# Properties

 Label 75.4 Level 75 Weight 4 Dimension 394 Nonzero newspaces 6 Newform subspaces 17 Sturm bound 1600 Trace bound 1

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## Defining parameters

 Level: $$N$$ = $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$6$$ Newform subspaces: $$17$$ Sturm bound: $$1600$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(75))$$.

Total New Old
Modular forms 656 434 222
Cusp forms 544 394 150
Eisenstein series 112 40 72

## Trace form

 $$394q - 8q^{2} + 2q^{3} + 28q^{4} - 6q^{5} - 18q^{6} + 20q^{7} + 72q^{8} + 26q^{9} + O(q^{10})$$ $$394q - 8q^{2} + 2q^{3} + 28q^{4} - 6q^{5} - 18q^{6} + 20q^{7} + 72q^{8} + 26q^{9} + 76q^{10} + 112q^{11} - 226q^{12} - 348q^{13} - 528q^{14} - 184q^{15} - 132q^{16} + 560q^{17} + 578q^{18} + 732q^{19} + 156q^{20} + 918q^{21} - 420q^{22} - 304q^{23} - 864q^{24} - 1546q^{25} - 1344q^{26} - 1414q^{27} - 2852q^{28} - 1736q^{29} - 886q^{30} - 580q^{31} + 2008q^{32} + 974q^{33} + 4416q^{34} + 2240q^{35} + 1886q^{36} + 3570q^{37} + 5764q^{38} + 3678q^{39} + 4232q^{40} + 1648q^{41} + 114q^{42} - 1484q^{43} - 4116q^{44} - 2456q^{45} - 5260q^{46} - 3904q^{47} - 4640q^{48} - 4710q^{49} - 8224q^{50} - 1960q^{51} - 7888q^{52} - 3554q^{53} - 3348q^{54} - 1516q^{55} - 840q^{56} - 1442q^{57} + 3088q^{58} + 2288q^{59} + 74q^{60} + 1052q^{61} + 10076q^{62} - 374q^{63} + 7192q^{64} + 1222q^{65} - 762q^{66} - 268q^{67} - 304q^{68} + 2782q^{69} + 1540q^{70} + 1376q^{71} + 13758q^{72} + 5940q^{73} + 6752q^{74} + 9396q^{75} + 8808q^{76} + 3456q^{77} + 8380q^{78} + 3020q^{79} - 1004q^{80} - 286q^{81} + 8716q^{82} + 11104q^{83} + 4358q^{84} + 12538q^{85} + 1192q^{86} + 1962q^{87} + 3532q^{88} + 3402q^{89} - 10894q^{90} - 7540q^{91} - 10712q^{92} - 13418q^{93} - 28932q^{94} - 15944q^{95} - 15150q^{96} - 31028q^{97} - 27968q^{98} - 2016q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(75))$$

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
75.4.a $$\chi_{75}(1, \cdot)$$ 75.4.a.a 1 1
75.4.a.b 1
75.4.a.c 2
75.4.a.d 2
75.4.a.e 2
75.4.a.f 2
75.4.b $$\chi_{75}(49, \cdot)$$ 75.4.b.a 2 1
75.4.b.b 2
75.4.b.c 4
75.4.e $$\chi_{75}(32, \cdot)$$ 75.4.e.a 4 2
75.4.e.b 4
75.4.e.c 8
75.4.e.d 16
75.4.g $$\chi_{75}(16, \cdot)$$ 75.4.g.a 28 4
75.4.g.b 28
75.4.i $$\chi_{75}(4, \cdot)$$ 75.4.i.a 64 4
75.4.l $$\chi_{75}(2, \cdot)$$ 75.4.l.a 224 8

