# Properties

 Label 90.2 Level 90 Weight 2 Dimension 53 Nonzero newspaces 6 Newform subspaces 12 Sturm bound 864 Trace bound 1

## Defining parameters

 Level: $$N$$ = $$90 = 2 \cdot 3^{2} \cdot 5$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$6$$ Newform subspaces: $$12$$ Sturm bound: $$864$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(90))$$.

Total New Old
Modular forms 280 53 227
Cusp forms 153 53 100
Eisenstein series 127 0 127

## Trace form

 $$53q + 3q^{2} + 6q^{3} + 3q^{4} + 5q^{5} - 6q^{6} + 12q^{7} - 3q^{8} - 14q^{9} + O(q^{10})$$ $$53q + 3q^{2} + 6q^{3} + 3q^{4} + 5q^{5} - 6q^{6} + 12q^{7} - 3q^{8} - 14q^{9} - 11q^{10} - 26q^{11} - 8q^{12} - 14q^{13} - 20q^{14} - 24q^{15} + 3q^{16} - 18q^{17} - 4q^{18} - 8q^{19} - 11q^{20} - 36q^{21} - 10q^{22} - 36q^{23} + 6q^{24} - 17q^{25} + 6q^{26} - 24q^{27} - 6q^{29} - 8q^{31} + 3q^{32} + 10q^{33} + 4q^{34} + 16q^{35} + 10q^{36} - 26q^{37} + 42q^{38} + 64q^{39} - 3q^{40} + 76q^{41} + 64q^{42} + 10q^{43} + 32q^{44} + 80q^{45} + 60q^{47} + 10q^{48} + 3q^{49} + 47q^{50} + 30q^{51} + 10q^{52} + 30q^{53} + 30q^{54} - 20q^{55} + 20q^{56} + 26q^{57} + 34q^{58} + 10q^{59} - 8q^{60} + 14q^{61} - 40q^{63} - 3q^{64} - 42q^{65} - 48q^{66} - 18q^{67} - 36q^{68} - 64q^{69} - 24q^{70} - 72q^{71} - 6q^{72} - 58q^{73} - 50q^{74} + 6q^{75} - 46q^{76} - 12q^{77} - 28q^{78} - 32q^{79} + 5q^{80} + 2q^{81} - 58q^{82} + 36q^{83} - 12q^{84} - 30q^{85} - 70q^{86} - 20q^{87} - 10q^{88} - 46q^{89} - 56q^{90} - 64q^{91} - 36q^{92} + 32q^{93} - 28q^{94} + 12q^{95} - 8q^{96} - 44q^{97} - 27q^{98} - 20q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(90))$$

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space $$S_k^{\mathrm{new}}(N, \chi)$$ we list the newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
90.2.a $$\chi_{90}(1, \cdot)$$ 90.2.a.a 1 1
90.2.a.b 1
90.2.a.c 1
90.2.c $$\chi_{90}(19, \cdot)$$ 90.2.c.a 2 1
90.2.e $$\chi_{90}(31, \cdot)$$ 90.2.e.a 2 2
90.2.e.b 2
90.2.e.c 4
90.2.f $$\chi_{90}(17, \cdot)$$ 90.2.f.a 4 2
90.2.i $$\chi_{90}(49, \cdot)$$ 90.2.i.a 4 2
90.2.i.b 8
90.2.l $$\chi_{90}(23, \cdot)$$ 90.2.l.a 8 4
90.2.l.b 16

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(90))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(90)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(45))$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 + T$$)($$1 - T$$)($$1 - T$$)($$1 + T^{2}$$)($$1 + T + T^{2}$$)($$1 - T + T^{2}$$)($$( 1 - T + T^{2} )^{2}$$)($$1 + T^{4}$$)($$1 - T^{2} + T^{4}$$)($$( 1 - T^{2} + T^{4} )^{2}$$)($$1 - T^{4} + T^{8}$$)($$( 1 - T^{4} + T^{8} )^{2}$$)
$3$ ()()()()($$1 - 3 T + 3 T^{2}$$)($$1 + 3 T + 3 T^{2}$$)($$( 1 - T + 3 T^{2} )^{2}$$)()($$1 + 3 T^{2} + 9 T^{4}$$)($$1 + 2 T^{2} - 5 T^{4} + 18 T^{6} + 81 T^{8}$$)($$1 - 4 T + 8 T^{2} - 8 T^{3} + 7 T^{4} - 24 T^{5} + 72 T^{6} - 108 T^{7} + 81 T^{8}$$)($$1 - 6 T^{4} - 45 T^{8} - 486 T^{12} + 6561 T^{16}$$)
$5$ ($$1 - T$$)($$1 + T$$)($$1 - T$$)($$1 - 4 T + 5 T^{2}$$)($$1 - T + T^{2}$$)($$1 + T + T^{2}$$)($$( 1 - T + T^{2} )^{2}$$)($$1 + 8 T^{2} + 25 T^{4}$$)($$1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4}$$)($$1 + 4 T + 8 T^{2} - 8 T^{3} - 41 T^{4} - 40 T^{5} + 200 T^{6} + 500 T^{7} + 625 T^{8}$$)($$( 1 - 6 T + 17 T^{2} - 30 T^{3} + 25 T^{4} )^{2}$$)($$1 + 12 T + 80 T^{2} + 384 T^{3} + 1480 T^{4} + 4860 T^{5} + 14048 T^{6} + 36420 T^{7} + 85471 T^{8} + 182100 T^{9} + 351200 T^{10} + 607500 T^{11} + 925000 T^{12} + 1200000 T^{13} + 1250000 T^{14} + 937500 T^{15} + 390625 T^{16}$$)
