# Properties

 Label 6030.2.d Level 6030 Weight 2 Character orbit d Rep. character $$\chi_{6030}(2411,\cdot)$$ Character field $$\Q$$ Dimension 96 Newform subspaces 12 Sturm bound 2448 Trace bound 11

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$6030 = 2 \cdot 3^{2} \cdot 5 \cdot 67$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6030.d (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$201$$ Character field: $$\Q$$ Newform subspaces: $$12$$ Sturm bound: $$2448$$ Trace bound: $$11$$ Distinguishing $$T_p$$: $$7$$, $$11$$, $$41$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(6030, [\chi])$$.

Total New Old
Modular forms 1240 96 1144
Cusp forms 1208 96 1112
Eisenstein series 32 0 32

## Trace form

 $$96q + 96q^{4} + O(q^{10})$$ $$96q + 96q^{4} + 96q^{16} - 16q^{19} + 96q^{25} - 48q^{37} - 144q^{49} + 48q^{55} + 96q^{64} - 56q^{67} - 16q^{73} - 16q^{76} + 64q^{91} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(6030, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
6030.2.d.a $$2$$ $$48.150$$ $$\Q(\sqrt{-2})$$ None $$-2$$ $$0$$ $$2$$ $$0$$ $$q-q^{2}+q^{4}+q^{5}-q^{8}-q^{10}-6q^{11}+\cdots$$
6030.2.d.b $$2$$ $$48.150$$ $$\Q(\sqrt{-2})$$ None $$-2$$ $$0$$ $$2$$ $$0$$ $$q-q^{2}+q^{4}+q^{5}+2\beta q^{7}-q^{8}-q^{10}+\cdots$$
6030.2.d.c $$2$$ $$48.150$$ $$\Q(\sqrt{-2})$$ None $$-2$$ $$0$$ $$2$$ $$0$$ $$q-q^{2}+q^{4}+q^{5}-q^{8}-q^{10}+q^{16}+\cdots$$
6030.2.d.d $$2$$ $$48.150$$ $$\Q(\sqrt{-2})$$ None $$-2$$ $$0$$ $$2$$ $$0$$ $$q-q^{2}+q^{4}+q^{5}+\beta q^{7}-q^{8}-q^{10}+\cdots$$
6030.2.d.e $$2$$ $$48.150$$ $$\Q(\sqrt{-2})$$ None $$2$$ $$0$$ $$-2$$ $$0$$ $$q+q^{2}+q^{4}-q^{5}+\beta q^{7}+q^{8}-q^{10}+\cdots$$
6030.2.d.f $$2$$ $$48.150$$ $$\Q(\sqrt{-2})$$ None $$2$$ $$0$$ $$-2$$ $$0$$ $$q+q^{2}+q^{4}-q^{5}+q^{8}-q^{10}+q^{16}+\cdots$$
6030.2.d.g $$2$$ $$48.150$$ $$\Q(\sqrt{-2})$$ None $$2$$ $$0$$ $$-2$$ $$0$$ $$q+q^{2}+q^{4}-q^{5}+2\beta q^{7}+q^{8}-q^{10}+\cdots$$
6030.2.d.h $$2$$ $$48.150$$ $$\Q(\sqrt{-2})$$ None $$2$$ $$0$$ $$-2$$ $$0$$ $$q+q^{2}+q^{4}-q^{5}+q^{8}-q^{10}+6q^{11}+\cdots$$
6030.2.d.i $$16$$ $$48.150$$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$-16$$ $$0$$ $$16$$ $$0$$ $$q-q^{2}+q^{4}+q^{5}+\beta _{12}q^{7}-q^{8}-q^{10}+\cdots$$
6030.2.d.j $$16$$ $$48.150$$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$16$$ $$0$$ $$-16$$ $$0$$ $$q+q^{2}+q^{4}-q^{5}+\beta _{12}q^{7}+q^{8}-q^{10}+\cdots$$
6030.2.d.k $$24$$ $$48.150$$ None $$-24$$ $$0$$ $$-24$$ $$0$$
6030.2.d.l $$24$$ $$48.150$$ None $$24$$ $$0$$ $$24$$ $$0$$

## Decomposition of $$S_{2}^{\mathrm{old}}(6030, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(6030, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(201, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(402, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(603, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1005, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1206, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(2010, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(3015, [\chi])$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$( 1 + T )^{2}$$)($$( 1 + T )^{2}$$)($$( 1 + T )^{2}$$)($$( 1 + T )^{2}$$)($$( 1 - T )^{2}$$)($$( 1 - T )^{2}$$)($$( 1 - T )^{2}$$)($$( 1 - T )^{2}$$)($$( 1 + T )^{16}$$)($$( 1 - T )^{16}$$)
$3$ 1