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(75))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(75)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 + 3 T + 8 T^{2}$$)($$1 + T + 8 T^{2}$$)($$1 + 3 T + 8 T^{2} + 24 T^{3} + 64 T^{4}$$)($$1 + 2 T - 2 T^{2} + 16 T^{3} + 64 T^{4}$$)($$1 - 2 T - 2 T^{2} - 16 T^{3} + 64 T^{4}$$)($$1 - 3 T + 8 T^{2} - 24 T^{3} + 64 T^{4}$$)($$1 - 7 T^{2} + 64 T^{4}$$)($$1 - 15 T^{2} + 64 T^{4}$$)($$1 + 8 T^{2} + 68 T^{4} + 512 T^{6} + 4096 T^{8}$$)($$( 1 - 4 T + 8 T^{2} )^{2}( 1 + 4 T + 8 T^{2} )^{2}$$)($$1 - 7 T^{4} + 4096 T^{8}$$)($$1 - 79 T^{4} + 2496 T^{8} - 323584 T^{12} + 16777216 T^{16}$$)($$( 1 - 59 T^{4} + 6756 T^{8} - 241664 T^{12} + 16777216 T^{16} )^{2}$$)
$3$ ($$1 - 3 T$$)($$1 + 3 T$$)($$( 1 + 3 T )^{2}$$)($$( 1 - 3 T )^{2}$$)($$( 1 + 3 T )^{2}$$)($$( 1 - 3 T )^{2}$$)($$1 + 9 T^{2}$$)($$1 + 9 T^{2}$$)($$( 1 + 9 T^{2} )^{2}$$)($$1 + 729 T^{4}$$)($$1 + 729 T^{4}$$)($$1 - 6 T + 18 T^{2} + 198 T^{3} - 1422 T^{4} + 5346 T^{5} + 13122 T^{6} - 118098 T^{7} + 531441 T^{8}$$)($$1 - 1719 T^{4} + 1614816 T^{8} - 913547079 T^{12} + 282429536481 T^{16}$$)
$5$ 1
$7$ ($$1 + 20 T + 343 T^{2}$$)($$1 - 24 T + 343 T^{2}$$)($$1 + 6 T + 326 T^{2} + 2058 T^{3} + 117649 T^{4}$$)($$1 - 26 T + 551 T^{2} - 8918 T^{3} + 117649 T^{4}$$)($$1 + 26 T + 551 T^{2} + 8918 T^{3} + 117649 T^{4}$$)($$1 - 6 T + 326 T^{2} - 2058 T^{3} + 117649 T^{4}$$)($$1 - 286 T^{2} + 117649 T^{4}$$)($$1 - 110 T^{2} + 117649 T^{4}$$)($$1 - 426 T^{2} + 75163 T^{4} - 50118474 T^{6} + 13841287201 T^{8}$$)($$1 - 153502 T^{4} + 13841287201 T^{8}$$)($$( 1 + 117649 T^{4} )^{2}$$)($$( 1 - 8 T + 32 T^{2} - 744 T^{3} - 45202 T^{4} - 255192 T^{5} + 3764768 T^{6} - 322828856 T^{7} + 13841287201 T^{8} )^{2}$$)($$( 1 - 30359 T^{4} - 16878186384 T^{8} - 420207638135159 T^{12} +$$$$19\!\cdots\!01$$$$T^{16} )^{2}$$)
$11$ ($$1 + 24 T + 1331 T^{2}$$)($$1 - 52 T + 1331 T^{2}$$)($$1 + 42 T + 2734 T^{2} + 55902 T^{3} + 1771561 T^{4}$$)($$1 - 28 T + 2554 T^{2} - 37268 T^{3} + 1771561 T^{4}$$)($$1 - 28 T + 2554 T^{2} - 37268 T^{3} + 1771561 T^{4}$$)($$1 + 42 T + 2734 T^{2} + 55902 T^{3} + 1771561 T^{4}$$)($$( 1 + 24 T + 1331 T^{2} )^{2}$$)($$( 1 - 52 T + 1331 T^{2} )^{2}$$)($$( 1 - 28 T + 2554 T^{2} - 37268 T^{3} + 1771561 T^{4} )^{2}$$)($$( 1 - 1331 T^{2} )^{4}$$)($$( 1 - 1331 T^{2} )^{4}$$)($$( 1 - 3334 T^{2} + 6292986 T^{4} - 5906384374 T^{6} + 3138428376721 T^{8} )^{2}$$)($$( 1 - 1409 T^{2} + 531636 T^{4} - 2496129449 T^{6} + 3138428376721 T^{8} )^{4}$$)
