Properties

Label 8.7
Level 8
Weight 7
Dimension 5
Nonzero newspaces 1
Newform subspaces 2
Sturm bound 28
Trace bound 0

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Defining parameters

Level: \( N \) = \( 8\( 8 = 2^{3} \) \)
Weight: \( k \) = \( 7 \)
Nonzero newspaces: \( 1 \)
Newform subspaces: \( 2 \)
Sturm bound: \(28\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(\Gamma_1(8))\).

Total New Old
Modular forms 15 7 8
Cusp forms 9 5 4
Eisenstein series 6 2 4

Trace form

\( 5q - 6q^{2} - 2q^{3} + 20q^{4} + 28q^{6} - 264q^{8} + 727q^{9} + O(q^{10}) \) \( 5q - 6q^{2} - 2q^{3} + 20q^{4} + 28q^{6} - 264q^{8} + 727q^{9} - 1920q^{10} - 1362q^{11} + 4312q^{12} + 5760q^{14} - 10480q^{16} + 2442q^{17} - 21506q^{18} - 3938q^{19} + 31680q^{20} + 43132q^{22} - 61808q^{24} - 8275q^{25} - 59520q^{26} + 32860q^{27} + 59520q^{28} + 90240q^{30} - 81696q^{32} - 23500q^{33} - 68108q^{34} - 49920q^{35} + 75868q^{36} + 46428q^{38} - 13440q^{40} + 16698q^{41} + 38400q^{42} + 122542q^{43} - 112488q^{44} - 213120q^{46} + 326368q^{48} + 119765q^{49} + 232650q^{50} - 465412q^{51} - 254400q^{52} - 495368q^{54} + 349440q^{56} + 12500q^{57} + 516480q^{58} + 846990q^{59} - 716160q^{60} - 407040q^{62} + 726080q^{64} - 205440q^{65} + 717608q^{66} - 1386818q^{67} - 324312q^{68} - 360960q^{70} + 95656q^{72} - 149222q^{73} - 32640q^{74} + 2483950q^{75} - 88232q^{76} + 324480q^{78} - 1032960q^{80} - 186839q^{81} - 672428q^{82} - 2786082q^{83} + 602880q^{84} + 1588860q^{86} - 753392q^{88} + 403962q^{89} - 1296000q^{90} + 3398400q^{91} + 2743680q^{92} + 971520q^{94} - 1760192q^{96} + 895978q^{97} - 3332454q^{98} - 5702086q^{99} + O(q^{100}) \)

Decomposition of \(S_{7}^{\mathrm{new}}(\Gamma_1(8))\)

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
8.7.c \(\chi_{8}(7, \cdot)\) None 0 1
8.7.d \(\chi_{8}(3, \cdot)\) 8.7.d.a 1 1
8.7.d.b 4

