## Defining parameters

 Level: $$N$$ = $$381 = 3 \cdot 127$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$12$$ Newform subspaces: $$26$$ Sturm bound: $$21504$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(381))$$.

Total New Old
Modular forms 5628 4157 1471
Cusp forms 5125 3905 1220
Eisenstein series 503 252 251

## Trace form

 $$3905q - 3q^{2} - 64q^{3} - 133q^{4} - 6q^{5} - 66q^{6} - 134q^{7} - 15q^{8} - 64q^{9} + O(q^{10})$$ $$3905q - 3q^{2} - 64q^{3} - 133q^{4} - 6q^{5} - 66q^{6} - 134q^{7} - 15q^{8} - 64q^{9} - 144q^{10} - 12q^{11} - 70q^{12} - 140q^{13} - 24q^{14} - 69q^{15} - 157q^{16} - 18q^{17} - 66q^{18} - 146q^{19} - 42q^{20} - 71q^{21} - 162q^{22} - 24q^{23} - 78q^{24} - 157q^{25} - 42q^{26} - 64q^{27} - 182q^{28} - 30q^{29} - 81q^{30} - 158q^{31} - 63q^{32} - 75q^{33} - 180q^{34} - 48q^{35} - 70q^{36} - 164q^{37} - 60q^{38} - 77q^{39} - 216q^{40} - 42q^{41} - 87q^{42} - 170q^{43} - 84q^{44} - 69q^{45} - 198q^{46} - 48q^{47} - 94q^{48} - 183q^{49} - 93q^{50} - 81q^{51} - 224q^{52} - 54q^{53} - 66q^{54} - 198q^{55} - 120q^{56} - 83q^{57} - 216q^{58} - 60q^{59} - 105q^{60} - 188q^{61} - 96q^{62} - 71q^{63} - 253q^{64} - 84q^{65} - 99q^{66} - 194q^{67} - 126q^{68} - 87q^{69} - 270q^{70} - 72q^{71} - 78q^{72} - 200q^{73} - 114q^{74} - 94q^{75} - 266q^{76} - 96q^{77} - 105q^{78} - 206q^{79} - 186q^{80} - 64q^{81} - 252q^{82} - 84q^{83} - 119q^{84} - 234q^{85} - 132q^{86} - 93q^{87} - 306q^{88} - 90q^{89} - 81q^{90} - 238q^{91} - 168q^{92} - 95q^{93} - 270q^{94} - 120q^{95} - 126q^{96} - 224q^{97} - 171q^{98} - 75q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(381))$$

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
381.2.a $$\chi_{381}(1, \cdot)$$ 381.2.a.a 1 1
381.2.a.b 1
381.2.a.c 5
381.2.a.d 5
381.2.a.e 9
381.2.c $$\chi_{381}(380, \cdot)$$ 381.2.c.a 40 1
381.2.e $$\chi_{381}(19, \cdot)$$ 381.2.e.a 4 2
381.2.e.b 16
381.2.e.c 24
381.2.g $$\chi_{381}(20, \cdot)$$ 381.2.g.a 4 2
381.2.g.b 4
381.2.g.c 72
381.2.i $$\chi_{381}(4, \cdot)$$ 381.2.i.a 60 6
381.2.i.b 72
381.2.j $$\chi_{381}(22, \cdot)$$ 381.2.j.a 60 6
381.2.j.b 66
381.2.l $$\chi_{381}(95, \cdot)$$ 381.2.l.a 240 6
381.2.n $$\chi_{381}(59, \cdot)$$ 381.2.n.a 6 6
381.2.n.b 240
381.2.q $$\chi_{381}(25, \cdot)$$ 381.2.q.a 120 12
381.2.q.b 144
381.2.s $$\chi_{381}(5, \cdot)$$ 381.2.s.a 480 12
381.2.u $$\chi_{381}(13, \cdot)$$ 381.2.u.a 360 36
381.2.u.b 396
381.2.x $$\chi_{381}(14, \cdot)$$ 381.2.x.a 36 36
381.2.x.b 1440

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(381))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(381)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(127))$$$$^{\oplus 2}$$