Properties

Label 8.5
Level 8
Weight 5
Dimension 3
Nonzero newspaces 1
Newform subspaces 2
Sturm bound 20
Trace bound 0

Downloads

Learn more about

Defining parameters

Level: \( N \) = \( 8\( 8 = 2^{3} \) \)
Weight: \( k \) = \( 5 \)
Nonzero newspaces: \( 1 \)
Newform subspaces: \( 2 \)
Sturm bound: \(20\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(\Gamma_1(8))\).

Total New Old
Modular forms 11 5 6
Cusp forms 5 3 2
Eisenstein series 6 2 4

Trace form

\( 3q + 2q^{2} - 2q^{3} - 12q^{4} - 68q^{6} + 152q^{8} + 25q^{9} + O(q^{10}) \) \( 3q + 2q^{2} - 2q^{3} - 12q^{4} - 68q^{6} + 152q^{8} + 25q^{9} + 240q^{10} - 98q^{11} - 392q^{12} - 480q^{14} + 528q^{16} - 122q^{17} + 550q^{18} + 702q^{19} - 480q^{20} - 132q^{22} - 368q^{24} - 45q^{25} - 240q^{26} - 1988q^{27} + 960q^{28} + 1440q^{30} - 928q^{32} + 332q^{33} - 2748q^{34} + 3840q^{35} + 3100q^{36} + 1468q^{38} - 2880q^{40} + 742q^{41} - 2880q^{42} - 7266q^{43} - 8q^{44} + 2400q^{46} - 1952q^{48} - 477q^{49} + 3170q^{50} + 10748q^{51} + 480q^{52} - 392q^{54} + 5760q^{56} - 4468q^{57} + 2640q^{58} - 10274q^{59} - 2880q^{60} - 9600q^{62} + 3648q^{64} + 1920q^{65} + 2888q^{66} + 10878q^{67} - 15512q^{68} - 3840q^{70} + 3400q^{72} + 10278q^{73} + 13680q^{74} - 12770q^{75} + 3192q^{76} - 1440q^{78} + 13440q^{80} - 4433q^{81} - 6972q^{82} + 6718q^{83} + 5760q^{84} - 10244q^{86} - 5232q^{88} - 14618q^{89} - 10800q^{90} - 3840q^{91} - 4800q^{92} + 16320q^{94} - 26048q^{96} + 7494q^{97} + 12482q^{98} - 2950q^{99} + O(q^{100}) \)

Decomposition of \(S_{5}^{\mathrm{new}}(\Gamma_1(8))\)

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
8.5.c \(\chi_{8}(7, \cdot)\) None 0 1
8.5.d \(\chi_{8}(3, \cdot)\) 8.5.d.a 1 1
8.5.d.b 2

Decomposition of \(S_{5}^{\mathrm{old}}(\Gamma_1(8))\) into lower level spaces

\( S_{5}^{\mathrm{old}}(\Gamma_1(8)) \cong \) \(S_{5}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 4 T \))(\( 1 + 2 T + 16 T^{2} \))
$3$ (\( 1 + 14 T + 81 T^{2} \))(\( ( 1 - 6 T + 81 T^{2} )^{2} \))
$5$ (\( ( 1 - 25 T )( 1 + 25 T ) \))(\( 1 - 290 T^{2} + 390625 T^{4} \))
$7$ (\( ( 1 - 49 T )( 1 + 49 T ) \))(\( 1 - 962 T^{2} + 5764801 T^{4} \))
$11$ (\( 1 + 46 T + 14641 T^{2} \))(\( ( 1 + 26 T + 14641 T^{2} )^{2} \))
$13$ (\( ( 1 - 169 T )( 1 + 169 T ) \))(\( 1 - 56162 T^{2} + 815730721 T^{4} \))
$17$ (\( 1 + 574 T + 83521 T^{2} \))(\( ( 1 - 226 T + 83521 T^{2} )^{2} \))
$19$ (\( 1 - 434 T + 130321 T^{2} \))(\( ( 1 - 134 T + 130321 T^{2} )^{2} \))
$23$ (\( ( 1 - 529 T )( 1 + 529 T ) \))(\( 1 - 463682 T^{2} + 78310985281 T^{4} \))
$29$ (\( ( 1 - 841 T )( 1 + 841 T ) \))(\( 1 - 1298402 T^{2} + 500246412961 T^{4} \))
$31$ (\( ( 1 - 961 T )( 1 + 961 T ) \))(\( 1 - 311042 T^{2} + 852891037441 T^{4} \))
$37$ (\( ( 1 - 1369 T )( 1 + 1369 T ) \))(\( 1 - 629282 T^{2} + 3512479453921 T^{4} \))
$41$ (\( 1 + 1246 T + 2825761 T^{2} \))(\( ( 1 - 994 T + 2825761 T^{2} )^{2} \))
$43$ (\( 1 + 3502 T + 3418801 T^{2} \))(\( ( 1 + 1882 T + 3418801 T^{2} )^{2} \))
$47$ (\( ( 1 - 2209 T )( 1 + 2209 T ) \))(\( 1 - 5320322 T^{2} + 23811286661761 T^{4} \))
$53$ (\( ( 1 - 2809 T )( 1 + 2809 T ) \))(\( 1 - 1257122 T^{2} + 62259690411361 T^{4} \))
$59$ (\( 1 + 238 T + 12117361 T^{2} \))(\( ( 1 + 5018 T + 12117361 T^{2} )^{2} \))
$61$ (\( ( 1 - 3721 T )( 1 + 3721 T ) \))(\( 1 - 23382242 T^{2} + 191707312997281 T^{4} \))
$67$ (\( 1 + 5134 T + 20151121 T^{2} \))(\( ( 1 - 8006 T + 20151121 T^{2} )^{2} \))
$71$ (\( ( 1 - 5041 T )( 1 + 5041 T ) \))(\( 1 - 50512322 T^{2} + 645753531245761 T^{4} \))
$73$ (\( 1 - 9506 T + 28398241 T^{2} \))(\( ( 1 - 386 T + 28398241 T^{2} )^{2} \))
$79$ (\( ( 1 - 6241 T )( 1 + 6241 T ) \))(\( 1 + 43766398 T^{2} + 1517108809906561 T^{4} \))
$83$ (\( 1 - 11186 T + 47458321 T^{2} \))(\( ( 1 + 2234 T + 47458321 T^{2} )^{2} \))
$89$ (\( 1 - 5474 T + 62742241 T^{2} \))(\( ( 1 + 10046 T + 62742241 T^{2} )^{2} \))
$97$ (\( 1 + 9982 T + 88529281 T^{2} \))(\( ( 1 - 8738 T + 88529281 T^{2} )^{2} \))
show more
show less