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Properties

Label 304.8
Level 304
Weight 8
Dimension 11201
Nonzero newspaces 12
Sturm bound 46080
Trace bound 7

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Defining parameters

Level: \( N \) = \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) = \( 8 \)
Nonzero newspaces: \( 12 \)
Sturm bound: \(46080\)
Trace bound: \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(304))\).

Total New Old
Modular forms 20412 11353 9059
Cusp forms 19908 11201 8707
Eisenstein series 504 152 352

Trace form

\( 11201 q - 32 q^{2} - 79 q^{3} + 332 q^{4} - 317 q^{5} + 316 q^{6} - 1355 q^{7} + 1972 q^{8} - 5159 q^{9} + O(q^{10}) \) \( 11201 q - 32 q^{2} - 79 q^{3} + 332 q^{4} - 317 q^{5} + 316 q^{6} - 1355 q^{7} + 1972 q^{8} - 5159 q^{9} - 25980 q^{10} - 11623 q^{11} + 54676 q^{12} + 6515 q^{13} - 44572 q^{14} + 100557 q^{15} - 26708 q^{16} + 25355 q^{17} + 58656 q^{18} - 101901 q^{19} - 92480 q^{20} - 2069 q^{21} + 88660 q^{22} + 303957 q^{23} + 118396 q^{24} - 2707 q^{25} + 71884 q^{26} - 299323 q^{27} - 440660 q^{28} - 136797 q^{29} - 55308 q^{30} + 161861 q^{31} + 497228 q^{32} - 171881 q^{33} - 1849356 q^{34} - 900595 q^{35} + 2030980 q^{36} - 208694 q^{37} + 5436 q^{38} + 2382202 q^{39} + 2243916 q^{40} + 334299 q^{41} - 249236 q^{42} - 2360311 q^{43} + 1517204 q^{44} - 147477 q^{45} - 4671356 q^{46} + 4302373 q^{47} - 8282612 q^{48} + 2758621 q^{49} + 1574352 q^{50} - 4424659 q^{51} - 820684 q^{52} - 3045997 q^{53} + 14808988 q^{54} - 2765771 q^{55} + 8186284 q^{56} + 1066631 q^{57} + 3721352 q^{58} + 11284425 q^{59} - 10294692 q^{60} - 6837497 q^{61} - 12548228 q^{62} + 1833765 q^{63} - 20904484 q^{64} - 4341929 q^{65} - 21875660 q^{66} - 7675675 q^{67} + 14807484 q^{68} - 46413 q^{69} + 44281564 q^{70} - 15466545 q^{71} + 50587860 q^{72} + 41566663 q^{73} + 26354052 q^{74} + 61306068 q^{75} - 17814296 q^{76} + 7120954 q^{77} - 75987868 q^{78} + 8748775 q^{79} - 80094580 q^{80} - 49309599 q^{81} - 45716516 q^{82} - 45953501 q^{83} + 60319372 q^{84} - 30318421 q^{85} + 108258900 q^{86} + 8112777 q^{87} + 118432748 q^{88} + 82430019 q^{89} + 67030572 q^{90} + 69930133 q^{91} - 43621908 q^{92} - 48849913 q^{93} - 154392260 q^{94} + 10595145 q^{95} - 233881000 q^{96} - 27096085 q^{97} - 43647768 q^{98} - 65082335 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(304))\)

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 available newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
304.8.a \(\chi_{304}(1, \cdot)\) 304.8.a.a 1 1
304.8.a.b 2
304.8.a.c 2
304.8.a.d 2
304.8.a.e 4
304.8.a.f 4
304.8.a.g 5
304.8.a.h 6
304.8.a.i 6
304.8.a.j 6
304.8.a.k 8
304.8.a.l 8
304.8.a.m 9
304.8.b \(\chi_{304}(151, \cdot)\) None 0 1
304.8.c \(\chi_{304}(153, \cdot)\) None 0 1
304.8.h \(\chi_{304}(303, \cdot)\) 304.8.h.a 2 1
304.8.h.b 4
304.8.h.c 20
304.8.h.d 44
304.8.i \(\chi_{304}(49, \cdot)\) n/a 138 2
304.8.k \(\chi_{304}(77, \cdot)\) n/a 504 2
304.8.m \(\chi_{304}(75, \cdot)\) n/a 556 2
304.8.n \(\chi_{304}(31, \cdot)\) n/a 140 2
304.8.s \(\chi_{304}(103, \cdot)\) None 0 2
304.8.t \(\chi_{304}(121, \cdot)\) None 0 2
304.8.u \(\chi_{304}(17, \cdot)\) n/a 414 6
304.8.v \(\chi_{304}(45, \cdot)\) n/a 1112 4
304.8.x \(\chi_{304}(27, \cdot)\) n/a 1112 4
304.8.bb \(\chi_{304}(9, \cdot)\) None 0 6
304.8.bd \(\chi_{304}(71, \cdot)\) None 0 6
304.8.be \(\chi_{304}(15, \cdot)\) n/a 420 6
304.8.bg \(\chi_{304}(3, \cdot)\) n/a 3336 12
304.8.bi \(\chi_{304}(5, \cdot)\) n/a 3336 12

"n/a" means that newforms for that character have not been added to the database yet

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_1(304))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_1(304)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(19))\)\(^{\oplus 5}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(38))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(76))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(152))\)\(^{\oplus 2}\)