Properties

Label 240.2
Level 240
Weight 2
Dimension 566
Nonzero newspaces 14
Newform subspaces 40
Sturm bound 6144
Trace bound 13

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Defining parameters

Level: \( N \) = \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 14 \)
Newform subspaces: \( 40 \)
Sturm bound: \(6144\)
Trace bound: \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(240))\).

Total New Old
Modular forms 1760 622 1138
Cusp forms 1313 566 747
Eisenstein series 447 56 391

Trace form

\( 566 q - 4 q^{3} + 2 q^{5} + 4 q^{7} + 24 q^{8} + 10 q^{9} + O(q^{10}) \) \( 566 q - 4 q^{3} + 2 q^{5} + 4 q^{7} + 24 q^{8} + 10 q^{9} - 8 q^{10} + 24 q^{11} - 8 q^{12} + 8 q^{13} - 24 q^{14} + 16 q^{15} - 64 q^{16} + 12 q^{17} - 24 q^{18} + 40 q^{19} - 16 q^{20} - 12 q^{21} - 48 q^{22} + 8 q^{23} - 64 q^{24} - 2 q^{25} - 40 q^{26} - 16 q^{27} - 80 q^{28} + 12 q^{29} - 60 q^{30} - 72 q^{31} - 80 q^{32} - 28 q^{33} - 128 q^{34} - 48 q^{35} - 64 q^{36} - 32 q^{37} - 96 q^{38} - 68 q^{39} - 152 q^{40} - 52 q^{41} - 44 q^{43} - 32 q^{44} - 22 q^{45} - 64 q^{46} - 24 q^{47} + 56 q^{48} - 6 q^{49} - 64 q^{50} - 112 q^{51} - 16 q^{52} - 44 q^{53} + 88 q^{54} - 64 q^{55} - 12 q^{57} - 48 q^{58} - 56 q^{59} + 16 q^{60} - 148 q^{61} + 24 q^{62} - 128 q^{63} - 96 q^{64} + 4 q^{65} - 56 q^{66} - 164 q^{67} + 48 q^{68} - 112 q^{69} + 56 q^{70} - 144 q^{71} + 40 q^{72} - 56 q^{73} + 72 q^{74} - 144 q^{75} - 32 q^{76} - 80 q^{77} + 64 q^{78} - 168 q^{79} + 152 q^{80} - 66 q^{81} + 64 q^{82} - 104 q^{83} + 160 q^{84} - 40 q^{85} + 160 q^{86} - 8 q^{87} + 224 q^{88} + 52 q^{89} + 136 q^{90} + 40 q^{91} + 176 q^{92} + 24 q^{93} + 176 q^{94} - 16 q^{95} + 160 q^{96} + 88 q^{97} + 128 q^{98} + 48 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(240))\)

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
240.2.a \(\chi_{240}(1, \cdot)\) 240.2.a.a 1 1
240.2.a.b 1
240.2.a.c 1
240.2.a.d 1
240.2.b \(\chi_{240}(71, \cdot)\) None 0 1
240.2.d \(\chi_{240}(169, \cdot)\) None 0 1
240.2.f \(\chi_{240}(49, \cdot)\) 240.2.f.a 2 1
240.2.f.b 2
240.2.f.c 2
240.2.h \(\chi_{240}(191, \cdot)\) 240.2.h.a 4 1
240.2.h.b 4
240.2.k \(\chi_{240}(121, \cdot)\) None 0 1
240.2.m \(\chi_{240}(119, \cdot)\) None 0 1
240.2.o \(\chi_{240}(239, \cdot)\) 240.2.o.a 4 1
240.2.o.b 8
240.2.s \(\chi_{240}(61, \cdot)\) 240.2.s.a 4 2
240.2.s.b 8
240.2.s.c 20
240.2.t \(\chi_{240}(59, \cdot)\) 240.2.t.a 8 2
240.2.t.b 80
240.2.v \(\chi_{240}(17, \cdot)\) 240.2.v.a 4 2
240.2.v.b 4
240.2.v.c 4
240.2.v.d 4
240.2.v.e 4
240.2.w \(\chi_{240}(127, \cdot)\) 240.2.w.a 4 2
240.2.w.b 8
240.2.y \(\chi_{240}(163, \cdot)\) 240.2.y.a 2 2
240.2.y.b 2
240.2.y.c 2
240.2.y.d 6
240.2.y.e 16
240.2.y.f 20
240.2.bb \(\chi_{240}(173, \cdot)\) 240.2.bb.a 88 2
240.2.bc \(\chi_{240}(43, \cdot)\) 240.2.bc.a 2 2
240.2.bc.b 2
240.2.bc.c 2
240.2.bc.d 6
240.2.bc.e 16
240.2.bc.f 20
240.2.bf \(\chi_{240}(53, \cdot)\) 240.2.bf.a 88 2
240.2.bh \(\chi_{240}(7, \cdot)\) None 0 2
240.2.bi \(\chi_{240}(137, \cdot)\) None 0 2
240.2.bk \(\chi_{240}(11, \cdot)\) 240.2.bk.a 4 2
240.2.bk.b 60
240.2.bl \(\chi_{240}(109, \cdot)\) 240.2.bl.a 48 2

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(240))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(240)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 20}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(48))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(60))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(80))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(120))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(240))\)\(^{\oplus 1}\)