$13$ ($$1 + 74 T + 2197 T^{2}$$)($$1 + 22 T + 2197 T^{2}$$)($$1 + 78 T + 5546 T^{2} + 171366 T^{3} + 4826809 T^{4}$$)($$1 - 18 T - 389 T^{2} - 39546 T^{3} + 4826809 T^{4}$$)($$1 + 18 T - 389 T^{2} + 39546 T^{3} + 4826809 T^{4}$$)($$1 - 78 T + 5546 T^{2} - 171366 T^{3} + 4826809 T^{4}$$)($$1 + 1082 T^{2} + 4826809 T^{4}$$)($$1 - 3910 T^{2} + 4826809 T^{4}$$)($$1 + 1102 T^{2} + 8381283 T^{4} + 5319143518 T^{6} + 23298085122481 T^{8}$$)($$1 - 9397582 T^{4} + 23298085122481 T^{8}$$)($$( 1 + 4826809 T^{4} )^{2}$$)($$( 1 + 34 T + 578 T^{2} + 77418 T^{3} + 10363058 T^{4} + 170087346 T^{5} + 2789895602 T^{6} + 360552978682 T^{7} + 23298085122481 T^{8} )^{2}$$)($$( 1 - 7771199 T^{4} + 29100532423056 T^{8} -$$$$18\!\cdots\!19$$$$T^{12} +$$$$54\!\cdots\!61$$$$T^{16} )^{2}$$)
$17$ ($$1 + 54 T + 4913 T^{2}$$)($$1 - 14 T + 4913 T^{2}$$)($$1 + 102 T + 11402 T^{2} + 501126 T^{3} + 24137569 T^{4}$$)($$1 + 68 T + 3382 T^{2} + 334084 T^{3} + 24137569 T^{4}$$)($$1 - 68 T + 3382 T^{2} - 334084 T^{3} + 24137569 T^{4}$$)($$1 - 102 T + 11402 T^{2} - 501126 T^{3} + 24137569 T^{4}$$)($$1 - 6910 T^{2} + 24137569 T^{4}$$)($$1 - 9630 T^{2} + 24137569 T^{4}$$)($$1 - 2140 T^{2} + 14277638 T^{4} - 51654397660 T^{6} + 582622237229761 T^{8}$$)($$( 1 + 24137569 T^{4} )^{2}$$)($$1 + 39972098 T^{4} + 582622237229761 T^{8}$$)($$1 - 10588864 T^{4} - 447530075229954 T^{8} -$$$$61\!\cdots\!04$$$$T^{12} +$$$$33\!\cdots\!21$$$$T^{16}$$)($$( 1 - 46422959 T^{4} + 1695160893047136 T^{8} -$$$$27\!\cdots\!99$$$$T^{12} +$$$$33\!\cdots\!21$$$$T^{16} )^{2}$$)
$19$ ($$1 + 124 T + 6859 T^{2}$$)($$1 + 20 T + 6859 T^{2}$$)($$1 - 56 T + 8598 T^{2} - 384104 T^{3} + 47045881 T^{4}$$)($$1 - 6 T + 13423 T^{2} - 41154 T^{3} + 47045881 T^{4}$$)($$1 - 6 T + 13423 T^{2} - 41154 T^{3} + 47045881 T^{4}$$)($$1 - 56 T + 8598 T^{2} - 384104 T^{3} + 47045881 T^{4}$$)($$( 1 - 124 T + 6859 T^{2} )^{2}$$)($$( 1 - 20 T + 6859 T^{2} )^{2}$$)($$( 1 + 6 T + 13423 T^{2} + 41154 T^{3} + 47045881 T^{4} )^{2}$$)($$( 1 - 10582 T^{2} + 47045881 T^{4} )^{2}$$)($$( 1 + 13178 T^{2} + 47045881 T^{4} )^{2}$$)($$( 1 - 24664 T^{2} + 245125086 T^{4} - 1160339608984 T^{6} + 2213314919066161 T^{8} )^{2}$$)($$( 1 - 18434 T^{2} + 161185251 T^{4} - 867243770354 T^{6} + 2213314919066161 T^{8} )^{4}$$)
$23$ ($$1 - 120 T + 12167 T^{2}$$)($$1 - 168 T + 12167 T^{2}$$)($$1 - 48 T + 24254 T^{2} - 584016 T^{3} + 148035889 T^{4}$$)($$1 - 132 T + 25954 T^{2} - 1606044 T^{3} + 148035889 T^{4}$$)($$1 + 132 T + 25954 T^{2} + 1606044 T^{3} + 148035889 T^{4}$$)($$1 + 48 T + 24254 T^{2} + 584016 T^{3} + 148035889 T^{4}$$)($$1 - 9934 T^{2} + 148035889 T^{4}$$)($$1 + 3890 T^{2} + 148035889 T^{4}$$)($$1 - 34484 T^{2} + 545686278 T^{4} - 5104869596276 T^{6} + 21914624432020321 T^{8}$$)($$( 1 + 148035889 T^{4} )^{2}$$)($$1 - 81332062 T^{4} + 21914624432020321 T^{8}$$)($$1 + 232150736 T^{4} + 54876630002867166 T^{8} +$$$$50\!\cdots\!56$$$$T^{12} +$$$$48\!\cdots\!41$$$$T^{16}$$)($$( 1 + 280972516 T^{4} + 61096766856220806 T^{8} +$$$$61\!\cdots\!36$$$$T^{12} +$$$$48\!\cdots\!41$$$$T^{16} )^{2}$$)
