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Properties

Label 406.2
Level 406
Weight 2
Dimension 1609
Nonzero newspaces 12
Newform subspaces 29
Sturm bound 20160
Trace bound 4

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Defining parameters

Level: \( N \) = \( 406 = 2 \cdot 7 \cdot 29 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 12 \)
Newform subspaces: \( 29 \)
Sturm bound: \(20160\)
Trace bound: \(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(406))\).

Total New Old
Modular forms 5376 1609 3767
Cusp forms 4705 1609 3096
Eisenstein series 671 0 671

Trace form

\( 1609 q + 3 q^{2} + 8 q^{3} - q^{4} + 6 q^{5} - q^{7} + 3 q^{8} + 11 q^{9} + O(q^{10}) \) \( 1609 q + 3 q^{2} + 8 q^{3} - q^{4} + 6 q^{5} - q^{7} + 3 q^{8} + 11 q^{9} + 6 q^{10} + 12 q^{11} + 8 q^{12} + 22 q^{13} + 3 q^{14} + 24 q^{15} - q^{16} + 6 q^{17} + 15 q^{18} + 16 q^{19} - 8 q^{20} - 48 q^{21} - 44 q^{22} - 32 q^{23} - 56 q^{24} - 71 q^{25} - 64 q^{26} - 136 q^{27} - q^{28} - 77 q^{29} - 144 q^{30} - 72 q^{31} + 3 q^{32} - 120 q^{33} - 40 q^{34} - 50 q^{35} - 45 q^{36} - 22 q^{37} - 32 q^{38} - 72 q^{39} - 8 q^{40} + 30 q^{41} + 28 q^{43} + 12 q^{44} + 8 q^{45} + 24 q^{46} + 16 q^{47} + 8 q^{48} - 57 q^{49} + 21 q^{50} - 16 q^{51} + 22 q^{52} - 84 q^{53} + 48 q^{54} - 152 q^{55} + 3 q^{56} - 24 q^{57} + 23 q^{58} + 16 q^{59} + 24 q^{60} - 66 q^{61} + 24 q^{62} - 101 q^{63} - q^{64} - 42 q^{65} + 48 q^{66} - 36 q^{67} + 6 q^{68} - 16 q^{69} - 22 q^{70} - 152 q^{71} + 15 q^{72} - 112 q^{73} - 126 q^{74} - 176 q^{75} - 96 q^{76} - 128 q^{77} - 152 q^{78} - 48 q^{79} + 6 q^{80} - 305 q^{81} - 58 q^{82} - 128 q^{83} - 76 q^{84} - 228 q^{85} - 220 q^{86} - 176 q^{87} + 12 q^{88} - 178 q^{89} - 202 q^{90} - 146 q^{91} - 144 q^{92} - 112 q^{93} - 88 q^{94} - 328 q^{95} - 8 q^{97} - 109 q^{98} - 236 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(406))\)

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
406.2.a \(\chi_{406}(1, \cdot)\) 406.2.a.a 1 1
406.2.a.b 1
406.2.a.c 1
406.2.a.d 1
406.2.a.e 2
406.2.a.f 3
406.2.a.g 4
406.2.c \(\chi_{406}(57, \cdot)\) 406.2.c.a 2 1
406.2.c.b 6
406.2.c.c 8
406.2.e \(\chi_{406}(233, \cdot)\) 406.2.e.a 10 2
406.2.e.b 10
406.2.e.c 10
406.2.e.d 10
406.2.g \(\chi_{406}(41, \cdot)\) 406.2.g.a 40 2
406.2.i \(\chi_{406}(289, \cdot)\) 406.2.i.a 40 2
406.2.k \(\chi_{406}(141, \cdot)\) 406.2.k.a 18 6
406.2.k.b 18
406.2.k.c 24
406.2.k.d 24
406.2.l \(\chi_{406}(17, \cdot)\) 406.2.l.a 80 4
406.2.o \(\chi_{406}(71, \cdot)\) 406.2.o.a 48 6
406.2.o.b 48
406.2.q \(\chi_{406}(23, \cdot)\) 406.2.q.a 12 12
406.2.q.b 108
406.2.q.c 120
406.2.r \(\chi_{406}(27, \cdot)\) 406.2.r.a 240 12
406.2.u \(\chi_{406}(9, \cdot)\) 406.2.u.a 240 12
406.2.x \(\chi_{406}(3, \cdot)\) 406.2.x.a 480 24

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(406))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(406)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(29))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(58))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(203))\)\(^{\oplus 2}\)