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Properties

Label 312.4
Level 312
Weight 4
Dimension 3320
Nonzero newspaces 18
Sturm bound 21504
Trace bound 10

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Defining parameters

Level: \( N \) = \( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 18 \)
Sturm bound: \(21504\)
Trace bound: \(10\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(312))\).

Total New Old
Modular forms 8352 3408 4944
Cusp forms 7776 3320 4456
Eisenstein series 576 88 488

Trace form

\( 3320 q - 4 q^{2} - 14 q^{3} - 64 q^{4} - 28 q^{5} + 16 q^{6} - 32 q^{7} + 152 q^{8} + 70 q^{9} + O(q^{10}) \) \( 3320 q - 4 q^{2} - 14 q^{3} - 64 q^{4} - 28 q^{5} + 16 q^{6} - 32 q^{7} + 152 q^{8} + 70 q^{9} - 96 q^{10} + 56 q^{11} + 100 q^{12} + 74 q^{13} + 200 q^{14} + 24 q^{15} + 168 q^{16} + 14 q^{17} - 344 q^{18} + 256 q^{19} - 112 q^{20} + 24 q^{21} - 920 q^{22} - 1128 q^{23} - 1076 q^{24} - 820 q^{25} - 56 q^{26} - 638 q^{27} - 376 q^{28} - 114 q^{29} + 492 q^{30} + 1208 q^{31} + 496 q^{32} + 1016 q^{33} + 2640 q^{34} + 2232 q^{35} + 2044 q^{36} + 582 q^{37} + 1552 q^{38} - 1362 q^{39} - 5792 q^{40} - 2098 q^{41} - 4332 q^{42} - 2904 q^{43} - 2904 q^{44} - 90 q^{45} + 1480 q^{46} + 2568 q^{47} - 860 q^{48} - 210 q^{49} + 6644 q^{50} + 1652 q^{51} + 9248 q^{52} + 3788 q^{53} + 4216 q^{54} + 4328 q^{55} + 10424 q^{56} - 220 q^{57} + 6656 q^{58} - 1552 q^{59} + 5052 q^{60} - 3074 q^{61} - 2864 q^{62} - 3240 q^{63} - 12040 q^{64} + 422 q^{65} - 10360 q^{66} - 416 q^{67} - 5952 q^{68} + 132 q^{69} - 13120 q^{70} + 1696 q^{71} - 7940 q^{72} - 388 q^{73} - 3136 q^{74} + 7414 q^{75} - 3752 q^{76} - 1344 q^{77} + 2076 q^{78} + 104 q^{79} + 4224 q^{80} + 1654 q^{81} + 10720 q^{82} + 3016 q^{83} + 16536 q^{84} - 6598 q^{85} + 21008 q^{86} - 14064 q^{87} + 21568 q^{88} - 6772 q^{89} + 5184 q^{90} - 15144 q^{91} + 720 q^{92} - 6852 q^{93} - 16296 q^{94} - 5344 q^{95} - 23012 q^{96} + 7604 q^{97} - 22476 q^{98} + 8204 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(312))\)

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
312.4.a \(\chi_{312}(1, \cdot)\) 312.4.a.a 2 1
312.4.a.b 2
312.4.a.c 2
312.4.a.d 2
312.4.a.e 2
312.4.a.f 2
312.4.a.g 3
312.4.a.h 3
312.4.c \(\chi_{312}(25, \cdot)\) 312.4.c.a 10 1
312.4.c.b 12
312.4.d \(\chi_{312}(287, \cdot)\) None 0 1
312.4.g \(\chi_{312}(157, \cdot)\) 312.4.g.a 32 1
312.4.g.b 40
312.4.h \(\chi_{312}(155, \cdot)\) n/a 164 1
312.4.j \(\chi_{312}(131, \cdot)\) n/a 144 1
312.4.m \(\chi_{312}(181, \cdot)\) 312.4.m.a 84 1
312.4.n \(\chi_{312}(311, \cdot)\) None 0 1
312.4.q \(\chi_{312}(217, \cdot)\) 312.4.q.a 4 2
312.4.q.b 6
312.4.q.c 8
312.4.q.d 10
312.4.q.e 12
312.4.t \(\chi_{312}(187, \cdot)\) n/a 168 2
312.4.u \(\chi_{312}(31, \cdot)\) None 0 2
312.4.x \(\chi_{312}(161, \cdot)\) 312.4.x.a 84 2
312.4.y \(\chi_{312}(5, \cdot)\) n/a 328 2
312.4.ba \(\chi_{312}(179, \cdot)\) n/a 328 2
312.4.bb \(\chi_{312}(61, \cdot)\) n/a 168 2
312.4.be \(\chi_{312}(191, \cdot)\) None 0 2
312.4.bf \(\chi_{312}(49, \cdot)\) 312.4.bf.a 20 2
312.4.bf.b 24
312.4.bj \(\chi_{312}(23, \cdot)\) None 0 2
312.4.bk \(\chi_{312}(205, \cdot)\) n/a 168 2
312.4.bn \(\chi_{312}(35, \cdot)\) n/a 328 2
312.4.bo \(\chi_{312}(149, \cdot)\) n/a 656 4
312.4.bp \(\chi_{312}(41, \cdot)\) n/a 168 4
312.4.bs \(\chi_{312}(7, \cdot)\) None 0 4
312.4.bt \(\chi_{312}(19, \cdot)\) n/a 336 4

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

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(312))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(312)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(13))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(24))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(26))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(39))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(52))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(78))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(104))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(156))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(312))\)\(^{\oplus 1}\)