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Properties

Label 370.4
Level 370
Weight 4
Dimension 3756
Nonzero newspaces 18
Sturm bound 32832
Trace bound 7

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Defining parameters

Level: \( N \) = \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 18 \)
Sturm bound: \(32832\)
Trace bound: \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(370))\).

Total New Old
Modular forms 12600 3756 8844
Cusp forms 12024 3756 8268
Eisenstein series 576 0 576

Trace form

\( 3756 q - 4 q^{2} + 16 q^{3} + 8 q^{4} + 10 q^{5} + 16 q^{6} + 8 q^{7} - 16 q^{8} - 166 q^{9} + O(q^{10}) \) \( 3756 q - 4 q^{2} + 16 q^{3} + 8 q^{4} + 10 q^{5} + 16 q^{6} + 8 q^{7} - 16 q^{8} - 166 q^{9} - 100 q^{10} + 88 q^{11} + 64 q^{12} + 116 q^{13} + 224 q^{14} + 160 q^{15} - 96 q^{16} - 132 q^{17} - 148 q^{18} - 40 q^{19} - 120 q^{20} - 272 q^{21} - 48 q^{22} - 264 q^{23} + 192 q^{24} + 250 q^{25} - 1196 q^{26} - 5024 q^{27} - 832 q^{28} - 900 q^{29} - 192 q^{30} + 1504 q^{31} - 64 q^{32} + 3648 q^{33} + 3544 q^{34} + 2672 q^{35} + 3312 q^{36} + 6082 q^{37} + 2848 q^{38} + 4208 q^{39} + 1320 q^{40} + 3364 q^{41} + 736 q^{42} - 928 q^{43} - 544 q^{44} - 3798 q^{45} - 7840 q^{46} - 4920 q^{47} - 2048 q^{48} - 9534 q^{49} - 614 q^{50} + 1568 q^{51} + 464 q^{52} - 444 q^{53} - 480 q^{54} - 680 q^{55} - 768 q^{56} - 1600 q^{57} + 360 q^{58} - 10784 q^{59} - 11442 q^{61} - 608 q^{62} - 7696 q^{63} + 128 q^{64} + 1141 q^{65} + 832 q^{66} + 4280 q^{67} - 528 q^{68} + 16912 q^{69} - 960 q^{70} + 13296 q^{71} - 592 q^{72} + 9788 q^{73} + 1012 q^{74} + 12380 q^{75} + 1760 q^{76} + 8880 q^{77} - 1856 q^{78} + 3632 q^{79} + 160 q^{80} + 3930 q^{81} + 1752 q^{82} - 1368 q^{83} + 576 q^{84} - 2879 q^{85} - 3024 q^{86} - 24984 q^{87} - 192 q^{88} - 18610 q^{89} - 2580 q^{90} - 24968 q^{91} - 17760 q^{92} - 39904 q^{93} - 20704 q^{94} - 16520 q^{95} + 256 q^{96} - 22948 q^{97} - 11940 q^{98} - 6232 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(370))\)

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
370.4.a \(\chi_{370}(1, \cdot)\) 370.4.a.a 1 1
370.4.a.b 1
370.4.a.c 1
370.4.a.d 3
370.4.a.e 4
370.4.a.f 5
370.4.a.g 5
370.4.a.h 5
370.4.a.i 5
370.4.a.j 6
370.4.b \(\chi_{370}(149, \cdot)\) 370.4.b.a 22 1
370.4.b.b 32
370.4.c \(\chi_{370}(369, \cdot)\) 370.4.c.a 28 1
370.4.c.b 28
370.4.d \(\chi_{370}(221, \cdot)\) 370.4.d.a 18 1
370.4.d.b 20
370.4.e \(\chi_{370}(121, \cdot)\) 370.4.e.a 2 2
370.4.e.b 16
370.4.e.c 18
370.4.e.d 20
370.4.e.e 20
370.4.g \(\chi_{370}(43, \cdot)\) n/a 114 2
370.4.h \(\chi_{370}(117, \cdot)\) n/a 114 2
370.4.l \(\chi_{370}(11, \cdot)\) 370.4.l.a 36 2
370.4.l.b 40
370.4.m \(\chi_{370}(159, \cdot)\) n/a 112 2
370.4.n \(\chi_{370}(269, \cdot)\) n/a 116 2
370.4.o \(\chi_{370}(71, \cdot)\) n/a 228 6
370.4.q \(\chi_{370}(97, \cdot)\) n/a 228 4
370.4.r \(\chi_{370}(23, \cdot)\) n/a 228 4
370.4.v \(\chi_{370}(99, \cdot)\) n/a 336 6
370.4.w \(\chi_{370}(21, \cdot)\) n/a 228 6
370.4.x \(\chi_{370}(9, \cdot)\) n/a 348 6
370.4.ba \(\chi_{370}(17, \cdot)\) n/a 684 12
370.4.bd \(\chi_{370}(13, \cdot)\) n/a 684 12

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

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(370))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(370)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(37))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(74))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(185))\)\(^{\oplus 2}\)