$7$ ($$1 - 2 T + 7 T^{2}$$)($$1 - 2 T + 7 T^{2}$$)($$1 + 4 T + 7 T^{2}$$)($$1 - 10 T^{2} + 49 T^{4}$$)($$( 1 - 5 T + 7 T^{2} )( 1 + 4 T + 7 T^{2} )$$)($$( 1 - 5 T + 7 T^{2} )( 1 + T + 7 T^{2} )$$)($$1 + T - 5 T^{2} - 8 T^{3} - 20 T^{4} - 56 T^{5} - 245 T^{6} + 343 T^{7} + 2401 T^{8}$$)($$( 1 - 4 T + 8 T^{2} - 28 T^{3} + 49 T^{4} )^{2}$$)($$( 1 + 2 T^{2} + 49 T^{4} )( 1 + 11 T^{2} + 49 T^{4} )$$)($$1 + 8 T^{2} + 46 T^{4} - 640 T^{6} - 5213 T^{8} - 31360 T^{10} + 110446 T^{12} + 941192 T^{14} + 5764801 T^{16}$$)($$1 + 8 T + 32 T^{2} + 64 T^{3} - 7 T^{4} - 464 T^{5} - 1440 T^{6} - 2472 T^{7} - 4016 T^{8} - 17304 T^{9} - 70560 T^{10} - 159152 T^{11} - 16807 T^{12} + 1075648 T^{13} + 3764768 T^{14} + 6588344 T^{15} + 5764801 T^{16}$$)($$1 - 8 T + 32 T^{2} - 32 T^{3} - 168 T^{4} + 752 T^{5} - 128 T^{6} - 4696 T^{7} + 9614 T^{8} + 32960 T^{9} - 159808 T^{10} + 211120 T^{11} + 686208 T^{12} - 2864720 T^{13} + 4398240 T^{14} + 5724096 T^{15} - 29956541 T^{16} + 40068672 T^{17} + 215513760 T^{18} - 982598960 T^{19} + 1647585408 T^{20} + 3548293840 T^{21} - 18801251392 T^{22} + 27143977280 T^{23} + 55422796814 T^{24} - 189500538472 T^{25} - 36156831872 T^{26} + 1486949710736 T^{27} - 2325336249768 T^{28} - 3100448333024 T^{29} + 21703138331168 T^{30} - 37980492079544 T^{31} + 33232930569601 T^{32}$$)
$11$ ($$1 - 6 T + 11 T^{2}$$)($$1 + 6 T + 11 T^{2}$$)($$1 + 11 T^{2}$$)($$( 1 + 2 T + 11 T^{2} )^{2}$$)($$1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4}$$)($$1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4}$$)($$1 - 3 T - 7 T^{2} + 18 T^{3} + 36 T^{4} + 198 T^{5} - 847 T^{6} - 3993 T^{7} + 14641 T^{8}$$)($$( 1 - 6 T + 11 T^{2} )^{2}( 1 + 6 T + 11 T^{2} )^{2}$$)($$( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} )^{2}$$)($$( 1 - 2 T - 13 T^{2} + 10 T^{3} + 124 T^{4} + 110 T^{5} - 1573 T^{6} - 2662 T^{7} + 14641 T^{8} )^{2}$$)($$( 1 + 12 T + 74 T^{2} + 312 T^{3} + 1083 T^{4} + 3432 T^{5} + 8954 T^{6} + 15972 T^{7} + 14641 T^{8} )^{2}$$)($$( 1 + 22 T^{2} + 199 T^{4} - 264 T^{5} + 1138 T^{6} - 6840 T^{7} + 10132 T^{8} - 75240 T^{9} + 137698 T^{10} - 351384 T^{11} + 2913559 T^{12} + 38974342 T^{14} + 214358881 T^{16} )^{2}$$)
$13$ ($$1 + 4 T + 13 T^{2}$$)($$1 + 4 T + 13 T^{2}$$)($$1 - 2 T + 13 T^{2}$$)($$( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )$$)($$( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )$$)($$1 - 4 T + 3 T^{2} - 52 T^{3} + 169 T^{4}$$)($$1 - 2 T + 10 T^{2} + 64 T^{3} - 185 T^{4} + 832 T^{5} + 1690 T^{6} - 4394 T^{7} + 28561 T^{8}$$)($$( 1 + 6 T + 18 T^{2} + 78 T^{3} + 169 T^{4} )^{2}$$)($$( 1 - 4 T + 3 T^{2} - 52 T^{3} + 169 T^{4} )( 1 + 4 T + 3 T^{2} + 52 T^{3} + 169 T^{4} )$$)($$( 1 + 20 T^{2} + 231 T^{4} + 3380 T^{6} + 28561 T^{8} )^{2}$$)($$1 + 142 T^{4} - 8397 T^{8} + 4055662 T^{12} + 815730721 T^{16}$$)($$1 + 48 T^{3} - 208 T^{4} + 648 T^{5} + 1152 T^{6} - 768 T^{7} + 58078 T^{8} + 38784 T^{9} + 412704 T^{10} - 164616 T^{11} + 6437888 T^{12} + 1602600 T^{13} + 36461952 T^{14} + 584300928 T^{15} - 665621597 T^{16} + 7595912064 T^{17} + 6162069888 T^{18} + 3520912200 T^{19} + 183872519168 T^{20} - 61120768488 T^{21} + 1992043381536 T^{22} + 2433638483328 T^{23} + 47376008814238 T^{24} - 8144255518464 T^{25} + 158812982610048 T^{26} + 1161319935335976 T^{27} - 4846001705476048 T^{28} + 14538005116428144 T^{29} + 665416609183179841 T^{32}$$)