$5$ ($$( 1 - T )^{2}$$)($$( 1 - T )^{2}$$)($$( 1 - T )^{2}$$)($$( 1 - T )^{2}$$)($$( 1 + T )^{2}$$)($$( 1 + T )^{2}$$)($$( 1 + T )^{2}$$)($$( 1 + T )^{2}$$)($$( 1 - T )^{16}$$)($$( 1 + T )^{16}$$)
$7$ ($$( 1 - 7 T^{2} )^{2}$$)($$1 - 6 T^{2} + 49 T^{4}$$)($$( 1 - 7 T^{2} )^{2}$$)($$1 - 12 T^{2} + 49 T^{4}$$)($$1 - 12 T^{2} + 49 T^{4}$$)($$( 1 - 7 T^{2} )^{2}$$)($$1 - 6 T^{2} + 49 T^{4}$$)($$( 1 - 7 T^{2} )^{2}$$)($$1 - 12 T^{2} + 65 T^{4} - 64 T^{6} + 70 T^{8} + 1000 T^{10} + 113791 T^{12} - 1570844 T^{14} + 15564210 T^{16} - 76971356 T^{18} + 273212191 T^{20} + 117649000 T^{22} + 403536070 T^{24} - 18078415936 T^{26} + 899683668065 T^{28} - 8138676874188 T^{30} + 33232930569601 T^{32}$$)($$1 - 12 T^{2} + 65 T^{4} - 64 T^{6} + 70 T^{8} + 1000 T^{10} + 113791 T^{12} - 1570844 T^{14} + 15564210 T^{16} - 76971356 T^{18} + 273212191 T^{20} + 117649000 T^{22} + 403536070 T^{24} - 18078415936 T^{26} + 899683668065 T^{28} - 8138676874188 T^{30} + 33232930569601 T^{32}$$)
$11$ ($$( 1 + 6 T + 11 T^{2} )^{2}$$)($$( 1 + 2 T + 11 T^{2} )^{2}$$)($$( 1 + 11 T^{2} )^{2}$$)($$( 1 - 4 T + 11 T^{2} )^{2}$$)($$( 1 + 4 T + 11 T^{2} )^{2}$$)($$( 1 + 11 T^{2} )^{2}$$)($$( 1 - 2 T + 11 T^{2} )^{2}$$)($$( 1 - 6 T + 11 T^{2} )^{2}$$)($$( 1 - 10 T + 89 T^{2} - 566 T^{3} + 3224 T^{4} - 15446 T^{5} + 67469 T^{6} - 258178 T^{7} + 910522 T^{8} - 2839958 T^{9} + 8163749 T^{10} - 20558626 T^{11} + 47202584 T^{12} - 91154866 T^{13} + 157668929 T^{14} - 194871710 T^{15} + 214358881 T^{16} )^{2}$$)($$( 1 + 10 T + 89 T^{2} + 566 T^{3} + 3224 T^{4} + 15446 T^{5} + 67469 T^{6} + 258178 T^{7} + 910522 T^{8} + 2839958 T^{9} + 8163749 T^{10} + 20558626 T^{11} + 47202584 T^{12} + 91154866 T^{13} + 157668929 T^{14} + 194871710 T^{15} + 214358881 T^{16} )^{2}$$)
$13$ ($$1 - 8 T^{2} + 169 T^{4}$$)($$1 - 24 T^{2} + 169 T^{4}$$)($$( 1 - 13 T^{2} )^{2}$$)($$1 + 6 T^{2} + 169 T^{4}$$)($$1 + 6 T^{2} + 169 T^{4}$$)($$( 1 - 13 T^{2} )^{2}$$)($$1 - 24 T^{2} + 169 T^{4}$$)($$1 - 8 T^{2} + 169 T^{4}$$)($$1 - 100 T^{2} + 5164 T^{4} - 182812 T^{6} + 4978036 T^{8} - 110574964 T^{10} + 2069794196 T^{12} - 33262092620 T^{14} + 463566233430 T^{16} - 5621293652780 T^{18} + 59115392031956 T^{20} - 533724231409876 T^{22} + 4060736895443956 T^{24} - 25202186611899388 T^{26} + 120311311572491884 T^{28} - 393737638569928900 T^{30} + 665416609183179841 T^{32}$$)($$1 - 100 T^{2} + 5164 T^{4} - 182812 T^{6} + 4978036 T^{8} - 110574964 T^{10} + 2069794196 T^{12} - 33262092620 T^{14} + 463566233430 T^{16} - 5621293652780 T^{18} + 59115392031956 T^{20} - 533724231409876 T^{22} + 4060736895443956 T^{24} - 25202186611899388 T^{26} + 120311311572491884 T^{28} - 393737638569928900 T^{30} + 665416609183179841 T^{32}$$)
