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Properties

Label 408.2
Level 408
Weight 2
Dimension 1870
Nonzero newspaces 15
Newform subspaces 42
Sturm bound 18432
Trace bound 6

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Defining parameters

Level: \( N \) = \( 408 = 2^{3} \cdot 3 \cdot 17 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 15 \)
Newform subspaces: \( 42 \)
Sturm bound: \(18432\)
Trace bound: \(6\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(408))\).

Total New Old
Modular forms 4992 1990 3002
Cusp forms 4225 1870 2355
Eisenstein series 767 120 647

Trace form

\( 1870 q + 4 q^{2} - 10 q^{3} - 24 q^{4} + 4 q^{5} - 20 q^{6} - 24 q^{7} - 8 q^{8} - 26 q^{9} + O(q^{10}) \) \( 1870 q + 4 q^{2} - 10 q^{3} - 24 q^{4} + 4 q^{5} - 20 q^{6} - 24 q^{7} - 8 q^{8} - 26 q^{9} - 40 q^{10} - 8 q^{11} - 32 q^{12} + 4 q^{13} - 8 q^{14} - 28 q^{15} - 32 q^{16} + 2 q^{17} - 20 q^{18} - 32 q^{19} + 16 q^{20} - 16 q^{22} + 8 q^{24} - 6 q^{25} + 16 q^{26} - 34 q^{27} - 32 q^{28} + 28 q^{29} - 8 q^{30} + 8 q^{31} - 16 q^{32} - 8 q^{33} - 52 q^{34} + 64 q^{35} - 24 q^{36} + 52 q^{37} - 16 q^{38} + 28 q^{39} - 48 q^{40} + 44 q^{41} - 24 q^{42} + 48 q^{43} + 12 q^{45} - 16 q^{46} + 48 q^{47} - 48 q^{48} - 66 q^{49} + 4 q^{50} + 2 q^{51} - 96 q^{52} - 20 q^{53} - 108 q^{54} - 112 q^{55} - 144 q^{56} - 72 q^{57} - 232 q^{58} - 104 q^{59} - 224 q^{60} - 92 q^{61} - 216 q^{62} - 168 q^{63} - 192 q^{64} - 64 q^{65} - 240 q^{66} - 208 q^{67} - 352 q^{68} - 112 q^{69} - 368 q^{70} - 160 q^{71} - 248 q^{72} - 92 q^{73} - 224 q^{74} - 182 q^{75} - 208 q^{76} - 96 q^{77} - 240 q^{78} - 152 q^{79} - 224 q^{80} - 34 q^{81} - 120 q^{82} - 88 q^{83} - 80 q^{84} - 20 q^{85} + 16 q^{86} - 28 q^{87} - 64 q^{88} + 52 q^{89} - 88 q^{90} - 32 q^{91} - 48 q^{93} - 80 q^{94} + 16 q^{95} - 64 q^{96} - 20 q^{97} - 12 q^{98} - 64 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(408))\)

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
408.2.a \(\chi_{408}(1, \cdot)\) 408.2.a.a 1 1
408.2.a.b 1
408.2.a.c 1
408.2.a.d 1
408.2.a.e 2
408.2.a.f 2
408.2.c \(\chi_{408}(169, \cdot)\) 408.2.c.a 4 1
408.2.c.b 6
408.2.e \(\chi_{408}(239, \cdot)\) None 0 1
408.2.f \(\chi_{408}(205, \cdot)\) 408.2.f.a 2 1
408.2.f.b 4
408.2.f.c 12
408.2.f.d 14
408.2.h \(\chi_{408}(203, \cdot)\) 408.2.h.a 2 1
408.2.h.b 2
408.2.h.c 64
408.2.j \(\chi_{408}(35, \cdot)\) 408.2.j.a 4 1
408.2.j.b 4
408.2.j.c 4
408.2.j.d 4
408.2.j.e 48
408.2.l \(\chi_{408}(373, \cdot)\) 408.2.l.a 18 1
408.2.l.b 18
408.2.o \(\chi_{408}(407, \cdot)\) None 0 1
408.2.q \(\chi_{408}(251, \cdot)\) 408.2.q.a 4 2
408.2.q.b 4
408.2.q.c 128
408.2.s \(\chi_{408}(13, \cdot)\) 408.2.s.a 4 2
408.2.s.b 68
408.2.v \(\chi_{408}(217, \cdot)\) 408.2.v.a 4 2
408.2.v.b 4
408.2.v.c 12
408.2.x \(\chi_{408}(47, \cdot)\) None 0 2
408.2.ba \(\chi_{408}(25, \cdot)\) 408.2.ba.a 16 4
408.2.ba.b 16
408.2.bb \(\chi_{408}(263, \cdot)\) None 0 4
408.2.bc \(\chi_{408}(229, \cdot)\) 408.2.bc.a 144 4
408.2.bd \(\chi_{408}(59, \cdot)\) 408.2.bd.a 4 4
408.2.bd.b 4
408.2.bd.c 4
408.2.bd.d 4
408.2.bd.e 256
408.2.bh \(\chi_{408}(41, \cdot)\) 408.2.bh.a 72 8
408.2.bh.b 72
408.2.bi \(\chi_{408}(7, \cdot)\) None 0 8
408.2.bl \(\chi_{408}(91, \cdot)\) 408.2.bl.a 288 8
408.2.bm \(\chi_{408}(5, \cdot)\) 408.2.bm.a 544 8

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(408))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(408)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(17))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(24))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(34))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(51))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(68))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(102))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(136))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(204))\)\(^{\oplus 2}\)