$17$ ($$1 + 6 T + 17 T^{2}$$)($$1 - 6 T + 17 T^{2}$$)($$1 + 6 T + 17 T^{2}$$)($$( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} )$$)($$( 1 + 17 T^{2} )^{2}$$)($$( 1 - 3 T + 17 T^{2} )^{2}$$)($$( 1 + 9 T + 46 T^{2} + 153 T^{3} + 289 T^{4} )^{2}$$)($$1 - 254 T^{4} + 83521 T^{8}$$)($$( 1 - 8 T + 17 T^{2} )^{2}( 1 + 8 T + 17 T^{2} )^{2}$$)($$( 1 - 18 T^{2} + 563 T^{4} - 5202 T^{6} + 83521 T^{8} )^{2}$$)($$1 + 188 T^{4} - 45306 T^{8} + 15701948 T^{12} + 6975757441 T^{16}$$)($$1 - 356 T^{4} + 24298 T^{8} + 31798576 T^{12} - 15547912973 T^{16} + 2655848866096 T^{20} + 169496954301418 T^{24} - 207413516453794916 T^{28} + 48661191875666868481 T^{32}$$)
$19$ ($$1 + 4 T + 19 T^{2}$$)($$1 + 4 T + 19 T^{2}$$)($$1 + 4 T + 19 T^{2}$$)($$( 1 + 19 T^{2} )^{2}$$)($$( 1 + 4 T + 19 T^{2} )^{2}$$)($$( 1 - 5 T + 19 T^{2} )^{2}$$)($$( 1 - T + 30 T^{2} - 19 T^{3} + 361 T^{4} )^{2}$$)($$( 1 + 26 T^{2} + 361 T^{4} )^{2}$$)($$( 1 + 6 T + 19 T^{2} )^{4}$$)($$( 1 - 6 T + 41 T^{2} - 114 T^{3} + 361 T^{4} )^{4}$$)($$( 1 - 32 T^{2} + 594 T^{4} - 11552 T^{6} + 130321 T^{8} )^{2}$$)($$( 1 - 100 T^{2} + 5038 T^{4} - 162424 T^{6} + 3656035 T^{8} - 58635064 T^{10} + 656557198 T^{12} - 4704588100 T^{14} + 16983563041 T^{16} )^{2}$$)
$23$ ($$1 + 23 T^{2}$$)($$1 + 23 T^{2}$$)($$1 + 23 T^{2}$$)($$1 - 30 T^{2} + 529 T^{4}$$)($$1 + 9 T + 58 T^{2} + 207 T^{3} + 529 T^{4}$$)($$1 + 6 T + 13 T^{2} + 138 T^{3} + 529 T^{4}$$)($$1 - 3 T - 31 T^{2} + 18 T^{3} + 864 T^{4} + 414 T^{5} - 16399 T^{6} - 36501 T^{7} + 279841 T^{8}$$)($$1 - 158 T^{4} + 279841 T^{8}$$)($$1 + 45 T^{2} + 1496 T^{4} + 23805 T^{6} + 279841 T^{8}$$)($$1 + 36 T^{2} + 298 T^{4} - 2160 T^{6} + 30579 T^{8} - 1142640 T^{10} + 83392618 T^{12} + 5329292004 T^{14} + 78310985281 T^{16}$$)($$1 - 967 T^{4} + 655248 T^{8} - 270606247 T^{12} + 78310985281 T^{16}$$)($$1 + 24 T + 288 T^{2} + 2304 T^{3} + 14956 T^{4} + 93528 T^{5} + 591552 T^{6} + 3510696 T^{7} + 18949162 T^{8} + 98431632 T^{9} + 520135776 T^{10} + 2742734952 T^{11} + 13919721136 T^{12} + 68549757432 T^{13} + 335617355232 T^{14} + 1641179221488 T^{15} + 7932015899923 T^{16} + 37747122094224 T^{17} + 177541580917728 T^{18} + 834044898675144 T^{19} + 3895308682419376 T^{20} + 17653182909160536 T^{21} + 76998762000864864 T^{22} + 335142525423339504 T^{23} + 1483927546469284522 T^{24} + 6323299443987508248 T^{25} + 24505935561456493248 T^{26} + 89114391038173764456 T^{27} +$$$$32\!\cdots\!76$$$$T^{28} +$$$$11\!\cdots\!32$$$$T^{29} +$$$$33\!\cdots\!92$$$$T^{30} +$$$$63\!\cdots\!68$$$$T^{31} +$$$$61\!\cdots\!61$$$$T^{32}$$)
