## Defining parameters

 Level: $$N$$ = $$38 = 2 \cdot 19$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$3$$ Newforms: $$5$$ 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

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

## 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 the 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

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