$17$ ($$1 - 26 T^{2} + 289 T^{4}$$)($$1 - 26 T^{2} + 289 T^{4}$$)($$1 - 32 T^{2} + 289 T^{4}$$)($$( 1 - 6 T + 17 T^{2} )( 1 + 6 T + 17 T^{2} )$$)($$( 1 - 6 T + 17 T^{2} )( 1 + 6 T + 17 T^{2} )$$)($$1 - 32 T^{2} + 289 T^{4}$$)($$1 - 26 T^{2} + 289 T^{4}$$)($$1 - 26 T^{2} + 289 T^{4}$$)($$1 - 72 T^{2} + 3388 T^{4} - 119208 T^{6} + 3472552 T^{8} - 87055608 T^{10} + 1924187636 T^{12} - 38031267608 T^{14} + 679448411470 T^{16} - 10991036338712 T^{18} + 160710075546356 T^{20} - 2101310744936952 T^{22} + 24223680453259432 T^{24} - 240322600884724392 T^{26} + 1973924139734430268 T^{28} - 12123203512276866888 T^{30} + 48661191875666868481 T^{32}$$)($$1 - 72 T^{2} + 3388 T^{4} - 119208 T^{6} + 3472552 T^{8} - 87055608 T^{10} + 1924187636 T^{12} - 38031267608 T^{14} + 679448411470 T^{16} - 10991036338712 T^{18} + 160710075546356 T^{20} - 2101310744936952 T^{22} + 24223680453259432 T^{24} - 240322600884724392 T^{26} + 1973924139734430268 T^{28} - 12123203512276866888 T^{30} + 48661191875666868481 T^{32}$$)
$19$ ($$( 1 + 4 T + 19 T^{2} )^{2}$$)($$( 1 + 19 T^{2} )^{2}$$)($$( 1 - 2 T + 19 T^{2} )^{2}$$)($$( 1 + 19 T^{2} )^{2}$$)($$( 1 + 19 T^{2} )^{2}$$)($$( 1 - 2 T + 19 T^{2} )^{2}$$)($$( 1 + 19 T^{2} )^{2}$$)($$( 1 + 4 T + 19 T^{2} )^{2}$$)($$( 1 + 4 T + 52 T^{2} + 132 T^{3} + 1720 T^{4} + 5684 T^{5} + 50044 T^{6} + 126740 T^{7} + 947150 T^{8} + 2408060 T^{9} + 18065884 T^{10} + 38986556 T^{11} + 224152120 T^{12} + 326845068 T^{13} + 2446385812 T^{14} + 3575486956 T^{15} + 16983563041 T^{16} )^{2}$$)($$( 1 + 4 T + 52 T^{2} + 132 T^{3} + 1720 T^{4} + 5684 T^{5} + 50044 T^{6} + 126740 T^{7} + 947150 T^{8} + 2408060 T^{9} + 18065884 T^{10} + 38986556 T^{11} + 224152120 T^{12} + 326845068 T^{13} + 2446385812 T^{14} + 3575486956 T^{15} + 16983563041 T^{16} )^{2}$$)
$23$ ($$1 - 44 T^{2} + 529 T^{4}$$)($$1 + 4 T^{2} + 529 T^{4}$$)($$1 + 4 T^{2} + 529 T^{4}$$)($$1 - 14 T^{2} + 529 T^{4}$$)($$1 - 14 T^{2} + 529 T^{4}$$)($$1 + 4 T^{2} + 529 T^{4}$$)($$1 + 4 T^{2} + 529 T^{4}$$)($$1 - 44 T^{2} + 529 T^{4}$$)($$1 - 236 T^{2} + 26820 T^{4} - 1953524 T^{6} + 102662900 T^{8} - 4171202524 T^{10} + 137809451964 T^{12} - 3858496068548 T^{14} + 94388796186262 T^{16} - 2041144420261892 T^{18} + 38564734847057724 T^{20} - 617487673839383836 T^{22} + 8039632850804774900 T^{24} - 80927683892132449076 T^{26} +$$$$58\!\cdots\!20$$$$T^{28} -$$$$27\!\cdots\!24$$$$T^{30} +$$$$61\!\cdots\!61$$$$T^{32}$$)($$1 - 236 T^{2} + 26820 T^{4} - 1953524 T^{6} + 102662900 T^{8} - 4171202524 T^{10} + 137809451964 T^{12} - 3858496068548 T^{14} + 94388796186262 T^{16} - 2041144420261892 T^{18} + 38564734847057724 T^{20} - 617487673839383836 T^{22} + 8039632850804774900 T^{24} - 80927683892132449076 T^{26} +$$$$58\!\cdots\!20$$$$T^{28} -$$$$27\!\cdots\!24$$$$T^{30} +$$$$61\!\cdots\!61$$$$T^{32}$$)
$29$ ($$1 - 8 T^{2} + 841 T^{4}$$)($$1 - 8 T^{2} + 841 T^{4}$$)($$1 - 56 T^{2} + 841 T^{4}$$)($$1 - 50 T^{2} + 841 T^{4}$$)($$1 - 50 T^{2} + 841 T^{4}$$)($$1 - 56 T^{2} + 841 T^{4}$$)($$1 - 8 T^{2} + 841 T^{4}$$)($$1 - 8 T^{2} + 841 T^{4}$$)($$1 - 116 T^{2} + 5532 T^{4} - 153676 T^{6} + 3478164 T^{8} - 93695876 T^{10} + 2804533284 T^{12} - 62769871100 T^{14} + 1298249772566 T^{16} - 52789461595100 T^{18} + 1983593105640804 T^{20} - 55732492126324196 T^{22} + 1739939064690083604 T^{24} - 64652604784641688876 T^{26} +$$$$19\!