$29$ ($$1 + 6 T + 29 T^{2}$$)($$1 - 6 T + 29 T^{2}$$)($$1 - 6 T + 29 T^{2}$$)($$( 1 + 29 T^{2} )^{2}$$)($$1 + 3 T - 20 T^{2} + 87 T^{3} + 841 T^{4}$$)($$1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4}$$)($$1 - 3 T - 43 T^{2} + 18 T^{3} + 1602 T^{4} + 522 T^{5} - 36163 T^{6} - 73167 T^{7} + 707281 T^{8}$$)($$( 1 + 56 T^{2} + 841 T^{4} )^{2}$$)($$( 1 - 9 T + 52 T^{2} - 261 T^{3} + 841 T^{4} )^{2}$$)($$( 1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4} )^{4}$$)($$1 - 106 T^{2} + 6769 T^{4} - 295210 T^{6} + 9797332 T^{8} - 248271610 T^{10} + 4787585089 T^{12} - 63051272026 T^{14} + 500246412961 T^{16}$$)($$1 - 152 T^{2} + 12036 T^{4} - 646864 T^{6} + 26386346 T^{8} - 870522600 T^{10} + 24736791952 T^{12} - 660065237288 T^{14} + 18313327590963 T^{16} - 555114864559208 T^{18} + 17495862948602512 T^{20} - 517807143937554600 T^{22} + 13199674937647830506 T^{24} -$$$$27\!\cdots\!64$$$$T^{26} +$$$$42\!\cdots\!76$$$$T^{28} -$$$$45\!\cdots\!12$$$$T^{30} +$$$$25\!\cdots\!21$$$$T^{32}$$)
$31$ ($$1 + 4 T + 31 T^{2}$$)($$1 + 4 T + 31 T^{2}$$)($$1 - 8 T + 31 T^{2}$$)($$( 1 + 8 T + 31 T^{2} )^{2}$$)($$( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} )$$)($$1 + 2 T - 27 T^{2} + 62 T^{3} + 961 T^{4}$$)($$1 - 2 T - 26 T^{2} + 64 T^{3} - 185 T^{4} + 1984 T^{5} - 24986 T^{6} - 59582 T^{7} + 923521 T^{8}$$)($$( 1 - 4 T + 31 T^{2} )^{4}$$)($$( 1 - 2 T - 27 T^{2} - 62 T^{3} + 961 T^{4} )^{2}$$)($$( 1 + 8 T - 8 T^{2} + 80 T^{3} + 2239 T^{4} + 2480 T^{5} - 7688 T^{6} + 238328 T^{7} + 923521 T^{8} )^{2}$$)($$( 1 - 4 T - 44 T^{2} + 8 T^{3} + 2143 T^{4} + 248 T^{5} - 42284 T^{6} - 119164 T^{7} + 923521 T^{8} )^{2}$$)($$( 1 + 4 T - 36 T^{2} - 280 T^{3} - 244 T^{4} + 4476 T^{5} + 26680 T^{6} + 13564 T^{7} - 525585 T^{8} + 420484 T^{9} + 25639480 T^{10} + 133344516 T^{11} - 225339124 T^{12} - 8016162280 T^{13} - 31950132516 T^{14} + 110050456444 T^{15} + 852891037441 T^{16} )^{2}$$)
$37$ ($$1 - 8 T + 37 T^{2}$$)($$1 - 8 T + 37 T^{2}$$)($$1 - 2 T + 37 T^{2}$$)($$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$)($$( 1 - 8 T + 37 T^{2} )^{2}$$)($$( 1 + 4 T + 37 T^{2} )^{2}$$)($$( 1 + 4 T + 37 T^{2} )^{4}$$)($$( 1 + 6 T + 18 T^{2} + 222 T^{3} + 1369 T^{4} )^{2}$$)($$( 1 - 12 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2}$$)($$( 1 - 10 T^{2} + 1369 T^{4} )^{4}$$)($$( 1 + 6 T + 18 T^{2} + 222 T^{3} + 1369 T^{4} )^{4}$$)($$( 1 - 576 T^{3} + 1060 T^{4} + 8640 T^{5} + 165888 T^{6} - 634752 T^{7} - 2979546 T^{8} - 23485824 T^{9} + 227100672 T^{10} + 437641920 T^{11} + 1986610660 T^{12} - 39942119232 T^{13} + 3512479453921 T^{16} )^{2}$$)
$41$ ($$1 + 41 T^{2}$$)($$1 + 41 T^{2}$$)($$1 - 6 T + 41 T^{2}$$)($$( 1 + 2 T + 41 T^{2} )^{2}$$)($$1 - 3 T - 32 T^{2} - 123 T^{3} + 1681 T^{4}$$)($$1 - 3 T - 32 T^{2} - 123 T^{3} + 1681 T^{4}$$)($$( 1 - 3 T - 32 T^{2} - 123 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 + 16 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 - 11 T + 80 T^{2} - 451 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 - T - 40 T^{2} - 41 T^{3} + 1681 T^{4} )^{4}$$)($$( 1 - 6 T + 65 T^{2} - 318 T^{3} + 1620 T^{4} - 13038 T^{5} + 109265 T^{6} - 413526 T^{7} + 2825761 T^{8} )^{2}$$)($$( 1 - 12 T + 154 T^{2} - 1272 T^{3} + 9001 T^{4} - 62400 T^{5} + 437338 T^{6} - 2866716 T^{7} + 21170932 T^{8} - 117535356 T^{9} + 735165178 T^{10} - 4300670400 T^{11} + 25434674761 T^{12} - 147369087672 T^{13} + 731516053114 T^{14} - 2337051286572 T^{15} + 7984925229121 T^{16} )^{2}$$)
