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Properties

Label 422.2
Level 422
Weight 2
Dimension 1854
Nonzero newspaces 8
Newform subspaces 23
Sturm bound 22260
Trace bound 1

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Defining parameters

Level: \( N \) = \( 422 = 2 \cdot 211 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 8 \)
Newform subspaces: \( 23 \)
Sturm bound: \(22260\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(422))\).

Total New Old
Modular forms 5775 1854 3921
Cusp forms 5356 1854 3502
Eisenstein series 419 0 419

Trace form

\( 1854 q - q^{2} - 4 q^{3} - q^{4} - 6 q^{5} - 4 q^{6} - 8 q^{7} - q^{8} - 13 q^{9} + O(q^{10}) \) \( 1854 q - q^{2} - 4 q^{3} - q^{4} - 6 q^{5} - 4 q^{6} - 8 q^{7} - q^{8} - 13 q^{9} - 6 q^{10} - 12 q^{11} - 4 q^{12} - 14 q^{13} - 8 q^{14} - 24 q^{15} - q^{16} - 18 q^{17} - 13 q^{18} - 20 q^{19} - 6 q^{20} - 32 q^{21} - 12 q^{22} - 24 q^{23} - 4 q^{24} - 31 q^{25} - 14 q^{26} - 40 q^{27} - 8 q^{28} - 30 q^{29} - 24 q^{30} - 32 q^{31} - q^{32} - 48 q^{33} - 18 q^{34} - 48 q^{35} - 13 q^{36} - 38 q^{37} - 20 q^{38} - 56 q^{39} - 6 q^{40} - 42 q^{41} - 32 q^{42} - 44 q^{43} - 12 q^{44} - 78 q^{45} - 24 q^{46} - 48 q^{47} - 4 q^{48} - 57 q^{49} - 31 q^{50} - 72 q^{51} - 14 q^{52} - 54 q^{53} - 40 q^{54} - 72 q^{55} - 8 q^{56} - 80 q^{57} - 30 q^{58} - 60 q^{59} - 24 q^{60} - 62 q^{61} - 32 q^{62} - 104 q^{63} - q^{64} - 84 q^{65} - 48 q^{66} - 68 q^{67} - 18 q^{68} - 96 q^{69} - 48 q^{70} - 72 q^{71} - 13 q^{72} - 74 q^{73} - 38 q^{74} - 124 q^{75} - 20 q^{76} - 96 q^{77} - 56 q^{78} - 80 q^{79} - 6 q^{80} - 121 q^{81} - 42 q^{82} - 84 q^{83} - 32 q^{84} - 108 q^{85} - 44 q^{86} - 120 q^{87} - 12 q^{88} - 90 q^{89} - 78 q^{90} - 112 q^{91} - 24 q^{92} - 128 q^{93} - 48 q^{94} - 120 q^{95} - 4 q^{96} - 98 q^{97} - 57 q^{98} - 156 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(422))\)

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
422.2.a \(\chi_{422}(1, \cdot)\) 422.2.a.a 1 1
422.2.a.b 2
422.2.a.c 3
422.2.a.d 3
422.2.a.e 3
422.2.a.f 6
422.2.c \(\chi_{422}(225, \cdot)\) 422.2.c.a 4 2
422.2.c.b 4
422.2.c.c 8
422.2.c.d 18
422.2.d \(\chi_{422}(55, \cdot)\) 422.2.d.a 16 4
422.2.d.b 20
422.2.d.c 40
422.2.f \(\chi_{422}(123, \cdot)\) 422.2.f.a 54 6
422.2.f.b 60
422.2.i \(\chi_{422}(19, \cdot)\) 422.2.i.a 64 8
422.2.i.b 72
422.2.j \(\chi_{422}(43, \cdot)\) 422.2.j.a 96 12
422.2.j.b 108
422.2.l \(\chi_{422}(5, \cdot)\) 422.2.l.a 216 24
422.2.l.b 240
422.2.o \(\chi_{422}(9, \cdot)\) 422.2.o.a 384 48
422.2.o.b 432

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(422))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(422)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(211))\)\(^{\oplus 2}\)