\cdots\!12$$$$T^{28} -$$$$34\!\cdots\!96$$$$T^{30} +$$$$25\!\cdots\!21$$$$T^{32}$$)($$1 - 116 T^{2} + 5532 T^{4} - 153676 T^{6} + 3478164 T^{8} - 93695876 T^{10} + 2804533284 T^{12} - 62769871100 T^{14} + 1298249772566 T^{16} - 52789461595100 T^{18} + 1983593105640804 T^{20} - 55732492126324196 T^{22} + 1739939064690083604 T^{24} - 64652604784641688876 T^{26} +$$$$19\!\cdots\!12$$$$T^{28} -$$$$34\!\cdots\!96$$$$T^{30} +$$$$25\!\cdots\!21$$$$T^{32}$$)
$31$ ($$1 - 44 T^{2} + 961 T^{4}$$)($$1 - 12 T^{2} + 961 T^{4}$$)($$( 1 - 31 T^{2} )^{2}$$)($$1 - 54 T^{2} + 961 T^{4}$$)($$1 - 54 T^{2} + 961 T^{4}$$)($$( 1 - 31 T^{2} )^{2}$$)($$1 - 12 T^{2} + 961 T^{4}$$)($$1 - 44 T^{2} + 961 T^{4}$$)($$1 - 356 T^{2} + 61544 T^{4} - 6865052 T^{6} + 554409616 T^{8} - 34496199012 T^{10} + 1717973499704 T^{12} - 70142618146140 T^{14} + 2379955389522206 T^{16} - 67407056038440540 T^{18} + 1586584604420137784 T^{20} - 30615503603658563172 T^{22} +$$$$47\!\cdots\!56$$$$T^{24} -$$$$56\!\cdots\!52$$$$T^{26} +$$$$48\!\cdots\!84$$$$T^{28} -$$$$26\!\cdots\!76$$$$T^{30} +$$$$72\!\cdots\!81$$$$T^{32}$$)($$1 - 356 T^{2} + 61544 T^{4} - 6865052 T^{6} + 554409616 T^{8} - 34496199012 T^{10} + 1717973499704 T^{12} - 70142618146140 T^{14} + 2379955389522206 T^{16} - 67407056038440540 T^{18} + 1586584604420137784 T^{20} - 30615503603658563172 T^{22} +$$$$47\!\cdots\!56$$$$T^{24} -$$$$56\!\cdots\!52$$$$T^{26} +$$$$48\!\cdots\!84$$$$T^{28} -$$$$26\!\cdots\!76$$$$T^{30} +$$$$72\!\cdots\!81$$$$T^{32}$$)
$37$ ($$( 1 + 10 T + 37 T^{2} )^{2}$$)($$( 1 + 6 T + 37 T^{2} )^{2}$$)($$( 1 + 4 T + 37 T^{2} )^{2}$$)($$( 1 + 37 T^{2} )^{2}$$)($$( 1 + 37 T^{2} )^{2}$$)($$( 1 + 4 T + 37 T^{2} )^{2}$$)($$( 1 + 6 T + 37 T^{2} )^{2}$$)($$( 1 + 10 T + 37 T^{2} )^{2}$$)($$( 1 - 16 T + 269 T^{2} - 2480 T^{3} + 24724 T^{4} - 168876 T^{5} + 1334745 T^{6} - 7746124 T^{7} + 54540506 T^{8} - 286606588 T^{9} + 1827265905 T^{10} - 8554076028 T^{11} + 46336756564 T^{12} - 171973013360 T^{13} + 690180404021 T^{14} - 1518910034128 T^{15} + 3512479453921 T^{16} )^{2}$$)($$( 1 - 16 T + 269 T^{2} - 2480 T^{3} + 24724 T^{4} - 168876 T^{5} + 1334745 T^{6} - 7746124 T^{7} + 54540506 T^{8} - 286606588 T^{9} + 1827265905 T^{10} - 8554076028 T^{11} + 46336756564 T^{12} - 171973013360 T^{13} + 690180404021 T^{14} - 1518910034128 T^{15} + 3512479453921 T^{16} )^{2}$$)
$41$ ($$( 1 - 6 T + 41 T^{2} )^{2}$$)($$( 1 + 6 T + 41 T^{2} )^{2}$$)($$( 1 + 6 T + 41 T^{2} )^{2}$$)($$( 1 - 6 T + 41 T^{2} )^{2}$$)($$( 1 + 6 T + 41 T^{2} )^{2}$$)($$( 1 - 6 T + 41 T^{2} )^{2}$$)($$( 1 - 6 T + 41 T^{2} )^{2}$$)($$( 1 + 6 T + 41 T^{2} )^{2}$$)($$( 1 - 4 T + 88 T^{2} - 192 T^{3} + 3730 T^{4} - 12008 T^{5} + 169712 T^{6} - 787124 T^{7} + 7094090 T^{8} - 32272084 T^{9} + 285285872 T^{10} - 827603368 T^{11} + 10540088530 T^{12} - 22244390592 T^{13} + 418009173208 T^{14} - 779017095524 T^{15} + 7984925229121 T^{16} )^{2}$$)($$( 1 + 4 T + 88 T^{2} + 192 T^{3} + 3730 T^{4} + 12008 T^{5} + 169712 T^{6} + 787124 T^{7} + 7094090 T^{8} + 32272084 T^{9} + 285285872 T^{10} + 827603368 T^{11} + 10540088530 T^{12} + 22244390592 T^{13} + 418009173208 T^{14} + 779017095524 T^{15} + 7984925229121 T^{16} )^{2}$$)