$43$ ($$1 - 8 T + 43 T^{2}$$)($$1 - 8 T + 43 T^{2}$$)($$1 + 4 T + 43 T^{2}$$)($$1 - 70 T^{2} + 1849 T^{4}$$)($$( 1 - 5 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} )$$)($$1 + 11 T + 78 T^{2} + 473 T^{3} + 1849 T^{4}$$)($$1 - 17 T + 139 T^{2} - 1088 T^{3} + 8224 T^{4} - 46784 T^{5} + 257011 T^{6} - 1351619 T^{7} + 3418801 T^{8}$$)($$( 1 + 1849 T^{4} )^{2}$$)($$1 + 70 T^{2} + 3051 T^{4} + 129430 T^{6} + 3418801 T^{8}$$)($$1 + 110 T^{2} + 5977 T^{4} + 266750 T^{6} + 11699428 T^{8} + 493220750 T^{10} + 20434173577 T^{12} + 695349935390 T^{14} + 11688200277601 T^{16}$$)($$1 - 1778 T^{4} - 257517 T^{8} - 6078628178 T^{12} + 11688200277601 T^{16}$$)($$1 + 96 T^{3} + 854 T^{4} + 1872 T^{5} + 4608 T^{6} + 53616 T^{7} + 4490401 T^{8} - 2573760 T^{9} + 2964096 T^{10} + 348774432 T^{11} - 9048958570 T^{12} + 6137398128 T^{13} + 6878498688 T^{14} - 451002697968 T^{15} - 494432039036 T^{16} - 19393116012624 T^{17} + 12718344074112 T^{18} + 487966112962896 T^{19} - 30936588608074570 T^{20} + 51272786206529376 T^{21} + 18737126928088704 T^{22} - 699595868522752320 T^{23} + 52484706214739808001 T^{24} + 26947005481605774288 T^{25} + 99585710499613819392 T^{26} +$$$$17\!\cdots\!04$$$$T^{27} +$$$$34\!\cdots\!54$$$$T^{28} +$$$$16\!\cdots\!28$$$$T^{29} +$$$$13\!\cdots\!01$$$$T^{32}$$)
$47$ ($$1 + 47 T^{2}$$)($$1 + 47 T^{2}$$)($$1 + 47 T^{2}$$)($$1 - 30 T^{2} + 2209 T^{4}$$)($$1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4}$$)($$1 - 47 T^{2} + 2209 T^{4}$$)($$1 - 9 T - 25 T^{2} - 108 T^{3} + 5220 T^{4} - 5076 T^{5} - 55225 T^{6} - 934407 T^{7} + 4879681 T^{8}$$)($$( 1 + 2209 T^{4} )^{2}$$)($$1 + 45 T^{2} - 184 T^{4} + 99405 T^{6} + 4879681 T^{8}$$)($$1 + 168 T^{2} + 16846 T^{4} + 1169280 T^{6} + 62232387 T^{8} + 2582939520 T^{10} + 82203106126 T^{12} + 1810908175272 T^{14} + 23811286661761 T^{16}$$)($$1 + 4249 T^{4} + 13174320 T^{8} + 20733764569 T^{12} + 23811286661761 T^{16}$$)($$1 - 48 T + 1152 T^{2} - 18432 T^{3} + 221336 T^{4} - 2122632 T^{5} + 16776576 T^{6} - 110138352 T^{7} + 584847502 T^{8} - 2167705440 T^{9} - 101278944 T^{10} + 103905054792 T^{11} - 1313297330560 T^{12} + 11748806482152 T^{13} - 89054821472256 T^{14} + 622050377830560 T^{15} - 4255385068735805 T^{16} + 29236367758036320 T^{17} - 196722100632213504 T^{18} + 1219796335396467096 T^{19} - 6408472031284351360 T^{20} + 23830105518606623544 T^{21} - 1091707545669732576 T^{22} -$$$$10\!\cdots\!20$$$$T^{23} +$$$$13\!\cdots\!22$$$$T^{24} -$$$$12\!\cdots\!84$$$$T^{25} +$$$$88\!\cdots\!24$$$$T^{26} -$$$$52\!\cdots\!96$$$$T^{27} +$$$$25\!\cdots\!76$$$$T^{28} -$$$$10\!\cdots\!64$$$$T^{29} +$$$$29\!\cdots\!88$$$$T^{30} -$$$$57\!\cdots\!64$$$$T^{31} +$$$$56\!\cdots\!21$$$$T^{32}$$)
$53$ ($$1 + 6 T + 53 T^{2}$$)($$1 - 6 T + 53 T^{2}$$)($$1 - 6 T + 53 T^{2}$$)($$1 - 70 T^{2} + 2809 T^{4}$$)($$( 1 - 6 T + 53 T^{2} )^{2}$$)($$( 1 - 6 T + 53 T^{2} )^{2}$$)($$( 1 - 26 T^{2} + 2809 T^{4} )^{2}$$)($$( 1 - 56 T^{2} + 2809 T^{4} )( 1 + 56 T^{2} + 2809 T^{4} )$$)($$( 1 - 53 T^{2} )^{4}$$)($$( 1 - 128 T^{2} + 8850 T^{4} - 359552 T^{6} + 7890481 T^{8} )^{2}$$)($$1 - 4900 T^{4} + 13820838 T^{8} - 38663356900 T^{12} + 62259690411361 T^{16}$$)($$1 + 4192 T^{4} + 3754180 T^{8} - 35115635936 T^{12} - 139198621476794 T^{16} - 277079258155925216 T^{20} +$$$$23\!\cdots\!80$$$$T^{24} +$$$$20\!\cdots\!72$$$$T^{28} +$$$$38\!\cdots\!21$$$$T^{32}$$)
