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

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