$43$ ($$( 1 - 43 T^{2} )^{2}$$)($$1 - 54 T^{2} + 1849 T^{4}$$)($$( 1 - 10 T + 43 T^{2} )( 1 + 10 T + 43 T^{2} )$$)($$1 - 54 T^{2} + 1849 T^{4}$$)($$1 - 54 T^{2} + 1849 T^{4}$$)($$( 1 - 10 T + 43 T^{2} )( 1 + 10 T + 43 T^{2} )$$)($$1 - 54 T^{2} + 1849 T^{4}$$)($$( 1 - 43 T^{2} )^{2}$$)($$1 - 88 T^{2} + 9648 T^{4} - 590152 T^{6} + 36665500 T^{8} - 1823293528 T^{10} + 84892264016 T^{12} - 3881157187976 T^{14} + 160891788941702 T^{16} - 7176259640567624 T^{18} + 290229757110164816 T^{20} - 11525700335380046872 T^{22} +$$$$42\!\cdots\!00$$$$T^{24} -$$$$12\!\cdots\!48$$$$T^{26} +$$$$38\!\cdots\!48$$$$T^{28} -$$$$65\!\cdots\!12$$$$T^{30} +$$$$13\!\cdots\!01$$$$T^{32}$$)($$1 - 88 T^{2} + 9648 T^{4} - 590152 T^{6} + 36665500 T^{8} - 1823293528 T^{10} + 84892264016 T^{12} - 3881157187976 T^{14} + 160891788941702 T^{16} - 7176259640567624 T^{18} + 290229757110164816 T^{20} - 11525700335380046872 T^{22} +$$$$42\!\cdots\!00$$$$T^{24} -$$$$12\!\cdots\!48$$$$T^{26} +$$$$38\!\cdots\!48$$$$T^{28} -$$$$65\!\cdots\!12$$$$T^{30} +$$$$13\!\cdots\!01$$$$T^{32}$$)
$47$ ($$1 - 92 T^{2} + 2209 T^{4}$$)($$1 + 4 T^{2} + 2209 T^{4}$$)($$1 - 44 T^{2} + 2209 T^{4}$$)($$1 - 86 T^{2} + 2209 T^{4}$$)($$1 - 86 T^{2} + 2209 T^{4}$$)($$1 - 44 T^{2} + 2209 T^{4}$$)($$1 + 4 T^{2} + 2209 T^{4}$$)($$1 - 92 T^{2} + 2209 T^{4}$$)($$1 - 236 T^{2} + 31140 T^{4} - 3102004 T^{6} + 253966836 T^{8} - 17838006620 T^{10} + 1103973200988 T^{12} - 60912719078276 T^{14} + 3016913541521174 T^{16} - 134556196443911684 T^{18} + 5387037053370324828 T^{20} -$$$$19\!\cdots\!80$$$$T^{22} +$$$$60\!\cdots\!96$$$$T^{24} -$$$$16\!\cdots\!96$$$$T^{26} +$$$$36\!\cdots\!40$$$$T^{28} -$$$$60\!\cdots\!84$$$$T^{30} +$$$$56\!\cdots\!21$$$$T^{32}$$)($$1 - 236 T^{2} + 31140 T^{4} - 3102004 T^{6} + 253966836 T^{8} - 17838006620 T^{10} + 1103973200988 T^{12} - 60912719078276 T^{14} + 3016913541521174 T^{16} - 134556196443911684 T^{18} + 5387037053370324828 T^{20} -$$$$19\!\cdots\!80$$$$T^{22} +$$$$60\!\cdots\!96$$$$T^{24} -$$$$16\!\cdots\!96$$$$T^{26} +$$$$36\!\cdots\!40$$$$T^{28} -$$$$60\!\cdots\!84$$$$T^{30} +$$$$56\!\cdots\!21$$$$T^{32}$$)
$53$ ($$( 1 + 53 T^{2} )^{2}$$)($$( 1 - 4 T + 53 T^{2} )^{2}$$)($$( 1 + 6 T + 53 T^{2} )^{2}$$)($$( 1 - 10 T + 53 T^{2} )^{2}$$)($$( 1 + 10 T + 53 T^{2} )^{2}$$)($$( 1 - 6 T + 53 T^{2} )^{2}$$)($$( 1 + 4 T + 53 T^{2} )^{2}$$)($$( 1 + 53 T^{2} )^{2}$$)($$( 1 + 4 T + 172 T^{2} + 124 T^{3} + 13704 T^{4} - 18028 T^{5} + 1035940 T^{6} - 527028 T^{7} + 68170350 T^{8} - 27932484 T^{9} + 2909955460 T^{10} - 2683954556 T^{11} + 108131151624 T^{12} + 51856241132 T^{13} + 3812270114188 T^{14} + 4698844559348 T^{15} + 62259690411361 T^{16} )^{2}$$)($$( 1 - 4 T + 172 T^{2} - 124 T^{3} + 13704 T^{4} + 18028 T^{5} + 1035940 T^{6} + 527028 T^{7} + 68170350 T^{8} + 27932484 T^{9} + 2909955460 T^{10} + 2683954556 T^{11} + 108131151624 T^{12} - 51856241132 T^{13} + 3812270114188 T^{14} - 4698844559348 T^{15} + 62259690411361 T^{16} )^{2}$$)
