## Defining parameters

 Level: $$N$$ = $$380 = 2^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$18$$ Newform subspaces: $$20$$ Sturm bound: $$25920$$ Trace bound: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(380))$$.

Total New Old
Modular forms 9000 4700 4300
Cusp forms 8280 4492 3788
Eisenstein series 720 208 512

## Trace form

 $$4492 q - 14 q^{2} - 4 q^{3} - 10 q^{4} - 34 q^{5} - 22 q^{6} + 28 q^{7} - 2 q^{8} - 20 q^{9} + O(q^{10})$$ $$4492 q - 14 q^{2} - 4 q^{3} - 10 q^{4} - 34 q^{5} - 22 q^{6} + 28 q^{7} - 2 q^{8} - 20 q^{9} - 39 q^{10} - 40 q^{11} - 98 q^{12} - 160 q^{13} - 130 q^{14} - 58 q^{15} - 150 q^{16} - 10 q^{17} - 44 q^{18} + 42 q^{19} + 34 q^{20} + 58 q^{21} + 142 q^{22} + 182 q^{23} + 270 q^{24} - 12 q^{25} + 162 q^{26} + 332 q^{27} + 548 q^{28} + 188 q^{29} + 334 q^{30} + 272 q^{31} + 176 q^{32} + 196 q^{33} - 60 q^{34} - 124 q^{35} - 58 q^{36} - 164 q^{37} - 224 q^{38} - 216 q^{39} - 168 q^{40} - 148 q^{41} - 678 q^{42} - 450 q^{43} - 488 q^{44} - 343 q^{45} - 592 q^{46} - 690 q^{47} - 1404 q^{48} - 426 q^{49} - 550 q^{50} - 710 q^{51} - 130 q^{52} + 128 q^{53} - 404 q^{54} + 84 q^{55} - 324 q^{56} + 70 q^{57} + 36 q^{58} + 270 q^{59} - 458 q^{60} + 1148 q^{61} - 1028 q^{62} + 884 q^{63} - 1474 q^{64} + 391 q^{65} - 1510 q^{66} + 1066 q^{67} - 628 q^{68} + 1450 q^{69} - 407 q^{70} + 562 q^{71} - 684 q^{72} - 334 q^{73} + 318 q^{74} + 124 q^{75} + 126 q^{76} - 470 q^{77} + 610 q^{78} + 204 q^{79} + 621 q^{80} + 588 q^{81} + 1640 q^{82} + 558 q^{83} + 1862 q^{84} + 82 q^{85} + 884 q^{86} + 176 q^{87} + 1534 q^{88} - 222 q^{89} + 1506 q^{90} - 48 q^{91} + 1752 q^{92} - 548 q^{93} + 1540 q^{94} + 503 q^{95} + 2852 q^{96} + 1350 q^{97} + 2748 q^{98} + 1044 q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(380))$$

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
380.3.b $$\chi_{380}(191, \cdot)$$ 380.3.b.a 72 1
380.3.e $$\chi_{380}(341, \cdot)$$ 380.3.e.a 12 1
380.3.g $$\chi_{380}(189, \cdot)$$ 380.3.g.a 4 1
380.3.g.b 4
380.3.g.c 12
380.3.h $$\chi_{380}(39, \cdot)$$ 380.3.h.a 108 1
380.3.j $$\chi_{380}(227, \cdot)$$ 380.3.j.a 232 2
380.3.m $$\chi_{380}(77, \cdot)$$ 380.3.m.a 36 2
380.3.o $$\chi_{380}(69, \cdot)$$ 380.3.o.a 40 2
380.3.p $$\chi_{380}(159, \cdot)$$ 380.3.p.a 232 2
380.3.q $$\chi_{380}(11, \cdot)$$ 380.3.q.a 160 2
380.3.t $$\chi_{380}(141, \cdot)$$ 380.3.t.a 24 2
380.3.w $$\chi_{380}(27, \cdot)$$ 380.3.w.a 464 4
380.3.x $$\chi_{380}(197, \cdot)$$ 380.3.x.a 80 4
380.3.z $$\chi_{380}(21, \cdot)$$ 380.3.z.a 84 6
380.3.ba $$\chi_{380}(99, \cdot)$$ 380.3.ba.a 696 6
380.3.bc $$\chi_{380}(29, \cdot)$$ 380.3.bc.a 120 6
380.3.bf $$\chi_{380}(111, \cdot)$$ 380.3.bf.a 480 6
380.3.bg $$\chi_{380}(17, \cdot)$$ 380.3.bg.a 240 12
380.3.bi $$\chi_{380}(3, \cdot)$$ 380.3.bi.a 1392 12

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(380))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(380)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(19))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(38))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(76))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(95))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(190))$$$$^{\oplus 2}$$