Properties

Label 8.8
Level 8
Weight 8
Dimension 8
Nonzero newspaces 2
Newform subspaces 3
Sturm bound 32
Trace bound 1

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Defining parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 8 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 3 \)
Sturm bound: \(32\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(8))\).

Total New Old
Modular forms 17 10 7
Cusp forms 11 8 3
Eisenstein series 6 2 4

Trace form

\( 8q + 6q^{2} - 40q^{3} + 116q^{4} + 348q^{5} + 268q^{6} - 2368q^{7} + 1512q^{8} + 1700q^{9} + O(q^{10}) \) \( 8q + 6q^{2} - 40q^{3} + 116q^{4} + 348q^{5} + 268q^{6} - 2368q^{7} + 1512q^{8} + 1700q^{9} - 1656q^{10} - 5688q^{11} - 4088q^{12} - 4660q^{13} + 12048q^{14} + 43680q^{15} + 35344q^{16} - 25968q^{17} - 89062q^{18} - 1352q^{19} - 114768q^{20} - 15552q^{21} + 152860q^{22} + 81984q^{23} + 282512q^{24} - 3940q^{25} - 316968q^{26} - 332560q^{27} - 480800q^{28} + 222636q^{29} + 821648q^{30} + 121600q^{31} + 817056q^{32} + 126680q^{33} - 1009108q^{34} - 488928q^{35} - 1253556q^{36} - 471780q^{37} + 974124q^{38} + 846336q^{39} + 954464q^{40} + 343248q^{41} - 1093088q^{42} + 20360q^{43} - 1096344q^{44} - 507188q^{45} + 929840q^{46} + 1168512q^{47} + 853920q^{48} - 452024q^{49} - 148626q^{50} + 220720q^{51} + 823952q^{52} + 2109180q^{53} - 1077064q^{54} - 4423808q^{55} - 2468928q^{56} - 2477256q^{57} + 3130744q^{58} - 1863576q^{59} + 5715168q^{60} + 1002860q^{61} - 7055808q^{62} + 3863776q^{63} - 4792768q^{64} + 4931016q^{65} + 7926264q^{66} + 7625560q^{67} + 6608040q^{68} - 6793792q^{69} - 7406912q^{70} - 12856128q^{71} - 11363944q^{72} - 5261616q^{73} + 7744200q^{74} + 10695784q^{75} + 9241288q^{76} + 5023680q^{77} - 9471184q^{78} + 14929792q^{79} - 12600384q^{80} + 3381600q^{81} + 10715932q^{82} - 6949320q^{83} + 4220608q^{84} - 6081800q^{85} - 5639076q^{86} - 40431936q^{87} + 1541200q^{88} + 7301712q^{89} - 121864q^{90} - 2573664q^{91} + 669600q^{92} + 14460160q^{93} + 15503712q^{94} + 45845088q^{95} + 21402176q^{96} + 3777552q^{97} - 14983242q^{98} - 11495192q^{99} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(8))\)

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
8.8.a \(\chi_{8}(1, \cdot)\) 8.8.a.a 1 1
8.8.a.b 1
8.8.b \(\chi_{8}(5, \cdot)\) 8.8.b.a 6 1

