\!\cdots\!50$$$$T^{38} -$$$$66\!\cdots\!28$$$$T^{39} +$$$$50\!\cdots\!01$$$$T^{40}$$)
$67$ ($$1 - 13 T + 12 T^{2} + 895 T^{3} - 9919 T^{4} + 59965 T^{5} + 53868 T^{6} - 3909919 T^{7} + 20151121 T^{8}$$)($$1 + 3 T - 311 T^{2} - 1039 T^{3} + 45007 T^{4} + 100608 T^{5} - 4203051 T^{6} + 2712957 T^{7} + 286251229 T^{8} - 1634937209 T^{9} - 12746581716 T^{10} + 220418749894 T^{11} + 98093010794 T^{12} - 19071644840712 T^{13} + 44213444347269 T^{14} + 1156522239768568 T^{15} - 5805695400666044 T^{16} - 49425850295776060 T^{17} + 521799371255402099 T^{18} + 1103857348682193140 T^{19} - 37995128081047056923 T^{20} + 73958442361706940380 T^{21} +$$$$23\!\cdots\!11$$$$T^{22} -$$$$14\!\cdots\!80$$$$T^{23} -$$$$11\!\cdots\!24$$$$T^{24} +$$$$15\!\cdots\!76$$$$T^{25} +$$$$39\!\cdots\!61$$$$T^{26} -$$$$11\!\cdots\!76$$$$T^{27} +$$$$39\!\cdots\!54$$$$T^{28} +$$$$59\!\cdots\!18$$$$T^{29} -$$$$23\!\cdots\!84$$$$T^{30} -$$$$19\!\cdots\!47$$$$T^{31} +$$$$23\!\cdots\!69$$$$T^{32} +$$$$14\!\cdots\!59$$$$T^{33} -$$$$15\!\cdots\!79$$$$T^{34} +$$$$24\!\cdots\!44$$$$T^{35} +$$$$74\!\cdots\!67$$$$T^{36} -$$$$11\!\cdots\!53$$$$T^{37} -$$$$23\!\cdots\!99$$$$T^{38} +$$$$14\!\cdots\!09$$$$T^{39} +$$$$33\!\cdots\!01$$$$T^{40}$$)
$71$ ($$1 - 13 T - 2 T^{2} + 349 T^{3} + 405 T^{4} + 24779 T^{5} - 10082 T^{6} - 4652843 T^{7} + 25411681 T^{8}$$)($$1 - 19 T + 23 T^{2} + 4108 T^{3} - 53279 T^{4} + 41926 T^{5} + 6067035 T^{6} - 66073585 T^{7} + 97395390 T^{8} + 4538324065 T^{9} - 50695622941 T^{10} + 177600768129 T^{11} + 1606952963487 T^{12} - 30747394363113 T^{13} + 239696534777059 T^{14} - 283347904916686 T^{15} - 18611857528547495 T^{16} + 235177810223507470 T^{17} - 873303489026431455 T^{18} - 10620340083202397445 T^{19} +$$$$16\!\cdots\!75$$$$T^{20} -$$$$75\!\cdots\!95$$$$T^{21} -$$$$44\!\cdots\!55$$$$T^{22} +$$$$84\!\cdots\!70$$$$T^{23} -$$$$47\!\cdots\!95$$$$T^{24} -$$$$51\!\cdots\!86$$$$T^{25} +$$$$30\!\cdots\!39$$$$T^{26} -$$$$27\!\cdots\!83$$$$T^{27} +$$$$10\!\cdots\!07$$$$T^{28} +$$$$81\!\cdots\!99$$$$T^{29} -$$$$16\!\cdots\!41$$$$T^{30} +$$$$10\!\cdots\!15$$$$T^{31} +$$$$15\!\cdots\!90$$$$T^{32} -$$$$76\!\cdots\!35$$$$T^{33} +$$$$50\!\cdots\!35$$$$T^{34} +$$$$24\!\cdots\!26$$$$T^{35} -$$$$22\!\cdots\!59$$$$T^{36} +$$$$12\!\cdots\!28$$$$T^{37} +$$$$48\!\cdots\!03$$$$T^{38} -$$$$28\!\cdots\!89$$$$T^{39} +$$$$10\!\cdots\!01$$$$T^{40}$$)
