# Properties

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

## 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

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

## 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 available 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

$$S_{2}^{\mathrm{old}}(\Gamma_1(338)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(13))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(26))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(169))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(338))$$$$^{\oplus 1}$$