$59$ ($$1 + 10 T^{2} + 3481 T^{4}$$)($$1 - 86 T^{2} + 3481 T^{4}$$)($$1 - 116 T^{2} + 3481 T^{4}$$)($$( 1 - 6 T + 59 T^{2} )( 1 + 6 T + 59 T^{2} )$$)($$( 1 - 6 T + 59 T^{2} )( 1 + 6 T + 59 T^{2} )$$)($$1 - 116 T^{2} + 3481 T^{4}$$)($$1 - 86 T^{2} + 3481 T^{4}$$)($$1 + 10 T^{2} + 3481 T^{4}$$)($$1 - 584 T^{2} + 167280 T^{4} - 31375000 T^{6} + 4345877148 T^{8} - 474929551176 T^{10} + 42593872173456 T^{12} - 3206861640064088 T^{14} + 204993910642000518 T^{16} - 11163085369063090328 T^{18} +$$$$51\!\cdots\!16$$$$T^{20} -$$$$20\!\cdots\!16$$$$T^{22} +$$$$63\!\cdots\!08$$$$T^{24} -$$$$16\!\cdots\!00$$$$T^{26} +$$$$29\!\cdots\!80$$$$T^{28} -$$$$36\!\cdots\!24$$$$T^{30} +$$$$21\!\cdots\!41$$$$T^{32}$$)($$1 - 584 T^{2} + 167280 T^{4} - 31375000 T^{6} + 4345877148 T^{8} - 474929551176 T^{10} + 42593872173456 T^{12} - 3206861640064088 T^{14} + 204993910642000518 T^{16} - 11163085369063090328 T^{18} +$$$$51\!\cdots\!16$$$$T^{20} -$$$$20\!\cdots\!16$$$$T^{22} +$$$$63\!\cdots\!08$$$$T^{24} -$$$$16\!\cdots\!00$$$$T^{26} +$$$$29\!\cdots\!80$$$$T^{28} -$$$$36\!\cdots\!24$$$$T^{30} +$$$$21\!\cdots\!41$$$$T^{32}$$)
$61$ ($$( 1 - 61 T^{2} )^{2}$$)($$1 - 50 T^{2} + 3721 T^{4}$$)($$( 1 - 61 T^{2} )^{2}$$)($$1 - 104 T^{2} + 3721 T^{4}$$)($$1 - 104 T^{2} + 3721 T^{4}$$)($$( 1 - 61 T^{2} )^{2}$$)($$1 - 50 T^{2} + 3721 T^{4}$$)($$( 1 - 61 T^{2} )^{2}$$)($$1 - 164 T^{2} + 23637 T^{4} - 2264768 T^{6} + 219737070 T^{8} - 17352386456 T^{10} + 1361333769007 T^{12} - 88851384720820 T^{14} + 5835679164863962 T^{16} - 330616002546171220 T^{18} + 18848810913601649887 T^{20} -$$$$89\!\cdots\!16$$$$T^{22} +$$$$42\!\cdots\!70$$$$T^{24} -$$$$16\!\cdots\!68$$$$T^{26} +$$$$62\!\cdots\!77$$$$T^{28} -$$$$16\!\cdots\!24$$$$T^{30} +$$$$36\!\cdots\!61$$$$T^{32}$$)($$1 - 164 T^{2} + 23637 T^{4} - 2264768 T^{6} + 219737070 T^{8} - 17352386456 T^{10} + 1361333769007 T^{12} - 88851384720820 T^{14} + 5835679164863962 T^{16} - 330616002546171220 T^{18} + 18848810913601649887 T^{20} -$$$$89\!\cdots\!16$$$$T^{22} +$$$$42\!\cdots\!70$$$$T^{24} -$$$$16\!\cdots\!68$$$$T^{26} +$$$$62\!\cdots\!77$$$$T^{28} -$$$$16\!\cdots\!24$$$$T^{30} +$$$$36\!\cdots\!61$$$$T^{32}$$)
$67$ ($$1 - 14 T + 67 T^{2}$$)($$1 + 14 T + 67 T^{2}$$)($$1 - 14 T + 67 T^{2}$$)($$1 + 14 T + 67 T^{2}$$)($$1 + 14 T + 67 T^{2}$$)($$1 - 14 T + 67 T^{2}$$)($$1 + 14 T + 67 T^{2}$$)($$1 - 14 T + 67 T^{2}$$)($$1 - 4 T - 144 T^{2} + 1052 T^{3} + 6556 T^{4} - 111204 T^{5} + 538512 T^{6} + 3644572 T^{7} - 74118810 T^{8} + 244186324 T^{9} + 2417380368 T^{10} - 33446048652 T^{11} + 132110749276 T^{12} + 1420331612564 T^{13} - 13026007032336 T^{14} - 24242846421292 T^{15} + 406067677556641 T^{16}$$)($$1 - 4 T - 144 T^{2} + 1052 T^{3} + 6556 T^{4} - 111204 T^{5} + 538512 T^{6} + 3644572 T^{7} - 74118810 T^{8} + 244186324 T^{9} + 2417380368 T^{10} - 33446048652 T^{11} + 132110749276 T^{12} + 1420331612564 T^{13} - 13026007032336 T^{14} - 24242846421292 T^{15} + 406067677556641 T^{16}$$)