$73$ ($$1 + 21 T + 93 T^{2} - 1405 T^{3} - 22044 T^{4} - 102565 T^{5} + 495597 T^{6} + 8169357 T^{7} + 28398241 T^{8}$$)($$1 - 31 T + 534 T^{2} - 7966 T^{3} + 111434 T^{4} - 1603743 T^{5} + 22351367 T^{6} - 281646142 T^{7} + 3353687638 T^{8} - 38445124926 T^{9} + 434550976103 T^{10} - 4822841787689 T^{11} + 51216166388444 T^{12} - 524844739003705 T^{13} + 5218228291241591 T^{14} - 50848328799375207 T^{15} + 487847407358487847 T^{16} - 4547910484654680503 T^{17} + 41187613457894448137 T^{18} -$$$$36\!\cdots\!40$$$$T^{19} +$$$$31\!\cdots\!33$$$$T^{20} -$$$$26\!\cdots\!20$$$$T^{21} +$$$$21\!\cdots\!73$$$$T^{22} -$$$$17\!\cdots\!51$$$$T^{23} +$$$$13\!\cdots\!27$$$$T^{24} -$$$$10\!\cdots\!51$$$$T^{25} +$$$$78\!\cdots\!99$$$$T^{26} -$$$$57\!\cdots\!85$$$$T^{27} +$$$$41\!\cdots\!64$$$$T^{28} -$$$$28\!\cdots\!57$$$$T^{29} +$$$$18\!\cdots\!47$$$$T^{30} -$$$$12\!\cdots\!02$$$$T^{31} +$$$$76\!\cdots\!98$$$$T^{32} -$$$$47\!\cdots\!86$$$$T^{33} +$$$$27\!\cdots\!03$$$$T^{34} -$$$$14\!\cdots\!51$$$$T^{35} +$$$$72\!\cdots\!74$$$$T^{36} -$$$$37\!\cdots\!98$$$$T^{37} +$$$$18\!\cdots\!46$$$$T^{38} -$$$$78\!\cdots\!47$$$$T^{39} +$$$$18\!\cdots\!01$$$$T^{40}$$)
$79$ ($$1 + 25 T + 231 T^{2} + 1505 T^{3} + 12896 T^{4} + 118895 T^{5} + 1441671 T^{6} + 12325975 T^{7} + 38950081 T^{8}$$)($$1 - 43 T + 1010 T^{2} - 15819 T^{3} + 185753 T^{4} - 1522733 T^{5} + 5028910 T^{6} + 100355235 T^{7} - 2341403015 T^{8} + 30188080151 T^{9} - 275046275717 T^{10} + 1696906291579 T^{11} - 1741732498173 T^{12} - 115433993429912 T^{13} + 1897867614882981 T^{14} - 18867870447247377 T^{15} + 133342713160758839 T^{16} - 441348550385955580 T^{17} - 3733471051129577399 T^{18} + 87409733638924456455 T^{19} -$$$$93\!\cdots\!19$$$$T^{20} +$$$$69\!\cdots\!45$$$$T^{21} -$$$$23\!\cdots\!59$$$$T^{22} -$$$$21\!\cdots\!20$$$$T^{23} +$$$$51\!\cdots\!59$$$$T^{24} -$$$$58\!\cdots\!23$$$$T^{25} +$$$$46\!\cdots\!01$$$$T^{26} -$$$$22\!\cdots\!08$$$$T^{27} -$$$$26\!\cdots\!53$$$$T^{28} +$$$$20\!\cdots\!01$$$$T^{29} -$$$$26\!\cdots\!17$$$$T^{30} +$$$$22\!\cdots\!29$$$$T^{31} -$$$$13\!\cdots\!15$$$$T^{32} +$$$$46\!\cdots\!65$$$$T^{33} +$$$$18\!\cdots\!10$$$$T^{34} -$$$$44\!\cdots\!67$$$$T^{35} +$$$$42\!\cdots\!13$$$$T^{36} -$$$$28\!\cdots\!21$$$$T^{37} +$$$$14\!\cdots\!10$$$$T^{38} -$$$$48\!\cdots\!17$$$$T^{39} +$$$$89\!\cdots\!01$$$$T^{40}$$)
$83$ ($$1 + 16 T + 173 T^{2} + 2200 T^{3} + 26921 T^{4} + 182600 T^{5} + 1191797 T^{6} + 9148592 T^{7} + 47458321 T^{8}$$)($$1 - 32 T + 406 T^{2} - 3496 T^{3} + 33502 T^{4} - 271710 T^{5} + 2069101 T^{6} - 15131563 T^{7} - 71284893 T^{8} + 1838127961 T^{9} - 6132967089 T^{10} + 19871695526 T^{11} - 29200068963 T^{12} - 5255246673042 T^{13} + 40994603541527 T^{14} - 656705281023150 T^{15} + 4381223353664725 T^{16} + 91171220352701247 T^{17} - 1466354427687287544 T^{18} + 14288651913681658093 T^{19} -$$$$14\!\cdots\!41$$$$T^{20} +$$$$11\!\cdots\!19$$$$T^{21} -$$$$10\!\cdots\!16$$$$T^{22} +$$$$52\!\cdots\!89$$$$T^{23} +$$$$20\!\cdots\!25$$$$T^{24} -$$$$25\!\cdots\!50$$$$T^{25} +$$$$13\!\cdots\!63$$$$T^{26} -$$$$14\!\cdots\!34$$$$T^{27} -$$$$65\!\cdots\!83$$$$T^{28} +$$$$37\!\cdots\!78$$$$T^{29} -$$$$95\!\cdots\!61$$$$T^{30} +$$$$23\!\cdots\!87$$$$T^{31} -$$$$76\!\cdots\!73$$$$T^{32} -$$$$13\!\cdots\!69$$$$T^{33} +$$$$15\!\cdots\!29$$$$T^{34} -$$$$16\!\cdots\!70$$$$T^{35} +$$$$16\!\cdots\!62$$$$T^{36} -$$$$14\!\cdots\!08$$$$T^{37} +$$$$14\!\cdots\!54$$$$T^{38} -$$$$92\!\cdots\!04$$$$T^{39} +$$$$24\!\cdots\!01$$$$T^{40}$$)