$29$ ($$1 + 78 T + 24389 T^{2}$$)($$1 - 230 T + 24389 T^{2}$$)($$1 + 318 T + 55978 T^{2} + 7755702 T^{3} + 594823321 T^{4}$$)($$1 - 92 T + 35998 T^{2} - 2243788 T^{3} + 594823321 T^{4}$$)($$1 - 92 T + 35998 T^{2} - 2243788 T^{3} + 594823321 T^{4}$$)($$1 + 318 T + 55978 T^{2} + 7755702 T^{3} + 594823321 T^{4}$$)($$( 1 - 78 T + 24389 T^{2} )^{2}$$)($$( 1 + 230 T + 24389 T^{2} )^{2}$$)($$( 1 + 92 T + 35998 T^{2} + 2243788 T^{3} + 594823321 T^{4} )^{2}$$)($$( 1 + 24389 T^{2} )^{4}$$)($$( 1 + 24389 T^{2} )^{4}$$)($$( 1 + 79166 T^{2} + 2712313506 T^{4} + 47089783030286 T^{6} + 353814783205469041 T^{8} )^{2}$$)($$( 1 + 24116 T^{2} + 1313787606 T^{4} + 14344759209236 T^{6} + 353814783205469041 T^{8} )^{4}$$)
$31$ ($$1 - 200 T + 29791 T^{2}$$)($$1 + 288 T + 29791 T^{2}$$)($$1 - 52 T + 58782 T^{2} - 1549132 T^{3} + 887503681 T^{4}$$)($$1 - 122 T + 48407 T^{2} - 3634502 T^{3} + 887503681 T^{4}$$)($$1 - 122 T + 48407 T^{2} - 3634502 T^{3} + 887503681 T^{4}$$)($$1 - 52 T + 58782 T^{2} - 1549132 T^{3} + 887503681 T^{4}$$)($$( 1 - 200 T + 29791 T^{2} )^{2}$$)($$( 1 + 288 T + 29791 T^{2} )^{2}$$)($$( 1 - 122 T + 48407 T^{2} - 3634502 T^{3} + 887503681 T^{4} )^{2}$$)($$( 1 + 308 T + 29791 T^{2} )^{4}$$)($$( 1 - 232 T + 29791 T^{2} )^{4}$$)($$( 1 - 154 T + 36486 T^{2} - 4587814 T^{3} + 887503681 T^{4} )^{4}$$)($$( 1 + 151 T + 62976 T^{2} + 4498441 T^{3} + 887503681 T^{4} )^{8}$$)
$37$ ($$1 - 70 T + 50653 T^{2}$$)($$1 - 34 T + 50653 T^{2}$$)($$1 + 306 T + 94826 T^{2} + 15499818 T^{3} + 2565726409 T^{4}$$)($$1 + 284 T + 110526 T^{2} + 14385452 T^{3} + 2565726409 T^{4}$$)($$1 - 284 T + 110526 T^{2} - 14385452 T^{3} + 2565726409 T^{4}$$)($$1 - 306 T + 94826 T^{2} - 15499818 T^{3} + 2565726409 T^{4}$$)($$1 - 96406 T^{2} + 2565726409 T^{4}$$)($$1 - 100150 T^{2} + 2565726409 T^{4}$$)($$1 - 140396 T^{2} + 9176512758 T^{4} - 360217724917964 T^{6} + 6582952005840035281 T^{8}$$)($$1 + 2826257618 T^{4} + 6582952005840035281 T^{8}$$)($$( 1 + 2565726409 T^{4} )^{2}$$)($$( 1 - 578 T + 167042 T^{2} - 53079474 T^{3} + 15170807378 T^{4} - 2688634596522 T^{5} + 428584070812178 T^{6} - 75117885601554506 T^{7} + 6582952005840035281 T^{8} )^{2}$$)($$( 1 + 4470281476 T^{4} + 10032918734992624806 T^{8} +$$$$29\!\cdots\!56$$$$T^{12} +$$$$43\!\cdots\!61$$$$T^{16} )^{2}$$)
