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

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

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 the 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

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