$71$ ($$1 - 110 T^{2} + 5041 T^{4}$$)($$1 + 58 T^{2} + 5041 T^{4}$$)($$1 - 92 T^{2} + 5041 T^{4}$$)($$1 - 92 T^{2} + 5041 T^{4}$$)($$1 - 92 T^{2} + 5041 T^{4}$$)($$1 - 92 T^{2} + 5041 T^{4}$$)($$1 + 58 T^{2} + 5041 T^{4}$$)($$1 - 110 T^{2} + 5041 T^{4}$$)($$1 - 436 T^{2} + 103865 T^{4} - 17205272 T^{6} + 2228671174 T^{8} - 240033414624 T^{10} + 22427523978183 T^{12} - 1860695785363924 T^{14} + 139019003009222450 T^{16} - 9379767454019540884 T^{18} +$$$$56\!\cdots\!23$$$$T^{20} -$$$$30\!\cdots\!04$$$$T^{22} +$$$$14\!\cdots\!14$$$$T^{24} -$$$$56\!\cdots\!72$$$$T^{26} +$$$$17\!\cdots\!65$$$$T^{28} -$$$$36\!\cdots\!16$$$$T^{30} +$$$$41\!\cdots\!21$$$$T^{32}$$)($$1 - 436 T^{2} + 103865 T^{4} - 17205272 T^{6} + 2228671174 T^{8} - 240033414624 T^{10} + 22427523978183 T^{12} - 1860695785363924 T^{14} + 139019003009222450 T^{16} - 9379767454019540884 T^{18} +$$$$56\!\cdots\!23$$$$T^{20} -$$$$30\!\cdots\!04$$$$T^{22} +$$$$14\!\cdots\!14$$$$T^{24} -$$$$56\!\cdots\!72$$$$T^{26} +$$$$17\!\cdots\!65$$$$T^{28} -$$$$36\!\cdots\!16$$$$T^{30} +$$$$41\!\cdots\!21$$$$T^{32}$$)
$73$ ($$( 1 - 2 T + 73 T^{2} )^{2}$$)($$( 1 - 2 T + 73 T^{2} )^{2}$$)($$( 1 + 4 T + 73 T^{2} )^{2}$$)($$( 1 + 10 T + 73 T^{2} )^{2}$$)($$( 1 + 10 T + 73 T^{2} )^{2}$$)($$( 1 + 4 T + 73 T^{2} )^{2}$$)($$( 1 - 2 T + 73 T^{2} )^{2}$$)($$( 1 - 2 T + 73 T^{2} )^{2}$$)($$( 1 - 8 T + 448 T^{2} - 2888 T^{3} + 94100 T^{4} - 507992 T^{5} + 12229600 T^{6} - 55637848 T^{7} + 1073875926 T^{8} - 4061562904 T^{9} + 65171538400 T^{10} - 197617523864 T^{11} + 2672274478100 T^{12} - 5987030760584 T^{13} + 67797733377472 T^{14} - 88379188152776 T^{15} + 806460091894081 T^{16} )^{2}$$)($$( 1 - 8 T + 448 T^{2} - 2888 T^{3} + 94100 T^{4} - 507992 T^{5} + 12229600 T^{6} - 55637848 T^{7} + 1073875926 T^{8} - 4061562904 T^{9} + 65171538400 T^{10} - 197617523864 T^{11} + 2672274478100 T^{12} - 5987030760584 T^{13} + 67797733377472 T^{14} - 88379188152776 T^{15} + 806460091894081 T^{16} )^{2}$$)
$79$ ($$1 - 140 T^{2} + 6241 T^{4}$$)($$1 + 84 T^{2} + 6241 T^{4}$$)($$1 + 130 T^{2} + 6241 T^{4}$$)($$1 - 150 T^{2} + 6241 T^{4}$$)($$1 - 150 T^{2} + 6241 T^{4}$$)($$1 + 130 T^{2} + 6241 T^{4}$$)($$1 + 84 T^{2} + 6241 T^{4}$$)($$1 - 140 T^{2} + 6241 T^{4}$$)($$1 - 692 T^{2} + 251368 T^{4} - 62358860 T^{6} + 11696423312 T^{8} - 1746760099060 T^{10} + 213912626620344 T^{12} - 21851789495031500 T^{14} + 1878932976025351710 T^{16} -$$$$13\!\cdots\!00$$$$T^{18} +$$$$83\!\cdots\!64$$$$T^{20} -$$$$42\!\cdots\!60$$$$T^{22} +$$$$17\!\cdots\!32$$$$T^{24} -$$$$59\!\cdots\!60$$$$T^{26} +$$$$14\!\cdots\!88$$$$T^{28} -$$$$25\!\cdots\!52$$$$T^{30} +$$$$23\!\cdots\!21$$$$T^{32}$$)($$1 - 692 T^{2} + 251368 T^{4} - 62358860 T^{6} + 11696423312 T^{8} - 1746760099060 T^{10} + 213912626620344 T^{12} - 21851789495031500 T^{14} + 1878932976025351710 T^{16} -$$$$13\!\cdots\!00$$$$T^{18} +$$$$83\!\cdots\!64$$$$T^{20} -$$$$42\!\cdots\!60$$$$T^{22} +$$$$17\!\cdots\!32$$$$T^{24} -$$$$59\!\cdots\!60$$$$T^{26} +$$$$14\!\cdots\!88$$$$T^{28} -$$$$25\!\cdots\!52$$$$T^{30} +$$$$23\!\cdots\!21$$$$T^{32}$$)