$41$ ($$1 - 330 T + 68921 T^{2}$$)($$1 - 122 T + 68921 T^{2}$$)($$1 + 408 T + 177982 T^{2} + 28119768 T^{3} + 4750104241 T^{4}$$)($$1 - 392 T + 124882 T^{2} - 27017032 T^{3} + 4750104241 T^{4}$$)($$1 - 392 T + 124882 T^{2} - 27017032 T^{3} + 4750104241 T^{4}$$)($$1 + 408 T + 177982 T^{2} + 28119768 T^{3} + 4750104241 T^{4}$$)($$( 1 - 330 T + 68921 T^{2} )^{2}$$)($$( 1 - 122 T + 68921 T^{2} )^{2}$$)($$( 1 - 392 T + 124882 T^{2} - 27017032 T^{3} + 4750104241 T^{4} )^{2}$$)($$( 1 - 68921 T^{2} )^{4}$$)($$( 1 - 68921 T^{2} )^{4}$$)($$( 1 - 115444 T^{2} + 7614039366 T^{4} - 548371033998004 T^{6} + 22563490300366186081 T^{8} )^{2}$$)($$( 1 - 271769 T^{2} + 27961298016 T^{4} - 1290931079472329 T^{6} + 22563490300366186081 T^{8} )^{4}$$)
$43$ ($$1 + 92 T + 79507 T^{2}$$)($$1 - 188 T + 79507 T^{2}$$)($$1 + 120 T + 68150 T^{2} + 9540840 T^{3} + 6321363049 T^{4}$$)($$1 - 690 T + 277735 T^{2} - 54859830 T^{3} + 6321363049 T^{4}$$)($$1 + 690 T + 277735 T^{2} + 54859830 T^{3} + 6321363049 T^{4}$$)($$1 - 120 T + 68150 T^{2} - 9540840 T^{3} + 6321363049 T^{4}$$)($$1 - 150550 T^{2} + 6321363049 T^{4}$$)($$1 - 123670 T^{2} + 6321363049 T^{4}$$)($$1 - 79370 T^{2} + 14072890923 T^{4} - 501726585199130 T^{6} + 39959630797262576401 T^{8}$$)($$1 - 235885102 T^{4} + 39959630797262576401 T^{8}$$)($$( 1 + 6321363049 T^{4} )^{2}$$)($$( 1 + 274 T + 37538 T^{2} + 17976318 T^{3} + 8415346898 T^{4} + 1429243115226 T^{5} + 237291326133362 T^{6} + 137710375670694982 T^{7} + 39959630797262576401 T^{8} )^{2}$$)($$( 1 + 12685147801 T^{4} +$$$$10\!\cdots\!96$$$$T^{8} +$$$$50\!\cdots\!01$$$$T^{12} +$$$$15\!\cdots\!01$$$$T^{16} )^{2}$$)
$47$ ($$1 - 24 T + 103823 T^{2}$$)($$1 + 256 T + 103823 T^{2}$$)($$1 + 180 T + 128990 T^{2} + 18688140 T^{3} + 10779215329 T^{4}$$)($$1 + 620 T + 284290 T^{2} + 64370260 T^{3} + 10779215329 T^{4}$$)($$1 - 620 T + 284290 T^{2} - 64370260 T^{3} + 10779215329 T^{4}$$)($$1 - 180 T + 128990 T^{2} - 18688140 T^{3} + 10779215329 T^{4}$$)($$1 - 207070 T^{2} + 10779215329 T^{4}$$)($$1 - 142110 T^{2} + 10779215329 T^{4}$$)($$1 - 184180 T^{2} + 22560112358 T^{4} - 1985315879295220 T^{6} +$$$$11\!\cdots\!41$$$$T^{8}$$)($$( 1 + 10779215329 T^{4} )^{2}$$)($$1 - 13452309502 T^{4} +$$$$11\!\cdots\!41$$$$T^{8}$$)($$1 + 29984206736 T^{4} +$$$$42\!\cdots\!06$$$$T^{8} +$$$$34\!\cdots\!76$$$$T^{12} +$$$$13\!\cdots\!81$$$$T^{16}$$)($$( 1 + 4831777156 T^{4} - 33598912467430419834 T^{8} +$$$$56\!\cdots\!96$$$$T^{12} +$$$$13\!\cdots\!81$$$$T^{16} )^{2}$$)
$53$ ($$1 + 450 T + 148877 T^{2}$$)($$1 - 338 T + 148877 T^{2}$$)($$1 + 402 T + 269234 T^{2} + 59848554 T^{3} + 22164361129 T^{4}$$)($$1 + 848 T + 462634 T^{2} + 126247696 T^{3} + 22164361129 T^{4}$$)($$1 - 848 T + 462634 T^{2} - 126247696 T^{3} + 22164361129 T^{4}$$)($$1 - 402 T + 269234 T^{2} - 59848554 T^{3} + 22164361129 T^{4}$$)($$1 - 95254 T^{2} + 22164361129 T^{4}$$)($$1 - 183510 T^{2} + 22164361129 T^{4}$$)($$1 - 206164 T^{2} + 44242847798 T^{4} - 4569493347799156 T^{6} +$$$$49\!