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_1(8))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_1(8)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 3}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( \))(\( \))(\( 1 - 6 T - 40 T^{2} - 192 T^{3} - 5120 T^{4} - 98304 T^{5} + 2097152 T^{6} \))
$3$ (\( 1 + 84 T + 2187 T^{2} \))(\( 1 - 44 T + 2187 T^{2} \))(\( 1 - 5102 T^{2} + 14791383 T^{4} - 32202570276 T^{6} + 70746726356127 T^{8} - 116717395105211022 T^{10} + \)\(10\!\cdots\!09\)\( T^{12} \))
$5$ (\( 1 + 82 T + 78125 T^{2} \))(\( 1 - 430 T + 78125 T^{2} \))(\( 1 - 214718 T^{2} + 31745660775 T^{4} - 2880459231122500 T^{6} + \)\(19\!\cdots\!75\)\( T^{8} - \)\(79\!\cdots\!50\)\( T^{10} + \)\(22\!\cdots\!25\)\( T^{12} \))
$7$ (\( 1 + 456 T + 823543 T^{2} \))(\( 1 + 1224 T + 823543 T^{2} \))(\( ( 1 + 344 T + 1422245 T^{2} + 124668368 T^{3} + 1171279914035 T^{4} + 233308737060056 T^{5} + 558545864083284007 T^{6} )^{2} \))
$11$ (\( 1 + 2524 T + 19487171 T^{2} \))(\( 1 + 3164 T + 19487171 T^{2} \))(\( 1 - 64629022 T^{2} + 2363172524483783 T^{4} - \)\(55\!\cdots\!04\)\( T^{6} + \)\(89\!\cdots\!03\)\( T^{8} - \)\(93\!\cdots\!82\)\( T^{10} + \)\(54\!\cdots\!21\)\( T^{12} \))
$13$ (\( 1 + 10778 T + 62748517 T^{2} \))(\( 1 - 6118 T + 62748517 T^{2} \))(\( 1 - 205410958 T^{2} + 21942186159527543 T^{4} - \)\(16\!\cdots\!64\)\( T^{6} + \)\(86\!\cdots\!27\)\( T^{8} - \)\(31\!\cdots\!18\)\( T^{10} + \)\(61\!\cdots\!69\)\( T^{12} \))
$17$ (\( 1 + 11150 T + 410338673 T^{2} \))(\( 1 + 16270 T + 410338673 T^{2} \))(\( ( 1 - 726 T + 323301023 T^{2} - 9708009717300 T^{3} + 132662912757362479 T^{4} - \)\(12\!\cdots\!54\)\( T^{5} + \)\(69\!\cdots\!17\)\( T^{6} )^{2} \))
$19$ (\( 1 - 4124 T + 893871739 T^{2} \))(\( 1 + 5476 T + 893871739 T^{2} \))(\( 1 - 2002416334 T^{2} + 2872909317854295863 T^{4} - \)\(28\!\cdots\!20\)\( T^{6} + \)\(22\!\cdots\!23\)\( T^{8} - \)\(12\!\cdots\!94\)\( T^{10} + \)\(51\!\cdots\!61\)\( T^{12} \))
$23$ (\( 1 - 81704 T + 3404825447 T^{2} \))(\( 1 - 1576 T + 3404825447 T^{2} \))(\( ( 1 + 648 T + 9708521717 T^{2} + 6547475963760 T^{3} + 33055821794793732499 T^{4} + \)\(75\!\cdots\!32\)\( T^{5} + \)\(39\!\cdots\!23\)\( T^{6} )^{2} \))
$29$ (\( 1 - 99798 T + 17249876309 T^{2} \))(\( 1 - 122838 T + 17249876309 T^{2} \))(\( 1 - 47836636078 T^{2} + \)\(14\!\cdots\!79\)\( T^{4} - \)\(30\!\cdots\!44\)\( T^{6} + \)\(41\!\cdots\!99\)\( T^{8} - \)\(42\!\cdots\!58\)\( T^{10} + \)\(26\!\cdots\!41\)\( T^{12} \))
$31$ (\( 1 + 40480 T + 27512614111 T^{2} \))(\( 1 - 251360 T + 27512614111 T^{2} \))(\( ( 1 + 44640 T + 48354349725 T^{2} + 4324137771289408 T^{3} + \)\(13\!\cdots\!75\)\( T^{4} + \)\(33\!\cdots\!40\)\( T^{5} + \)\(20\!\cdots\!31\)\( T^{6} )^{2} \))
$37$ (\( 1 + 419442 T + 94931877133 T^{2} \))(\( 1 + 52338 T + 94931877133 T^{2} \))(\( 1 - 78937168126 T^{2} + \)\(11\!\cdots\!51\)\( T^{4} - \)\(18\!\cdots\!48\)\( T^{6} + \)\(10\!\cdots\!39\)\( T^{8} - \)\(64\!\cdots\!46\)\( T^{10} + \)\(73\!\cdots\!69\)\( T^{12} \))
$41$ (\( 1 - 141402 T + 194754273881 T^{2} \))(\( 1 + 319398 T + 194754273881 T^{2} \))(\( ( 1 - 260622 T + 351195126263 T^{2} - 83834535574878564 T^{3} + \)\(68\!\cdots\!03\)\( T^{4} - \)\(98\!\cdots\!42\)\( T^{5} + \)\(73\!\cdots\!41\)\( T^{6} )^{2} \))