$89$ ($$1 + 10 T - 29 T^{2} + 200 T^{3} + 10101 T^{4} + 17800 T^{5} - 229709 T^{6} + 7049690 T^{7} + 62742241 T^{8}$$)($$1 + 47 T + 1272 T^{2} + 26378 T^{3} + 462421 T^{4} + 7095712 T^{5} + 97695631 T^{6} + 1229767604 T^{7} + 14280412620 T^{8} + 153298352320 T^{9} + 1534900173114 T^{10} + 14514559954090 T^{11} + 131455544472509 T^{12} + 1167821070144865 T^{13} + 10554337797718648 T^{14} + 99660514974136256 T^{15} + 988915111864240255 T^{16} + 10175025404199544427 T^{17} +$$$$10\!\cdots\!12$$$$T^{18} +$$$$10\!\cdots\!71$$$$T^{19} +$$$$10\!\cdots\!15$$$$T^{20} +$$$$94\!\cdots\!19$$$$T^{21} +$$$$83\!\cdots\!52$$$$T^{22} +$$$$71\!\cdots\!63$$$$T^{23} +$$$$62\!\cdots\!55$$$$T^{24} +$$$$55\!\cdots\!44$$$$T^{25} +$$$$52\!\cdots\!28$$$$T^{26} +$$$$51\!\cdots\!85$$$$T^{27} +$$$$51\!\cdots\!29$$$$T^{28} +$$$$50\!\cdots\!10$$$$T^{29} +$$$$47\!\cdots\!14$$$$T^{30} +$$$$42\!\cdots\!80$$$$T^{31} +$$$$35\!\cdots\!20$$$$T^{32} +$$$$27\!\cdots\!76$$$$T^{33} +$$$$19\!\cdots\!71$$$$T^{34} +$$$$12\!\cdots\!88$$$$T^{35} +$$$$71\!\cdots\!81$$$$T^{36} +$$$$36\!\cdots\!62$$$$T^{37} +$$$$15\!\cdots\!32$$$$T^{38} +$$$$51\!\cdots\!23$$$$T^{39} +$$$$97\!\cdots\!01$$$$T^{40}$$)
$97$ ($$1 - 3 T + 12 T^{2} - 865 T^{3} + 11511 T^{4} - 83905 T^{5} + 112908 T^{6} - 2738019 T^{7} + 88529281 T^{8}$$)($$1 + 45 T + 424 T^{2} - 9536 T^{3} - 203356 T^{4} + 44605 T^{5} + 35372400 T^{6} + 374377366 T^{7} - 1208346180 T^{8} - 73904977716 T^{9} - 534049014451 T^{10} + 5845040218546 T^{11} + 113608841594705 T^{12} + 304677990329790 T^{13} - 9760410996893029 T^{14} - 128654933544812038 T^{15} + 42375802710219630 T^{16} + 15203142043841323748 T^{17} +$$$$10\!\cdots\!85$$$$T^{18} -$$$$65\!\cdots\!30$$$$T^{19} -$$$$14\!\cdots\!02$$$$T^{20} -$$$$63\!\cdots\!10$$$$T^{21} +$$$$99\!\cdots\!65$$$$T^{22} +$$$$13\!\cdots\!04$$$$T^{23} +$$$$37\!\cdots\!30$$$$T^{24} -$$$$11\!\cdots\!66$$$$T^{25} -$$$$81\!\cdots\!41$$$$T^{26} +$$$$24\!\cdots\!70$$$$T^{27} +$$$$89\!\cdots\!05$$$$T^{28} +$$$$44\!\cdots\!82$$$$T^{29} -$$$$39\!\cdots\!99$$$$T^{30} -$$$$52\!\cdots\!48$$$$T^{31} -$$$$83\!\cdots\!80$$$$T^{32} +$$$$25\!\cdots\!82$$$$T^{33} +$$$$23\!\cdots\!00$$$$T^{34} +$$$$28\!\cdots\!65$$$$T^{35} -$$$$12\!\cdots\!76$$$$T^{36} -$$$$56\!\cdots\!32$$$$T^{37} +$$$$24\!\cdots\!36$$$$T^{38} +$$$$25\!\cdots\!85$$$$T^{39} +$$$$54\!\cdots\!01$$$$T^{40}$$)
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