\cdots\!41$$$$T^{8}$$)($$( 1 + 22164361129 T^{4} )^{2}$$)($$1 - 36467062702 T^{4} +$$$$49\!\cdots\!41$$$$T^{8}$$)($$1 - 24371904064 T^{4} +$$$$31\!\cdots\!06$$$$T^{8} -$$$$11\!\cdots\!24$$$$T^{12} +$$$$24\!\cdots\!81$$$$T^{16}$$)($$( 1 - 45122118044 T^{4} +$$$$13\!\cdots\!66$$$$T^{8} -$$$$22\!\cdots\!04$$$$T^{12} +$$$$24\!\cdots\!81$$$$T^{16} )^{2}$$)
$59$ ($$1 - 24 T + 205379 T^{2}$$)($$1 - 100 T + 205379 T^{2}$$)($$1 + 186 T + 419038 T^{2} + 38200494 T^{3} + 42180533641 T^{4}$$)($$1 - 124 T + 336778 T^{2} - 25466996 T^{3} + 42180533641 T^{4}$$)($$1 - 124 T + 336778 T^{2} - 25466996 T^{3} + 42180533641 T^{4}$$)($$1 + 186 T + 419038 T^{2} + 38200494 T^{3} + 42180533641 T^{4}$$)($$( 1 + 24 T + 205379 T^{2} )^{2}$$)($$( 1 + 100 T + 205379 T^{2} )^{2}$$)($$( 1 + 124 T + 336778 T^{2} + 25466996 T^{3} + 42180533641 T^{4} )^{2}$$)($$( 1 + 205379 T^{2} )^{4}$$)($$( 1 + 205379 T^{2} )^{4}$$)($$( 1 + 112106 T^{2} + 56049165066 T^{4} + 4728690904357946 T^{6} +$$$$17\!\cdots\!81$$$$T^{8} )^{2}$$)($$( 1 + 654656 T^{2} + 191262585966 T^{4} + 27613739431282496 T^{6} +$$$$17\!\cdots\!81$$$$T^{8} )^{4}$$)
$61$ ($$1 + 322 T + 226981 T^{2}$$)($$1 - 742 T + 226981 T^{2}$$)($$1 - 340 T + 388398 T^{2} - 77173540 T^{3} + 51520374361 T^{4}$$)($$1 - 750 T + 535003 T^{2} - 170235750 T^{3} + 51520374361 T^{4}$$)($$1 - 750 T + 535003 T^{2} - 170235750 T^{3} + 51520374361 T^{4}$$)($$1 - 340 T + 388398 T^{2} - 77173540 T^{3} + 51520374361 T^{4}$$)($$( 1 + 322 T + 226981 T^{2} )^{2}$$)($$( 1 - 742 T + 226981 T^{2} )^{2}$$)($$( 1 - 750 T + 535003 T^{2} - 170235750 T^{3} + 51520374361 T^{4} )^{2}$$)($$( 1 - 182 T + 226981 T^{2} )^{4}$$)($$( 1 + 358 T + 226981 T^{2} )^{4}$$)($$( 1 - 2 T + 226981 T^{2} )^{8}$$)($$( 1 + 191 T + 350076 T^{2} + 43353371 T^{3} + 51520374361 T^{4} )^{8}$$)
$67$ ($$1 - 196 T + 300763 T^{2}$$)($$1 - 84 T + 300763 T^{2}$$)($$1 + 732 T + 698582 T^{2} + 220158516 T^{3} + 90458382169 T^{4}$$)($$1 + 358 T + 443567 T^{2} + 107673154 T^{3} + 90458382169 T^{4}$$)($$1 - 358 T + 443567 T^{2} - 107673154 T^{3} + 90458382169 T^{4}$$)($$1 - 732 T + 698582 T^{2} - 220158516 T^{3} + 90458382169 T^{4}$$)($$1 - 563110 T^{2} + 90458382169 T^{4}$$)($$1 - 594470 T^{2} + 90458382169 T^{4}$$)($$1 - 758970 T^{2} + 300574469563 T^{4} - 68655198314805930 T^{6} +$$$$81\!\cdots\!61$$$$T^{8}$$)($$1 - 151031344462 T^{4} +$$$$81\!\cdots\!61$$$$T^{8}$$)($$( 1 + 90458382169 T^{4} )^{2}$$)($$( 1 + 202 T + 20402 T^{2} + 3297246 T^{3} - 80373232942 T^{4} + 991689598698 T^{5} + 1845531913011938 T^{6} + 5495719948051579294 T^{7} +$$$$81\!\cdots\!61$$$$T^{8} )^{2}$$)($$( 1 - 145878207314 T^{4} +$$$$19\!\cdots\!71$$$$T^{8} -$$$$11\!\cdots\!54$$$$T^{12} +$$$$66\!\cdots\!21$$$$T^{16} )^{2}$$)
