## Defining parameters

 Level: $$N$$ = $$38 = 2 \cdot 19$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$3$$ Newform subspaces: $$3$$ Sturm bound: $$270$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(38))$$.

Total New Old
Modular forms 108 30 78
Cusp forms 72 30 42
Eisenstein series 36 0 36

## Trace form

 $$30q + O(q^{10})$$ $$30q - 36q^{12} - 120q^{13} - 72q^{14} - 108q^{15} - 18q^{17} + 42q^{19} + 36q^{20} + 126q^{21} + 108q^{22} + 90q^{23} + 252q^{25} + 144q^{26} + 126q^{27} + 60q^{28} - 144q^{29} - 108q^{31} - 216q^{33} - 72q^{35} + 108q^{39} + 72q^{41} + 138q^{43} - 108q^{44} - 162q^{45} - 360q^{46} - 306q^{47} - 72q^{48} - 78q^{49} - 432q^{50} - 324q^{51} - 24q^{52} - 216q^{53} - 108q^{54} - 72q^{55} + 90q^{57} + 72q^{58} + 270q^{59} + 144q^{60} + 276q^{61} + 324q^{62} + 216q^{63} + 48q^{64} + 630q^{65} + 720q^{66} + 174q^{67} + 108q^{68} + 702q^{69} + 504q^{70} + 738q^{71} + 144q^{72} + 48q^{73} - 270q^{77} - 576q^{78} - 372q^{79} - 774q^{81} - 432q^{82} - 450q^{83} - 540q^{84} - 648q^{85} - 288q^{86} - 576q^{87} - 18q^{89} - 360q^{90} + 168q^{91} - 72q^{92} - 36q^{93} - 594q^{95} - 414q^{97} + 288q^{98} - 162q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(38))$$

We only show spaces with odd 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
38.3.b $$\chi_{38}(37, \cdot)$$ 38.3.b.a 2 1
38.3.d $$\chi_{38}(27, \cdot)$$ 38.3.d.a 4 2
38.3.f $$\chi_{38}(3, \cdot)$$ 38.3.f.a 24 6

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(38))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(38)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(19))$$$$^{\oplus 2}$$