Properties

Label 22.7
Level 22
Weight 7
Dimension 30
Nonzero newspaces 2
Newform subspaces 2
Sturm bound 210
Trace bound 1

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Defining parameters

Level: \( N \) = \( 22 = 2 \cdot 11 \)
Weight: \( k \) = \( 7 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 2 \)
Sturm bound: \(210\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(\Gamma_1(22))\).

Total New Old
Modular forms 100 30 70
Cusp forms 80 30 50
Eisenstein series 20 0 20

Trace form

\( 30q + 400q^{6} + 720q^{7} - 4160q^{9} + O(q^{10}) \) \( 30q + 400q^{6} + 720q^{7} - 4160q^{9} + 3200q^{11} + 5120q^{12} + 3600q^{13} - 2080q^{14} - 8660q^{15} - 5880q^{17} + 18400q^{18} - 23850q^{19} - 37940q^{23} + 12800q^{24} + 129840q^{25} + 50400q^{26} - 53250q^{27} - 32640q^{28} - 204800q^{29} - 134160q^{30} - 6900q^{31} + 153450q^{33} + 117600q^{34} + 469520q^{35} + 36800q^{36} + 103200q^{37} + 28080q^{38} - 328640q^{39} - 107520q^{40} - 541660q^{41} - 521920q^{42} + 251200q^{44} + 1180380q^{45} + 213120q^{46} + 193640q^{47} - 645420q^{49} - 364480q^{50} - 1347330q^{51} - 291840q^{52} + 81120q^{53} + 714960q^{55} + 1416050q^{57} + 1061760q^{58} + 94170q^{59} + 414720q^{60} - 1184160q^{61} - 226640q^{62} - 921200q^{63} + 321920q^{66} + 22320q^{67} - 188160q^{68} + 1356080q^{69} + 304080q^{70} - 242800q^{71} - 896000q^{72} - 590340q^{73} - 610400q^{74} - 2419830q^{75} - 1972220q^{77} + 1361600q^{78} + 93240q^{79} + 573440q^{80} + 1519730q^{81} + 741600q^{82} + 4588350q^{83} + 2357760q^{84} + 4132920q^{85} + 2113440q^{86} - 806400q^{88} - 2260900q^{89} - 6535040q^{90} - 8984700q^{91} - 2419200q^{92} + 143240q^{93} - 2410800q^{94} + 661320q^{95} + 1009290q^{97} + 2419140q^{99} + O(q^{100}) \)

Decomposition of \(S_{7}^{\mathrm{new}}(\Gamma_1(22))\)

We only show spaces with odd 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
22.7.b \(\chi_{22}(21, \cdot)\) 22.7.b.a 6 1
22.7.d \(\chi_{22}(7, \cdot)\) 22.7.d.a 24 4