$43$ (\( 1 + 690428 T + 271818611107 T^{2} \))(\( 1 - 710788 T + 271818611107 T^{2} \))(\( 1 - 1505999929054 T^{2} + \)\(97\!\cdots\!71\)\( T^{4} - \)\(34\!\cdots\!32\)\( T^{6} + \)\(71\!\cdots\!79\)\( T^{8} - \)\(82\!\cdots\!54\)\( T^{10} + \)\(40\!\cdots\!49\)\( T^{12} \))
$47$ (\( 1 + 682032 T + 506623120463 T^{2} \))(\( 1 - 284112 T + 506623120463 T^{2} \))(\( ( 1 - 783216 T + 1470820452333 T^{2} - 778339576797138720 T^{3} + \)\(74\!\cdots\!79\)\( T^{4} - \)\(20\!\cdots\!04\)\( T^{5} + \)\(13\!\cdots\!47\)\( T^{6} )^{2} \))
$53$ (\( 1 - 1813118 T + 1174711139837 T^{2} \))(\( 1 - 296062 T + 1174711139837 T^{2} \))(\( 1 - 3131635055902 T^{2} + \)\(64\!\cdots\!63\)\( T^{4} - \)\(86\!\cdots\!56\)\( T^{6} + \)\(89\!\cdots\!47\)\( T^{8} - \)\(59\!\cdots\!22\)\( T^{10} + \)\(26\!\cdots\!09\)\( T^{12} \))
$59$ (\( 1 + 966028 T + 2488651484819 T^{2} \))(\( 1 + 897548 T + 2488651484819 T^{2} \))(\( 1 - 8312323804862 T^{2} + \)\(38\!\cdots\!03\)\( T^{4} - \)\(11\!\cdots\!64\)\( T^{6} + \)\(23\!\cdots\!83\)\( T^{8} - \)\(31\!\cdots\!02\)\( T^{10} + \)\(23\!\cdots\!81\)\( T^{12} \))
$61$ (\( 1 - 1887670 T + 3142742836021 T^{2} \))(\( 1 + 884810 T + 3142742836021 T^{2} \))(\( 1 - 14350504934382 T^{2} + \)\(96\!\cdots\!03\)\( T^{4} - \)\(38\!\cdots\!24\)\( T^{6} + \)\(95\!\cdots\!23\)\( T^{8} - \)\(13\!\cdots\!42\)\( T^{10} + \)\(96\!\cdots\!21\)\( T^{12} \))
$67$ (\( 1 - 2965868 T + 6060711605323 T^{2} \))(\( 1 - 4659692 T + 6060711605323 T^{2} \))(\( 1 - 35072892237678 T^{2} + \)\(51\!\cdots\!43\)\( T^{4} - \)\(41\!\cdots\!64\)\( T^{6} + \)\(19\!\cdots\!47\)\( T^{8} - \)\(47\!\cdots\!98\)\( T^{10} + \)\(49\!\cdots\!89\)\( T^{12} \))
$71$ (\( 1 + 2548232 T + 9095120158391 T^{2} \))(\( 1 + 2710792 T + 9095120158391 T^{2} \))(\( ( 1 + 3798552 T + 17820666583269 T^{2} + 72937737977373055056 T^{3} + \)\(16\!\cdots\!79\)\( T^{4} + \)\(31\!\cdots\!12\)\( T^{5} + \)\(75\!\cdots\!71\)\( T^{6} )^{2} \))
$73$ (\( 1 + 1680326 T + 11047398519097 T^{2} \))(\( 1 + 5670854 T + 11047398519097 T^{2} \))(\( ( 1 - 1044782 T + 23207418111255 T^{2} - 22868192285089705636 T^{3} + \)\(25\!\cdots\!35\)\( T^{4} - \)\(12\!\cdots\!38\)\( T^{5} + \)\(13\!\cdots\!73\)\( T^{6} )^{2} \))
$79$ (\( 1 - 4038064 T + 19203908986159 T^{2} \))(\( 1 + 5124176 T + 19203908986159 T^{2} \))(\( ( 1 - 8007952 T + 49828330384013 T^{2} - \)\(25\!\cdots\!96\)\( T^{3} + \)\(95\!\cdots\!67\)\( T^{4} - \)\(29\!\cdots\!12\)\( T^{5} + \)\(70\!\cdots\!79\)\( T^{6} )^{2} \))
$83$ (\( 1 + 5385764 T + 27136050989627 T^{2} \))(\( 1 + 1563556 T + 27136050989627 T^{2} \))(\( 1 - 124932236904014 T^{2} + \)\(70\!\cdots\!51\)\( T^{4} - \)\(23\!\cdots\!92\)\( T^{6} + \)\(51\!\cdots\!79\)\( T^{8} - \)\(67\!\cdots\!74\)\( T^{10} + \)\(39\!\cdots\!89\)\( T^{12} \))
$89$ (\( 1 + 6473046 T + 44231334895529 T^{2} \))(\( 1 - 11605674 T + 44231334895529 T^{2} \))(\( ( 1 - 1084542 T + 130289056617383 T^{2} - 94766843887733093316 T^{3} + \)\(57\!\cdots\!07\)\( T^{4} - \)\(21\!\cdots\!22\)\( T^{5} + \)\(86\!\cdots\!89\)\( T^{6} )^{2} \))
$97$ (\( 1 + 6065758 T + 80798284478113 T^{2} \))(\( 1 - 10931618 T + 80798284478113 T^{2} \))(\( ( 1 + 544154 T + 184934786492783 T^{2} + 86271378317799707180 T^{3} + \)\(14\!\cdots\!79\)\( T^{4} + \)\(35\!\cdots\!26\)\( T^{5} + \)\(52\!\cdots\!97\)\( T^{6} )^{2} \))
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