# Properties

 Label 38.2 Level 38 Weight 2 Dimension 14 Nonzero newspaces 3 Newform subspaces 5 Sturm bound 180 Trace bound 2

## Defining parameters

 Level: $$N$$ = $$38 = 2 \cdot 19$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$3$$ Newform subspaces: $$5$$ Sturm bound: $$180$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(38))$$.

Total New Old
Modular forms 63 14 49
Cusp forms 28 14 14
Eisenstein series 35 0 35

## Trace form

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

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(38))$$

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
38.2.a $$\chi_{38}(1, \cdot)$$ 38.2.a.a 1 1
38.2.a.b 1
38.2.c $$\chi_{38}(7, \cdot)$$ 38.2.c.a 2 2
38.2.c.b 4
38.2.e $$\chi_{38}(5, \cdot)$$ 38.2.e.a 6 6

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(38))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(38)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(19))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(38))$$$$^{\oplus 1}$$