Decomposition of \(S_{7}^{\mathrm{old}}(\Gamma_1(22))\) into lower level spaces

\( S_{7}^{\mathrm{old}}(\Gamma_1(22)) \cong \) \(S_{7}^{\mathrm{new}}(\Gamma_1(11))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + 32 T^{2} )^{3} \))
$3$ (\( ( 1 + 26 T + 1518 T^{2} + 36870 T^{3} + 1106622 T^{4} + 13817466 T^{5} + 387420489 T^{6} )^{2} \))
$5$ (\( ( 1 - 184 T + 25400 T^{2} - 1721050 T^{3} + 396875000 T^{4} - 44921875000 T^{5} + 3814697265625 T^{6} )^{2} \))
$7$ (\( 1 - 217566 T^{2} + 32540011983 T^{4} - 4783143212564132 T^{6} + \)\(45\!\cdots\!83\)\( T^{8} - \)\(41\!\cdots\!66\)\( T^{10} + \)\(26\!\cdots\!01\)\( T^{12} \))
$11$ (\( 1 + 1166 T - 830665 T^{2} - 3785518396 T^{3} - 1471573718065 T^{4} + 3659407487256686 T^{5} + 5559917313492231481 T^{6} \))
$13$ (\( 1 - 12318414 T^{2} + 100269018276927 T^{4} - \)\(52\!\cdots\!44\)\( T^{6} + \)\(23\!\cdots\!87\)\( T^{8} - \)\(66\!\cdots\!54\)\( T^{10} + \)\(12\!\cdots\!41\)\( T^{12} \))
$17$ (\( 1 - 61535358 T^{2} + 1899521995563567 T^{4} - \)\(45\!\cdots\!16\)\( T^{6} + \)\(11\!\cdots\!87\)\( T^{8} - \)\(20\!\cdots\!18\)\( T^{10} + \)\(19\!\cdots\!81\)\( T^{12} \))
$19$ (\( 1 - 26275086 T^{2} + 6822221013915327 T^{4} - \)\(11\!\cdots\!76\)\( T^{6} + \)\(15\!\cdots\!47\)\( T^{8} - \)\(12\!\cdots\!06\)\( T^{10} + \)\(10\!\cdots\!81\)\( T^{12} \))
$23$ (\( ( 1 - 1078 T + 415875950 T^{2} - 312925939186 T^{3} + 61564565971969550 T^{4} - 23623965137717906038 T^{5} + \)\(32\!\cdots\!69\)\( T^{6} )^{2} \))
$29$ (\( 1 - 2760181398 T^{2} + 3539740269101774751 T^{4} - \)\(26\!\cdots\!36\)\( T^{6} + \)\(12\!\cdots\!91\)\( T^{8} - \)\(34\!\cdots\!38\)\( T^{10} + \)\(44\!\cdots\!21\)\( T^{12} \))
$31$ (\( ( 1 + 39234 T + 2560490982 T^{2} + 68508627283486 T^{3} + 2272445171692304742 T^{4} + \)\(30\!\cdots\!74\)\( T^{5} + \)\(69\!\cdots\!41\)\( T^{6} )^{2} \))
$37$ (\( ( 1 + 102960 T + 10729446096 T^{2} + 554698638636454 T^{3} + 27528823202449149264 T^{4} + \)\(67\!\cdots\!60\)\( T^{5} + \)\(16\!\cdots\!29\)\( T^{6} )^{2} \))
$41$ (\( 1 - 13389830046 T^{2} + 86823147879328529295 T^{4} - \)\(41\!\cdots\!72\)\( T^{6} + \)\(19\!\cdots\!95\)\( T^{8} - \)\(68\!\cdots\!06\)\( T^{10} + \)\(11\!\cdots\!41\)\( T^{12} \))
$43$ (\( 1 - 19691349462 T^{2} + \)\(15\!\cdots\!31\)\( T^{4} - \)\(85\!\cdots\!04\)\( T^{6} + \)\(60\!\cdots\!31\)\( T^{8} - \)\(31\!\cdots\!62\)\( T^{10} + \)\(63\!\cdots\!01\)\( T^{12} \))
$47$ (\( ( 1 - 246730 T + 50981717951 T^{2} - 5751342041687596 T^{3} + \)\(54\!\cdots\!79\)\( T^{4} - \)\(28\!\cdots\!30\)\( T^{5} + \)\(12\!\cdots\!89\)\( T^{6} )^{2} \))
$53$ (\( ( 1 + 265850 T + 61006376519 T^{2} + 8665677288711308 T^{3} + \)\(13\!\cdots\!51\)\( T^{4} + \)\(13\!\cdots\!50\)\( T^{5} + \)\(10\!\cdots\!89\)\( T^{6} )^{2} \))
$59$ (\( ( 1 + 416690 T + 126613451990 T^{2} + 26137140602762390 T^{3} + \)\(53\!\cdots\!90\)\( T^{4} + \)\(74\!\cdots\!90\)\( T^{5} + \)\(75\!\cdots\!21\)\( T^{6} )^{2} \))
$61$ (\( 1 - 162819828342 T^{2} + \)\(15\!\cdots\!15\)\( T^{4} - \)\(96\!\cdots\!28\)\( T^{6} + \)\(40\!\cdots\!15\)\( T^{8} - \)\(11\!\cdots\!22\)\( T^{10} + \)\(18\!\cdots\!61\)\( T^{12} \))
$67$ (\( ( 1 - 268710 T + 276879935070 T^{2} - 46741488269575706 T^{3} + \)\(25\!\cdots\!30\)\( T^{4} - \)\(21\!\cdots\!10\)\( T^{5} + \)\(74\!\cdots\!09\)\( T^{6} )^{2} \))
$71$ (\( ( 1 + 230186 T + 260527573982 T^{2} + 68731919036734094 T^{3} + \)\(33\!\cdots\!22\)\( T^{4} + \)\(37\!\cdots\!26\)\( T^{5} + \)\(21\!\cdots\!61\)\( T^{6} )^{2} \))
$73$ (\( 1 - 203158701342 T^{2} + \)\(52\!\cdots\!79\)\( T^{4} - \)\(57\!\cdots\!36\)\( T^{6} + \)\(11\!\cdots\!59\)\( T^{8} - \)\(10\!\cdots\!22\)\( T^{10} + \)\(12\!\cdots\!61\)\( T^{12} \))
$79$ (\( 1 - 576022933062 T^{2} + \)\(21\!\cdots\!67\)\( T^{4} - \)\(62\!\cdots\!84\)\( T^{6} + \)\(12\!\cdots\!47\)\( T^{8} - \)\(20\!\cdots\!22\)\( T^{10} + \)\(20\!\cdots\!21\)\( T^{12} \))
$83$ (\( 1 - 217331496918 T^{2} + \)\(22\!\cdots\!43\)\( T^{4} - \)\(26\!\cdots\!84\)\( T^{6} + \)\(23\!\cdots\!23\)\( T^{8} - \)\(24\!\cdots\!78\)\( T^{10} + \)\(12\!\cdots\!81\)\( T^{12} \))
$89$ (\( ( 1 - 1188976 T + 1827007541360 T^{2} - 1203827383443407986 T^{3} + \)\(90\!\cdots\!60\)\( T^{4} - \)\(29\!\cdots\!96\)\( T^{5} + \)\(12\!\cdots\!81\)\( T^{6} )^{2} \))
$97$ (\( ( 1 - 675816 T + 466099093224 T^{2} + 583457088977364286 T^{3} + \)\(38\!\cdots\!96\)\( T^{4} - \)\(46\!\cdots\!56\)\( T^{5} + \)\(57\!\cdots\!89\)\( T^{6} )^{2} \))
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