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Properties

Label 750.2
Level 750
Weight 2
Dimension 3456
Nonzero newspaces 9
Newform subspaces 36
Sturm bound 60000
Trace bound 4

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Defining parameters

Level: \( N \) = \( 750 = 2 \cdot 3 \cdot 5^{3} \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 9 \)
Newform subspaces: \( 36 \)
Sturm bound: \(60000\)
Trace bound: \(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(750))\).

Total New Old
Modular forms 15720 3456 12264
Cusp forms 14281 3456 10825
Eisenstein series 1439 0 1439

Trace form

\( 3456 q - 2 q^{2} - 4 q^{3} - 2 q^{4} - 4 q^{6} - 16 q^{7} - 2 q^{8} - 2 q^{9} + O(q^{10}) \) \( 3456 q - 2 q^{2} - 4 q^{3} - 2 q^{4} - 4 q^{6} - 16 q^{7} - 2 q^{8} - 2 q^{9} - 8 q^{11} + 4 q^{12} - 12 q^{13} - 2 q^{16} + 68 q^{17} + 26 q^{18} + 136 q^{19} + 10 q^{20} + 72 q^{21} + 144 q^{22} + 136 q^{23} + 36 q^{24} + 120 q^{25} + 52 q^{26} - 4 q^{27} + 64 q^{28} + 124 q^{29} + 40 q^{30} + 112 q^{31} + 18 q^{32} + 64 q^{33} + 80 q^{34} + 40 q^{35} - 2 q^{36} + 8 q^{37} - 16 q^{38} + 80 q^{39} + 36 q^{41} + 136 q^{43} - 16 q^{44} + 80 q^{45} + 112 q^{47} + 4 q^{48} + 138 q^{49} + 96 q^{51} - 12 q^{52} + 120 q^{53} - 4 q^{54} + 80 q^{55} - 8 q^{56} + 120 q^{57} - 44 q^{58} + 80 q^{59} - 20 q^{60} - 12 q^{61} - 40 q^{62} - 136 q^{63} - 2 q^{64} + 10 q^{65} - 120 q^{66} - 88 q^{67} - 12 q^{68} - 312 q^{69} - 48 q^{71} - 10 q^{72} - 92 q^{73} - 36 q^{74} - 160 q^{75} - 8 q^{76} - 96 q^{77} - 216 q^{78} - 72 q^{79} - 130 q^{81} - 52 q^{82} + 8 q^{83} - 128 q^{84} + 50 q^{85} - 32 q^{86} - 136 q^{87} - 16 q^{88} - 32 q^{89} - 60 q^{90} + 64 q^{91} - 24 q^{92} + 24 q^{93} - 64 q^{94} + 80 q^{95} + 4 q^{96} - 4 q^{97} - 66 q^{98} - 16 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(750))\)

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
750.2.a \(\chi_{750}(1, \cdot)\) 750.2.a.a 2 1
750.2.a.b 2
750.2.a.c 2
750.2.a.d 2
750.2.a.e 2
750.2.a.f 2
750.2.a.g 2
750.2.a.h 2
750.2.c \(\chi_{750}(499, \cdot)\) 750.2.c.a 4 1
750.2.c.b 4
750.2.c.c 4
750.2.c.d 4
750.2.e \(\chi_{750}(443, \cdot)\) 750.2.e.a 32 2
750.2.e.b 32
750.2.g \(\chi_{750}(151, \cdot)\) 750.2.g.a 4 4
750.2.g.b 4
750.2.g.c 8
750.2.g.d 8
750.2.g.e 8
750.2.g.f 16
750.2.g.g 16
750.2.h \(\chi_{750}(49, \cdot)\) 750.2.h.a 8 4
750.2.h.b 8
750.2.h.c 8
750.2.h.d 16
750.2.h.e 16
750.2.l \(\chi_{750}(107, \cdot)\) 750.2.l.a 80 8
750.2.l.b 80
750.2.l.c 80
750.2.m \(\chi_{750}(31, \cdot)\) 750.2.m.a 100 20
750.2.m.b 120
750.2.m.c 120
750.2.m.d 140
750.2.o \(\chi_{750}(19, \cdot)\) 750.2.o.a 240 20
750.2.o.b 280
750.2.r \(\chi_{750}(17, \cdot)\) 750.2.r.a 2000 40

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(750))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(750)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(25))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(30))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(75))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(125))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(150))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(250))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(375))\)\(^{\oplus 2}\)