# Properties

 Label 38.4 Level 38 Weight 4 Dimension 45 Nonzero newspaces 3 Newform subspaces 8 Sturm bound 360 Trace bound 1

## Defining parameters

 Level: $$N$$ = $$38 = 2 \cdot 19$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$3$$ Newform subspaces: $$8$$ Sturm bound: $$360$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(38))$$.

Total New Old
Modular forms 153 45 108
Cusp forms 117 45 72
Eisenstein series 36 0 36

## Trace form

 $$45 q + O(q^{10})$$ $$45 q + 144 q^{12} + 288 q^{13} + 36 q^{14} - 216 q^{15} - 288 q^{17} - 486 q^{18} - 756 q^{19} - 288 q^{20} - 504 q^{21} - 162 q^{22} + 36 q^{23} + 864 q^{25} + 684 q^{26} + 1953 q^{27} + 432 q^{28} + 630 q^{29} + 90 q^{31} - 918 q^{33} - 1116 q^{35} - 666 q^{37} - 1890 q^{39} - 450 q^{41} + 504 q^{43} + 900 q^{44} + 5130 q^{45} + 1944 q^{46} + 2790 q^{47} + 144 q^{48} + 1710 q^{49} - 1008 q^{50} - 1035 q^{51} - 216 q^{52} - 3168 q^{53} - 2700 q^{54} - 3888 q^{55} - 2016 q^{56} - 5040 q^{57} - 1944 q^{58} - 4500 q^{59} - 1584 q^{60} - 1782 q^{61} - 1404 q^{62} + 936 q^{63} + 2070 q^{65} + 2448 q^{66} + 2160 q^{67} + 684 q^{68} + 5400 q^{69} + 4536 q^{70} + 7326 q^{71} + 1656 q^{72} - 567 q^{73} - 3150 q^{75} + 2502 q^{77} + 6408 q^{78} + 2610 q^{79} + 5121 q^{81} + 2232 q^{82} + 1710 q^{83} + 864 q^{84} - 288 q^{85} - 792 q^{86} - 3420 q^{87} - 1530 q^{89} - 6660 q^{90} - 1944 q^{91} - 3168 q^{92} - 11862 q^{93} - 6768 q^{94} + 3168 q^{95} + 972 q^{97} - 5904 q^{98} - 4545 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(38))$$

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
38.4.a $$\chi_{38}(1, \cdot)$$ 38.4.a.a 1 1
38.4.a.b 2
38.4.a.c 2
38.4.c $$\chi_{38}(7, \cdot)$$ 38.4.c.a 2 2
38.4.c.b 2
38.4.c.c 6
38.4.e $$\chi_{38}(5, \cdot)$$ 38.4.e.a 12 6
38.4.e.b 18

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(38))$$ into lower level spaces

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