$83$ ($$1 - 116 T^{2} + 6889 T^{4}$$)($$1 - 116 T^{2} + 6889 T^{4}$$)($$1 - 68 T^{2} + 6889 T^{4}$$)($$1 - 68 T^{2} + 6889 T^{4}$$)($$1 - 68 T^{2} + 6889 T^{4}$$)($$1 - 68 T^{2} + 6889 T^{4}$$)($$1 - 116 T^{2} + 6889 T^{4}$$)($$1 - 116 T^{2} + 6889 T^{4}$$)($$1 - 472 T^{2} + 119705 T^{4} - 21970144 T^{6} + 3253765858 T^{8} - 410263950336 T^{10} + 45393778890011 T^{12} - 4461334566964584 T^{14} + 391452390769526978 T^{16} - 30734133831819019176 T^{18} +$$$$21\!\cdots\!31$$$$T^{20} -$$$$13\!\cdots\!84$$$$T^{22} +$$$$73\!\cdots\!78$$$$T^{24} -$$$$34\!\cdots\!56$$$$T^{26} +$$$$12\!\cdots\!05$$$$T^{28} -$$$$34\!\cdots\!88$$$$T^{30} +$$$$50\!\cdots\!81$$$$T^{32}$$)($$1 - 472 T^{2} + 119705 T^{4} - 21970144 T^{6} + 3253765858 T^{8} - 410263950336 T^{10} + 45393778890011 T^{12} - 4461334566964584 T^{14} + 391452390769526978 T^{16} - 30734133831819019176 T^{18} +$$$$21\!\cdots\!31$$$$T^{20} -$$$$13\!\cdots\!84$$$$T^{22} +$$$$73\!\cdots\!78$$$$T^{24} -$$$$34\!\cdots\!56$$$$T^{26} +$$$$12\!\cdots\!05$$$$T^{28} -$$$$34\!\cdots\!88$$$$T^{30} +$$$$50\!\cdots\!81$$$$T^{32}$$)
$89$ ($$1 - 128 T^{2} + 7921 T^{4}$$)($$1 - 160 T^{2} + 7921 T^{4}$$)($$1 + 64 T^{2} + 7921 T^{4}$$)($$1 - 16 T^{2} + 7921 T^{4}$$)($$1 - 16 T^{2} + 7921 T^{4}$$)($$1 + 64 T^{2} + 7921 T^{4}$$)($$1 - 160 T^{2} + 7921 T^{4}$$)($$1 - 128 T^{2} + 7921 T^{4}$$)($$1 - 496 T^{2} + 136613 T^{4} - 27528624 T^{6} + 4522731274 T^{8} - 631608552016 T^{10} + 76556075269003 T^{12} - 8155835267046416 T^{14} + 770110258891235850 T^{16} - 64602371150274661136 T^{18} +$$$$48\!\cdots\!23$$$$T^{20} -$$$$31\!\cdots\!76$$$$T^{22} +$$$$17\!\cdots\!94$$$$T^{24} -$$$$85\!\cdots\!24$$$$T^{26} +$$$$33\!\cdots\!73$$$$T^{28} -$$$$97\!\cdots\!36$$$$T^{30} +$$$$15\!\cdots\!61$$$$T^{32}$$)($$1 - 496 T^{2} + 136613 T^{4} - 27528624 T^{6} + 4522731274 T^{8} - 631608552016 T^{10} + 76556075269003 T^{12} - 8155835267046416 T^{14} + 770110258891235850 T^{16} - 64602371150274661136 T^{18} +$$$$48\!\cdots\!23$$$$T^{20} -$$$$31\!\cdots\!76$$$$T^{22} +$$$$17\!\cdots\!94$$$$T^{24} -$$$$85\!\cdots\!24$$$$T^{26} +$$$$33\!\cdots\!73$$$$T^{28} -$$$$97\!\cdots\!36$$$$T^{30} +$$$$15\!\cdots\!61$$$$T^{32}$$)
$97$ ($$1 - 176 T^{2} + 9409 T^{4}$$)($$1 - 96 T^{2} + 9409 T^{4}$$)($$( 1 - 10 T + 97 T^{2} )( 1 + 10 T + 97 T^{2} )$$)($$1 - 144 T^{2} + 9409 T^{4}$$)($$1 - 144 T^{2} + 9409 T^{4}$$)($$( 1 - 10 T + 97 T^{2} )( 1 + 10 T + 97 T^{2} )$$)($$1 - 96 T^{2} + 9409 T^{4}$$)($$1 - 176 T^{2} + 9409 T^{4}$$)($$1 - 832 T^{2} + 338597 T^{4} - 90113976 T^{6} + 17744840082 T^{8} - 2779884709240 T^{10} + 364561254235739 T^{12} - 41610271866017936 T^{14} + 4246834307168352730 T^{16} -$$$$39\!\cdots\!24$$$$T^{18} +$$$$32\!\cdots\!59$$$$T^{20} -$$$$23\!\cdots\!60$$$$T^{22} +$$$$13\!\cdots\!02$$$$T^{24} -$$$$66\!\cdots\!24$$$$T^{26} +$$$$23\!\cdots\!77$$$$T^{28} -$$$$54\!\cdots\!08$$$$T^{30} +$$$$61\!\cdots\!21$$$$T^{32}$$)($$1 - 832 T^{2} + 338597 T^{4} - 90113976 T^{6} + 17744840082 T^{8} - 2779884709240 T^{10} + 364561254235739 T^{12} - 41610271866017936 T^{14} + 4246834307168352730 T^{16} -$$$$39\!\cdots\!24$$$$T^{18} +$$$$32\!\cdots\!59$$$$T^{20} -$$$$23\!\cdots\!60$$$$T^{22} +$$$$13\!\cdots\!02$$$$T^{24} -$$$$66\!\cdots\!24$$$$T^{26} +$$$$23\!\cdots\!77$$$$T^{28} -$$$$54\!\cdots\!08$$$$T^{30} +$$$$61\!\cdots\!21$$$$T^{32}$$)