$59$ ($$1 - 6 T + 59 T^{2}$$)($$1 + 6 T + 59 T^{2}$$)($$1 + 59 T^{2}$$)($$( 1 - 10 T + 59 T^{2} )^{2}$$)($$1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4}$$)($$1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4}$$)($$1 + 3 T - 103 T^{2} - 18 T^{3} + 8532 T^{4} - 1062 T^{5} - 358543 T^{6} + 616137 T^{7} + 12117361 T^{8}$$)($$( 1 + 110 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 + 4 T - 43 T^{2} + 236 T^{3} + 3481 T^{4} )^{2}$$)($$( 1 - 2 T + 35 T^{2} + 298 T^{3} - 2756 T^{4} + 17582 T^{5} + 121835 T^{6} - 410758 T^{7} + 12117361 T^{8} )^{2}$$)($$1 - 16 T^{2} - 5906 T^{4} + 12800 T^{6} + 24961747 T^{8} + 44556800 T^{10} - 71565134066 T^{12} - 674888538256 T^{14} + 146830437604321 T^{16}$$)($$1 - 428 T^{2} + 100818 T^{4} - 16624408 T^{6} + 2120510801 T^{8} - 219902811120 T^{10} + 19093571371522 T^{12} - 1411764424766636 T^{14} + 89719050225623076 T^{16} - 4914351962612659916 T^{18} +$$$$23\!\cdots\!42$$$$T^{20} -$$$$92\!\cdots\!20$$$$T^{22} +$$$$31\!\cdots\!21$$$$T^{24} -$$$$84\!\cdots\!08$$$$T^{26} +$$$$17\!\cdots\!58$$$$T^{28} -$$$$26\!\cdots\!08$$$$T^{30} +$$$$21\!\cdots\!41$$$$T^{32}$$)
$61$ ($$1 - 2 T + 61 T^{2}$$)($$1 - 2 T + 61 T^{2}$$)($$1 + 10 T + 61 T^{2}$$)($$( 1 - 2 T + 61 T^{2} )^{2}$$)($$( 1 - 14 T + 61 T^{2} )( 1 + T + 61 T^{2} )$$)($$1 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4}$$)($$1 + T - 47 T^{2} - 74 T^{3} - 1478 T^{4} - 4514 T^{5} - 174887 T^{6} + 226981 T^{7} + 13845841 T^{8}$$)($$( 1 + 61 T^{2} )^{4}$$)($$( 1 - 7 T - 12 T^{2} - 427 T^{3} + 3721 T^{4} )^{2}$$)($$( 1 + 4 T - 104 T^{2} - 8 T^{3} + 9703 T^{4} - 488 T^{5} - 386984 T^{6} + 907924 T^{7} + 13845841 T^{8} )^{2}$$)($$( 1 - 6 T - 89 T^{2} - 18 T^{3} + 9708 T^{4} - 1098 T^{5} - 331169 T^{6} - 1361886 T^{7} + 13845841 T^{8} )^{2}$$)($$( 1 + 12 T + 8 T^{2} + 1176 T^{3} + 13384 T^{4} - 12108 T^{5} + 574064 T^{6} + 5555196 T^{7} - 14824577 T^{8} + 338866956 T^{9} + 2136092144 T^{10} - 2748285948 T^{11} + 185312735944 T^{12} + 993245249976 T^{13} + 412162994888 T^{14} + 37712914032252 T^{15} + 191707312997281 T^{16} )^{2}$$)
$67$ ($$1 + 4 T + 67 T^{2}$$)($$1 + 4 T + 67 T^{2}$$)($$1 + 4 T + 67 T^{2}$$)($$1 - 70 T^{2} + 4489 T^{4}$$)($$1 - 13 T + 102 T^{2} - 871 T^{3} + 4489 T^{4}$$)($$( 1 - 11 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} )$$)($$( 1 - 7 T - 18 T^{2} - 469 T^{3} + 4489 T^{4} )^{2}$$)($$( 1 + 8 T + 32 T^{2} + 536 T^{3} + 4489 T^{4} )^{2}$$)($$( 1 - 109 T^{2} + 4489 T^{4} )( 1 + 122 T^{2} + 4489 T^{4} )$$)($$1 + 158 T^{2} + 10921 T^{4} + 800270 T^{6} + 64991332 T^{8} + 3592412030 T^{10} + 220070392441 T^{12} + 14292424382702 T^{14} + 406067677556641 T^{16}$$)($$1 - 4 T + 8 T^{2} + 304 T^{3} - 3511 T^{4} + 10624 T^{5} + 31800 T^{6} - 1654308 T^{7} - 4277600 T^{8} - 110838636 T^{9} + 142750200 T^{10} + 3195306112 T^{11} - 70750585831 T^{12} + 410438032528 T^{13} + 723667057352 T^{14} - 24242846421292 T^{15} + 406067677556641 T^{16}$$)($$1 + 16 T + 128 T^{2} - 128 T^{3} - 5562 T^{4} - 6640 T^{5} + 613888 T^{6} + 5277488 T^{7} + 23820881 T^{8} + 40207664 T^{9} + 355221248 T^{10} + 2724412528 T^{11} + 197658905958 T^{12} + 3038720900128 T^{13} + 23981818128000 T^{14} - 20130978078192 T^{15} - 1000499923621916 T^{16} - 1348775531238864 T^{17} + 107654381576592000 T^{18} + 913934814085197664 T^{19} + 3983048530687278918 T^{20} + 3678297755878140496 T^{21} + 32132739406133126912 T^{22} +$$$$24\!\cdots\!72$$$$T^{23} +$$$$96\!\cdots\!21$$$$T^{24} +$$$$14\!\cdots\!36$$$$T^{25} +$$$$11\!\cdots\!12$$$$T^{26} -$$$$81\!\cdots\!20$$$$T^{27} -$$$$45\!\cdots\!82$$$$T^{28} -$$$$70\!\cdots\!36$$$$T^{29} +$$$$47\!\cdots\!12$$$$T^{30} +$$$$39\!\cdots\!88$$$$T^{31} +$$$$16\!\cdots\!81$$$$T^{32}$$)