$71$ ($$1 + 288 T + 357911 T^{2}$$)($$1 + 328 T + 357911 T^{2}$$)($$1 + 36 T + 384046 T^{2} + 12884796 T^{3} + 128100283921 T^{4}$$)($$1 - 824 T + 877966 T^{2} - 294918664 T^{3} + 128100283921 T^{4}$$)($$1 - 824 T + 877966 T^{2} - 294918664 T^{3} + 128100283921 T^{4}$$)($$1 + 36 T + 384046 T^{2} + 12884796 T^{3} + 128100283921 T^{4}$$)($$( 1 + 288 T + 357911 T^{2} )^{2}$$)($$( 1 + 328 T + 357911 T^{2} )^{2}$$)($$( 1 - 824 T + 877966 T^{2} - 294918664 T^{3} + 128100283921 T^{4} )^{2}$$)($$( 1 - 357911 T^{2} )^{4}$$)($$( 1 - 357911 T^{2} )^{4}$$)($$( 1 - 863584 T^{2} + 377860054206 T^{4} - 110625355589632864 T^{6} +$$$$16\!\cdots\!41$$$$T^{8} )^{2}$$)($$( 1 + 266116 T^{2} + 166761568806 T^{4} + 34089535155920836 T^{6} +$$$$16\!\cdots\!41$$$$T^{8} )^{4}$$)
$73$ ($$1 - 430 T + 389017 T^{2}$$)($$1 - 38 T + 389017 T^{2}$$)($$1 + 1332 T + 1102034 T^{2} + 518170644 T^{3} + 151334226289 T^{4}$$)($$1 + 108 T + 779734 T^{2} + 42013836 T^{3} + 151334226289 T^{4}$$)($$1 - 108 T + 779734 T^{2} - 42013836 T^{3} + 151334226289 T^{4}$$)($$1 - 1332 T + 1102034 T^{2} - 518170644 T^{3} + 151334226289 T^{4}$$)($$1 - 593134 T^{2} + 151334226289 T^{4}$$)($$1 - 776590 T^{2} + 151334226289 T^{4}$$)($$1 - 1547804 T^{2} + 901578574758 T^{4} - 234235720787019356 T^{6} +$$$$22\!\cdots\!21$$$$T^{8}$$)($$1 + 104459767778 T^{4} +$$$$22\!\cdots\!21$$$$T^{8}$$)($$( 1 + 151334226289 T^{4} )^{2}$$)($$( 1 - 1256 T + 788768 T^{2} - 733362072 T^{3} + 643873740638 T^{4} - 285290313163224 T^{5} + 119367595001521952 T^{6} - 73942712905584498728 T^{7} +$$$$22\!\cdots\!21$$$$T^{8} )^{2}$$)($$( 1 - 117295922639 T^{4} -$$$$55\!\cdots\!84$$$$T^{8} -$$$$26\!\cdots\!19$$$$T^{12} +$$$$52\!\cdots\!41$$$$T^{16} )^{2}$$)
$79$ ($$1 + 520 T + 493039 T^{2}$$)($$1 + 240 T + 493039 T^{2}$$)($$1 - 380 T + 99678 T^{2} - 187354820 T^{3} + 243087455521 T^{4}$$)($$1 + 880 T + 693278 T^{2} + 433874320 T^{3} + 243087455521 T^{4}$$)($$1 + 880 T + 693278 T^{2} + 433874320 T^{3} + 243087455521 T^{4}$$)($$1 - 380 T + 99678 T^{2} - 187354820 T^{3} + 243087455521 T^{4}$$)($$( 1 - 520 T + 493039 T^{2} )^{2}$$)($$( 1 - 240 T + 493039 T^{2} )^{2}$$)($$( 1 - 880 T + 693278 T^{2} - 433874320 T^{3} + 243087455521 T^{4} )^{2}$$)($$( 1 - 204622 T^{2} + 243087455521 T^{4} )^{2}$$)($$( 1 - 893662 T^{2} + 243087455521 T^{4} )^{2}$$)($$( 1 - 1624084 T^{2} + 1116095863206 T^{4} - 394794447112367764 T^{6} +$$$$59\!\cdots\!41$$$$T^{8} )^{2}$$)($$( 1 - 1471304 T^{2} + 984238875246 T^{4} - 357655545657869384 T^{6} +$$$$59\!\cdots\!41$$$$T^{8} )^{4}$$)
