Properties

Label 338.2
Level 338
Weight 2
Dimension 1170
Nonzero newspaces 8
Newform subspaces 32
Sturm bound 14196
Trace bound 4

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Defining parameters

Level: \( N \) = \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 8 \)
Newform subspaces: \( 32 \)
Sturm bound: \(14196\)
Trace bound: \(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(338))\).

Total New Old
Modular forms 3777 1170 2607
Cusp forms 3322 1170 2152
Eisenstein series 455 0 455

Trace form

\( 1170q + q^{2} + 4q^{3} + q^{4} + 6q^{5} + 4q^{6} - 5q^{8} - 19q^{9} + O(q^{10}) \) \( 1170q + q^{2} + 4q^{3} + q^{4} + 6q^{5} + 4q^{6} - 5q^{8} - 19q^{9} - 24q^{10} - 12q^{11} - 4q^{12} - 24q^{13} - 16q^{14} - 24q^{15} - 7q^{16} - 12q^{17} - 17q^{18} - 36q^{19} - 24q^{21} + 12q^{22} + 4q^{24} + q^{25} - 32q^{27} - 48q^{29} - 24q^{30} - 48q^{31} + q^{32} - 72q^{33} - 30q^{34} - 96q^{35} - 35q^{36} - 24q^{37} - 52q^{38} - 52q^{39} + 6q^{40} - 60q^{41} - 40q^{42} - 44q^{43} - 36q^{44} - 72q^{45} - 24q^{46} - 48q^{47} + 4q^{48} - 7q^{49} - 23q^{50} - 72q^{51} - 7q^{52} - 18q^{53} - 32q^{54} - 48q^{55} - 16q^{56} - 48q^{57} - 24q^{58} - 36q^{59} - 24q^{60} - 16q^{61} - 40q^{62} - 144q^{63} - 5q^{64} - 87q^{65} - 48q^{66} - 84q^{67} + 12q^{68} - 144q^{69} - 24q^{70} - 72q^{71} - 35q^{72} - 102q^{73} - 40q^{74} - 172q^{75} - 36q^{76} - 144q^{77} - 60q^{78} - 112q^{79} - 127q^{81} - 60q^{82} - 108q^{83} - 24q^{84} - 138q^{85} - 4q^{86} - 120q^{87} + 12q^{88} - 102q^{89} - 42q^{90} - 76q^{91} - 24q^{92} - 96q^{93} - 72q^{94} - 24q^{95} + 4q^{96} - 102q^{97} - 39q^{98} - 60q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(338))\)

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
338.2.a \(\chi_{338}(1, \cdot)\) 338.2.a.a 1 1
338.2.a.b 1
338.2.a.c 1
338.2.a.d 1
338.2.a.e 1
338.2.a.f 1
338.2.a.g 3
338.2.a.h 3
338.2.b \(\chi_{338}(337, \cdot)\) 338.2.b.a 2 1
338.2.b.b 2
338.2.b.c 2
338.2.b.d 6
338.2.c \(\chi_{338}(191, \cdot)\) 338.2.c.a 2 2
338.2.c.b 2
338.2.c.c 2
338.2.c.d 2
338.2.c.e 2
338.2.c.f 2
338.2.c.g 2
338.2.c.h 6
338.2.c.i 6
338.2.e \(\chi_{338}(23, \cdot)\) 338.2.e.a 4 2
338.2.e.b 4
338.2.e.c 4
338.2.e.d 4
338.2.e.e 12
338.2.g \(\chi_{338}(27, \cdot)\) 338.2.g.a 96 12
338.2.g.b 108
338.2.h \(\chi_{338}(25, \cdot)\) 338.2.h.a 192 12
338.2.i \(\chi_{338}(3, \cdot)\) 338.2.i.a 168 24
338.2.i.b 192
338.2.k \(\chi_{338}(17, \cdot)\) 338.2.k.a 336 24

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(338))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(338)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(13))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(26))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(169))\)\(^{\oplus 2}\)