$71$ ($$1 + 12 T + 71 T^{2}$$)($$1 - 12 T + 71 T^{2}$$)($$1 + 71 T^{2}$$)($$( 1 + 12 T + 71 T^{2} )^{2}$$)($$( 1 + 6 T + 71 T^{2} )^{2}$$)($$( 1 - 6 T + 71 T^{2} )^{2}$$)($$( 1 + 6 T + 71 T^{2} )^{4}$$)($$( 1 - 110 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 + 6 T + 71 T^{2} )^{4}$$)($$( 1 + 136 T^{2} + 5041 T^{4} )^{4}$$)($$( 1 - 244 T^{2} + 24582 T^{4} - 1230004 T^{6} + 25411681 T^{8} )^{2}$$)($$( 1 - 296 T^{2} + 51256 T^{4} - 5830568 T^{6} + 486270130 T^{8} - 29391893288 T^{10} + 1302501121336 T^{12} - 37917684040616 T^{14} + 645753531245761 T^{16} )^{2}$$)
$73$ ($$1 + 10 T + 73 T^{2}$$)($$1 + 10 T + 73 T^{2}$$)($$1 - 2 T + 73 T^{2}$$)($$1 - 130 T^{2} + 5329 T^{4}$$)($$( 1 + 4 T + 73 T^{2} )^{2}$$)($$( 1 + 7 T + 73 T^{2} )^{2}$$)($$( 1 + 11 T + 102 T^{2} + 803 T^{3} + 5329 T^{4} )^{2}$$)($$( 1 - 2 T + 2 T^{2} - 146 T^{3} + 5329 T^{4} )^{2}$$)($$( 1 - 130 T^{2} + 5329 T^{4} )^{2}$$)($$( 1 - 50 T^{2} + 1683 T^{4} - 266450 T^{6} + 28398241 T^{8} )^{2}$$)($$( 1 + 8 T + 32 T^{2} + 264 T^{3} + 578 T^{4} + 19272 T^{5} + 170528 T^{6} + 3112136 T^{7} + 28398241 T^{8} )^{2}$$)($$( 1 - 8 T + 32 T^{2} - 736 T^{3} + 2366 T^{4} + 45704 T^{5} - 170496 T^{6} + 3682392 T^{7} - 78261341 T^{8} + 268814616 T^{9} - 908573184 T^{10} + 17779632968 T^{11} + 67190238206 T^{12} - 1525780692448 T^{13} + 4842695241248 T^{14} - 88379188152776 T^{15} + 806460091894081 T^{16} )^{2}$$)
$79$ ($$1 + 4 T + 79 T^{2}$$)($$1 + 4 T + 79 T^{2}$$)($$1 - 8 T + 79 T^{2}$$)($$( 1 + 79 T^{2} )^{2}$$)($$1 - 10 T + 21 T^{2} - 790 T^{3} + 6241 T^{4}$$)($$1 + 14 T + 117 T^{2} + 1106 T^{3} + 6241 T^{4}$$)($$( 1 + 2 T - 75 T^{2} + 158 T^{3} + 6241 T^{4} )^{2}$$)($$( 1 - 14 T^{2} + 6241 T^{4} )^{2}$$)($$( 1 + 12 T + 65 T^{2} + 948 T^{3} + 6241 T^{4} )^{2}$$)($$( 1 - 104 T^{2} + 4575 T^{4} - 649064 T^{6} + 38950081 T^{8} )^{2}$$)($$( 1 + 152 T^{2} + 16863 T^{4} + 948632 T^{6} + 38950081 T^{8} )^{2}$$)($$1 + 200 T^{2} + 7464 T^{4} - 683600 T^{6} + 16634510 T^{8} + 10268111400 T^{10} + 468563441536 T^{12} + 2801113391000 T^{14} + 270299704808259 T^{16} + 17481748673231000 T^{18} + 18250584001465964416 T^{20} +$$$$24\!\cdots\!00$$$$T^{22} +$$$$25\!\cdots\!10$$$$T^{24} -$$$$64\!\cdots\!00$$$$T^{26} +$$$$44\!\cdots\!24$$$$T^{28} +$$$$73\!\cdots\!00$$$$T^{30} +$$$$23\!\cdots\!21$$$$T^{32}$$)
$83$ ($$1 - 12 T + 83 T^{2}$$)($$1 + 12 T + 83 T^{2}$$)($$1 + 12 T + 83 T^{2}$$)($$1 - 150 T^{2} + 6889 T^{4}$$)($$1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4}$$)($$1 + 12 T + 61 T^{2} + 996 T^{3} + 6889 T^{4}$$)($$1 + 9 T - 97 T^{2} + 108 T^{3} + 18072 T^{4} + 8964 T^{5} - 668233 T^{6} + 5146083 T^{7} + 47458321 T^{8}$$)($$1 - 13294 T^{4} + 47458321 T^{8}$$)($$1 + 45 T^{2} - 4864 T^{4} + 310005 T^{6} + 47458321 T^{8}$$)($$( 1 + 150 T^{2} + 15611 T^{4} + 1033350 T^{6} + 47458321 T^{8} )^{2}$$)($$1 - 12 T + 72 T^{2} - 288 T^{3} - 