Properties

Label 414.2
Level 414
Weight 2
Dimension 1250
Nonzero newspaces 8
Newform subspaces 26
Sturm bound 19008
Trace bound 3

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Defining parameters

Level: \( N \) = \( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 8 \)
Newform subspaces: \( 26 \)
Sturm bound: \(19008\)
Trace bound: \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(414))\).

Total New Old
Modular forms 5104 1250 3854
Cusp forms 4401 1250 3151
Eisenstein series 703 0 703

Trace form

\( 1250 q + 2 q^{2} + 6 q^{3} + 2 q^{4} - 6 q^{6} + 4 q^{7} - 4 q^{8} - 6 q^{9} + O(q^{10}) \) \( 1250 q + 2 q^{2} + 6 q^{3} + 2 q^{4} - 6 q^{6} + 4 q^{7} - 4 q^{8} - 6 q^{9} - 6 q^{11} + 4 q^{13} + 4 q^{14} + 2 q^{16} + 34 q^{17} + 12 q^{18} + 26 q^{19} + 22 q^{20} - 12 q^{21} + 16 q^{22} + 38 q^{23} + 6 q^{24} + 34 q^{25} + 14 q^{26} + 14 q^{28} + 34 q^{29} + 14 q^{31} + 2 q^{32} + 18 q^{33} - 6 q^{34} + 22 q^{35} - 6 q^{36} + 60 q^{37} - 2 q^{38} + 40 q^{41} + 42 q^{43} + 12 q^{44} + 12 q^{46} + 32 q^{47} - 6 q^{48} + 60 q^{49} - 10 q^{50} - 18 q^{51} + 4 q^{52} - 26 q^{53} - 62 q^{54} - 88 q^{55} - 62 q^{56} - 138 q^{57} - 120 q^{58} - 236 q^{59} - 44 q^{60} - 248 q^{61} - 182 q^{62} - 196 q^{63} - 4 q^{64} - 462 q^{65} - 176 q^{66} - 122 q^{67} - 138 q^{68} - 220 q^{69} - 264 q^{70} - 282 q^{71} - 94 q^{72} - 176 q^{73} - 272 q^{74} - 278 q^{75} - 2 q^{76} - 342 q^{77} - 120 q^{78} - 250 q^{79} - 66 q^{80} - 158 q^{81} - 168 q^{82} - 152 q^{83} - 32 q^{84} - 110 q^{85} - 68 q^{86} - 36 q^{87} - 6 q^{88} - 2 q^{89} + 28 q^{91} - 6 q^{92} - 12 q^{94} + 66 q^{95} + 98 q^{97} + 56 q^{98} - 36 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(414))\)

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
414.2.a \(\chi_{414}(1, \cdot)\) 414.2.a.a 1 1
414.2.a.b 1
414.2.a.c 1
414.2.a.d 1
414.2.a.e 2
414.2.a.f 2
414.2.a.g 2
414.2.d \(\chi_{414}(413, \cdot)\) 414.2.d.a 8 1
414.2.e \(\chi_{414}(139, \cdot)\) 414.2.e.a 2 2
414.2.e.b 10
414.2.e.c 10
414.2.e.d 10
414.2.e.e 12
414.2.f \(\chi_{414}(137, \cdot)\) 414.2.f.a 48 2
414.2.i \(\chi_{414}(55, \cdot)\) 414.2.i.a 10 10
414.2.i.b 10
414.2.i.c 10
414.2.i.d 10
414.2.i.e 10
414.2.i.f 10
414.2.i.g 20
414.2.i.h 20
414.2.j \(\chi_{414}(17, \cdot)\) 414.2.j.a 80 10
414.2.m \(\chi_{414}(13, \cdot)\) 414.2.m.a 240 20
414.2.m.b 240
414.2.p \(\chi_{414}(5, \cdot)\) 414.2.p.a 480 20

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(414))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(414)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(23))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(46))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(69))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(138))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(207))\)\(^{\oplus 2}\)