$83$ ($$1 + 156 T + 571787 T^{2}$$)($$1 + 1212 T + 571787 T^{2}$$)($$1 - 984 T + 942182 T^{2} - 562638408 T^{3} + 326940373369 T^{4}$$)($$1 - 156 T + 449242 T^{2} - 89198772 T^{3} + 326940373369 T^{4}$$)($$1 + 156 T + 449242 T^{2} + 89198772 T^{3} + 326940373369 T^{4}$$)($$1 + 984 T + 942182 T^{2} + 562638408 T^{3} + 326940373369 T^{4}$$)($$1 - 1119238 T^{2} + 326940373369 T^{4}$$)($$1 + 325370 T^{2} + 326940373369 T^{4}$$)($$1 - 874148 T^{2} + 827869104438 T^{4} - 285794273499764612 T^{6} +$$$$10\!\cdots\!61$$$$T^{8}$$)($$( 1 + 326940373369 T^{4} )^{2}$$)($$1 - 433407300622 T^{4} +$$$$10\!\cdots\!61$$$$T^{8}$$)($$1 + 1005705244496 T^{4} +$$$$46\!\cdots\!26$$$$T^{8} +$$$$10\!\cdots\!56$$$$T^{12} +$$$$11\!\cdots\!21$$$$T^{16}$$)($$( 1 + 280364494441 T^{4} -$$$$71\!\cdots\!64$$$$T^{8} +$$$$29\!\cdots\!01$$$$T^{12} +$$$$11\!\cdots\!21$$$$T^{16} )^{2}$$)
$89$ ($$1 - 1026 T + 704969 T^{2}$$)($$1 - 330 T + 704969 T^{2}$$)($$1 - 1116 T + 1508758 T^{2} - 786745404 T^{3} + 496981290961 T^{4}$$)($$1 + 864 T + 1202578 T^{2} + 609093216 T^{3} + 496981290961 T^{4}$$)($$1 + 864 T + 1202578 T^{2} + 609093216 T^{3} + 496981290961 T^{4}$$)($$1 - 1116 T + 1508758 T^{2} - 786745404 T^{3} + 496981290961 T^{4}$$)($$( 1 + 1026 T + 704969 T^{2} )^{2}$$)($$( 1 + 330 T + 704969 T^{2} )^{2}$$)($$( 1 - 864 T + 1202578 T^{2} - 609093216 T^{3} + 496981290961 T^{4} )^{2}$$)($$( 1 + 704969 T^{2} )^{4}$$)($$( 1 + 704969 T^{2} )^{4}$$)($$( 1 - 806584 T^{2} + 1118488022286 T^{4} - 400857157588487224 T^{6} +$$$$24\!\cdots\!21$$$$T^{8} )^{2}$$)($$( 1 + 2449841 T^{2} + 2476709961936 T^{4} + 1217525142829187201 T^{6} +$$$$24\!\cdots\!21$$$$T^{8} )^{4}$$)
$97$ ($$1 - 286 T + 912673 T^{2}$$)($$1 + 866 T + 912673 T^{2}$$)($$1 - 768 T + 1382402 T^{2} - 700932864 T^{3} + 832972004929 T^{4}$$)($$( 1 - 521 T + 912673 T^{2} )^{2}$$)($$( 1 + 521 T + 912673 T^{2} )^{2}$$)($$1 + 768 T + 1382402 T^{2} + 700932864 T^{3} + 832972004929 T^{4}$$)($$1 - 1743550 T^{2} + 832972004929 T^{4}$$)($$1 - 1075390 T^{2} + 832972004929 T^{4}$$)($$( 1 - 1553905 T^{2} + 832972004929 T^{4} )^{2}$$)($$1 - 1662757858942 T^{4} +$$$$69\!\cdots\!41$$$$T^{8}$$)($$( 1 + 832972004929 T^{4} )^{2}$$)($$( 1 + 952 T + 453152 T^{2} + 121725576 T^{3} - 583228842562 T^{4} + 111095646624648 T^{5} + 377462929977586208 T^{6} +$$$$72\!\cdots\!84$$$$T^{7} +$$$$69\!\cdots\!41$$$$T^{8} )^{2}$$)($$( 1 + 3241877334001 T^{4} +$$$$40\!\cdots\!76$$$$T^{8} +$$$$22\!\cdots\!41$$$$T^{12} +$$$$48\!\cdots\!81$$$$T^{16} )^{2}$$)
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