8375 T^{4} + 56640 T^{5} - 35208 T^{6} - 3433884 T^{7} + 82789344 T^{8} - 285012372 T^{9} - 242547912 T^{10} + 32386015680 T^{11} - 397463438375 T^{12} - 1134443705184 T^{13} + 23539706882568 T^{14} - 325632611875524 T^{15} + 2252292232139041 T^{16}$$)($$1 - 48 T + 1152 T^{2} - 18432 T^{3} + 215516 T^{4} - 1906416 T^{5} + 13102848 T^{6} - 74216976 T^{7} + 433250986 T^{8} - 3712707360 T^{9} + 40409712000 T^{10} - 415323735120 T^{11} + 3731885962352 T^{12} - 29403431286576 T^{13} + 213762329771904 T^{14} - 1567513821393504 T^{15} + 12987012407281267 T^{16} - 130103647175660832 T^{17} + 1472608689798646656 T^{18} - 16812499765057431312 T^{19} +$$$$17\!\cdots\!92$$$$T^{20} -$$$$16\!\cdots\!60$$$$T^{21} +$$$$13\!\cdots\!00$$$$T^{22} -$$$$10\!\cdots\!20$$$$T^{23} +$$$$97\!\cdots\!26$$$$T^{24} -$$$$13\!\cdots\!28$$$$T^{25} +$$$$20\!\cdots\!52$$$$T^{26} -$$$$24\!\cdots\!72$$$$T^{27} +$$$$23\!\cdots\!76$$$$T^{28} -$$$$16\!\cdots\!16$$$$T^{29} +$$$$84\!\cdots\!08$$$$T^{30} -$$$$29\!\cdots\!36$$$$T^{31} +$$$$50\!\cdots\!81$$$$T^{32}$$)
$89$ ($$1 - 12 T + 89 T^{2}$$)($$1 + 12 T + 89 T^{2}$$)($$1 + 18 T + 89 T^{2}$$)($$( 1 + 10 T + 89 T^{2} )^{2}$$)($$( 1 - 9 T + 89 T^{2} )^{2}$$)($$( 1 - 6 T + 89 T^{2} )^{2}$$)($$( 1 - 15 T + 160 T^{2} - 1335 T^{3} + 7921 T^{4} )^{2}$$)($$( 1 + 80 T^{2} + 7921 T^{4} )^{2}$$)($$( 1 + T + 89 T^{2} )^{4}$$)($$( 1 + 16 T + 218 T^{2} + 1424 T^{3} + 7921 T^{4} )^{4}$$)($$( 1 + 286 T^{2} + 35427 T^{4} + 2265406 T^{6} + 62742241 T^{8} )^{2}$$)($$( 1 + 328 T^{2} + 42642 T^{4} + 2598088 T^{6} + 62742241 T^{8} )^{4}$$)
$97$ ($$1 - 2 T + 97 T^{2}$$)($$1 - 2 T + 97 T^{2}$$)($$1 - 2 T + 97 T^{2}$$)($$( 1 - 18 T + 97 T^{2} )( 1 + 18 T + 97 T^{2} )$$)($$1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4}$$)($$1 + 11 T + 24 T^{2} + 1067 T^{3} + 9409 T^{4}$$)($$1 - 11 T - 95 T^{2} - 242 T^{3} + 25510 T^{4} - 23474 T^{5} - 893855 T^{6} - 10039403 T^{7} + 88529281 T^{8}$$)($$( 1 + 6 T + 18 T^{2} + 582 T^{3} + 9409 T^{4} )^{2}$$)($$( 1 - 18 T + 227 T^{2} - 1746 T^{3} + 9409 T^{4} )( 1 + 18 T + 227 T^{2} + 1746 T^{3} + 9409 T^{4} )$$)($$( 1 + 25 T^{2} - 8784 T^{4} + 235225 T^{6} + 88529281 T^{8} )^{2}$$)($$1 - 12 T + 72 T^{2} + 744 T^{3} - 12542 T^{4} + 57996 T^{5} + 483840 T^{6} - 14345892 T^{7} + 128096019 T^{8} - 1391551524 T^{9} + 4552450560 T^{10} + 52931383308 T^{11} - 1110334242302 T^{12} + 6388981151208 T^{13} + 59973984354888 T^{14} - 969579413737356 T^{15} + 7837433594376961 T^{16}$$)($$1 + 48 T + 1152 T^{2} + 15456 T^{3} + 79442 T^{4} - 1258944 T^{5} - 32502528 T^{6} - 320382528 T^{7} - 602207375 T^{8} + 26997703632 T^{9} + 412336144512 T^{10} + 2856970349424 T^{11} + 3870433274354 T^{12} - 127396070447952 T^{13} - 1412601730375680 T^{14} - 7780535897090304 T^{15} - 41112146089827164 T^{16} - 754711982017759488 T^{17} - 13291169681104773120 T^{18} -$$$$11\!\cdots\!96$$$$T^{19} +$$$$34\!\cdots\!74$$$$T^{20} +$$$$24\!\cdots\!68$$$$T^{21} +$$$$34\!\cdots\!48$$$$T^{22} +$$$$21\!\cdots\!16$$$$T^{23} -$$$$47\!\cdots\!75$$$$T^{24} -$$$$24\!\cdots\!76$$$$T^{25} -$$$$23\!\cdots\!72$$$$T^{26} -$$$$90\!\cdots\!32$$$$T^{27} +$$$$55\!\cdots\!22$$$$T^{28} +$$$$10\!\cdots\!12$$$$T^{29} +$$$$75\!\cdots\!88$$$$T^{30} +$$$$30\!\cdots\!64$$$$T^{31} +$$$$61\!\cdots\!21$$$$T^{32}$$)