Decomposition of \(S_{7}^{\mathrm{old}}(\Gamma_1(8))\) into lower level spaces

\( S_{7}^{\mathrm{old}}(\Gamma_1(8)) \cong \) \(S_{7}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 8 T \))(\( 1 - 2 T + 24 T^{2} - 128 T^{3} + 4096 T^{4} \))
$3$ (\( 1 - 46 T + 729 T^{2} \))(\( ( 1 + 24 T + 1182 T^{2} + 17496 T^{3} + 531441 T^{4} )^{2} \))
$5$ (\( ( 1 - 125 T )( 1 + 125 T ) \))(\( 1 - 19300 T^{2} + 256155750 T^{4} - 4711914062500 T^{6} + 59604644775390625 T^{8} \))
$7$ (\( ( 1 - 343 T )( 1 + 343 T ) \))(\( 1 - 236356 T^{2} + 28021959366 T^{4} - 3271471277679556 T^{6} + \)\(19\!\cdots\!01\)\( T^{8} \))
$11$ (\( 1 + 2338 T + 1771561 T^{2} \))(\( ( 1 - 488 T + 2238078 T^{2} - 864521768 T^{3} + 3138428376721 T^{4} )^{2} \))
$13$ (\( ( 1 - 2197 T )( 1 + 2197 T ) \))(\( 1 - 4994596 T^{2} + 30260873415846 T^{4} - \)\(11\!\cdots\!76\)\( T^{6} + \)\(54\!\cdots\!61\)\( T^{8} \))
$17$ (\( 1 + 1726 T + 24137569 T^{2} \))(\( ( 1 - 2084 T + 32560902 T^{2} - 50302693796 T^{3} + 582622237229761 T^{4} )^{2} \))
$19$ (\( 1 + 2482 T + 47045881 T^{2} \))(\( ( 1 + 728 T + 92449758 T^{2} + 34249401368 T^{3} + 2213314919066161 T^{4} )^{2} \))
$23$ (\( ( 1 - 12167 T )( 1 + 12167 T ) \))(\( 1 - 212117956 T^{2} + 23026671552237126 T^{4} - \)\(46\!\cdots\!76\)\( T^{6} + \)\(48\!\cdots\!41\)\( T^{8} \))
$29$ (\( ( 1 - 24389 T )( 1 + 24389 T ) \))(\( 1 - 1409719204 T^{2} + 1160515330165289766 T^{4} - \)\(49\!\cdots\!64\)\( T^{6} + \)\(12\!\cdots\!81\)\( T^{8} \))
$31$ (\( ( 1 - 29791 T )( 1 + 29791 T ) \))(\( 1 - 2758360324 T^{2} + 3362667870952277766 T^{4} - \)\(21\!\cdots\!64\)\( T^{6} + \)\(62\!\cdots\!21\)\( T^{8} \))
$37$ (\( ( 1 - 50653 T )( 1 + 50653 T ) \))(\( 1 - 8121202276 T^{2} + 29600495645847907686 T^{4} - \)\(53\!\cdots\!56\)\( T^{6} + \)\(43\!\cdots\!61\)\( T^{8} \))
$41$ (\( 1 - 134642 T + 4750104241 T^{2} \))(\( ( 1 + 58972 T + 9857729958 T^{2} + 280123147300252 T^{3} + 22563490300366186081 T^{4} )^{2} \))
$43$ (\( 1 + 74914 T + 6321363049 T^{2} \))(\( ( 1 - 98728 T + 13190101374 T^{2} - 624095531101672 T^{3} + 39959630797262576401 T^{4} )^{2} \))
$47$ (\( ( 1 - 103823 T )( 1 + 103823 T ) \))(\( 1 - 13578767236 T^{2} + \)\(16\!\cdots\!86\)\( T^{4} - \)\(15\!\cdots\!76\)\( T^{6} + \)\(13\!\cdots\!81\)\( T^{8} \))
$53$ (\( ( 1 - 148877 T )( 1 + 148877 T ) \))(\( 1 - 42194545636 T^{2} + \)\(11\!\cdots\!86\)\( T^{4} - \)\(20\!\cdots\!76\)\( T^{6} + \)\(24\!\cdots\!81\)\( T^{8} \))
$59$ (\( 1 - 304958 T + 42180533641 T^{2} \))(\( ( 1 - 271016 T + 102678575166 T^{2} - 11431599505249256 T^{3} + \)\(17\!\cdots\!81\)\( T^{4} )^{2} \))
$61$ (\( ( 1 - 226981 T )( 1 + 226981 T ) \))(\( 1 - 160868902564 T^{2} + \)\(11\!\cdots\!46\)\( T^{4} - \)\(42\!\cdots\!44\)\( T^{6} + \)\(70\!\cdots\!41\)\( T^{8} \))
$67$ (\( 1 + 596626 T + 90458382169 T^{2} \))(\( ( 1 + 395096 T + 204987839262 T^{2} + 35739744961443224 T^{3} + \)\(81\!\cdots\!61\)\( T^{4} )^{2} \))
$71$ (\( ( 1 - 357911 T )( 1 + 357911 T ) \))(\( 1 - 164364621124 T^{2} + \)\(21\!\cdots\!06\)\( T^{4} - \)\(26\!\cdots\!84\)\( T^{6} + \)\(26\!\cdots\!81\)\( T^{8} \))
$73$ (\( 1 + 593134 T + 151334226289 T^{2} \))(\( ( 1 - 221956 T + 284381984742 T^{2} - 33589539530201284 T^{3} + \)\(22\!\cdots\!21\)\( T^{4} )^{2} \))
$79$ (\( ( 1 - 493039 T )( 1 + 493039 T ) \))(\( 1 - 828257339524 T^{2} + \)\(28\!\cdots\!06\)\( T^{4} - \)\(48\!\cdots\!84\)\( T^{6} + \)\(34\!\cdots\!81\)\( T^{8} \))
$83$ (\( 1 - 678926 T + 326940373369 T^{2} \))(\( ( 1 + 1732504 T + 1400265667422 T^{2} + 566425504623285976 T^{3} + \)\(10\!\cdots\!61\)\( T^{4} )^{2} \))
$89$ (\( 1 + 357262 T + 496981290961 T^{2} \))(\( ( 1 - 380612 T + 970422538278 T^{2} - 189157043115248132 T^{3} + \)\(24\!\cdots\!21\)\( T^{4} )^{2} \))
$97$ (\( 1 - 1822754 T + 832972004929 T^{2} \))(\( ( 1 + 463388 T + 953987784774 T^{2} + 385989231420039452 T^{3} + \)\(69\!\cdots\!41\)\( T^{4} )^{2} \))
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