## Defining parameters

 Level: $$N$$ = $$1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$24$$ Sturm bound: $$202752$$ Trace bound: $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(1380))$$.

Total New Old
Modular forms 52448 20508 31940
Cusp forms 48929 19996 28933
Eisenstein series 3519 512 3007

## Trace form

 $$19996q - 4q^{3} - 28q^{4} - 4q^{5} - 50q^{6} + 8q^{7} + 24q^{8} - 24q^{9} + O(q^{10})$$ $$19996q - 4q^{3} - 28q^{4} - 4q^{5} - 50q^{6} + 8q^{7} + 24q^{8} - 24q^{9} - 26q^{10} + 16q^{11} - 14q^{12} - 40q^{13} + 17q^{15} - 132q^{16} - 4q^{17} - 54q^{18} - 44q^{19} - 40q^{20} - 174q^{21} - 16q^{22} - 88q^{23} - 92q^{24} - 148q^{25} - 32q^{26} - 94q^{27} - 92q^{28} - 68q^{29} - 105q^{30} - 92q^{31} - 40q^{32} - 90q^{33} - 4q^{34} + 12q^{35} - 22q^{36} + 88q^{37} + 188q^{38} + 88q^{39} + 78q^{40} + 96q^{41} + 182q^{42} + 200q^{43} + 308q^{44} - 40q^{45} + 244q^{46} + 176q^{47} + 194q^{48} + 212q^{49} + 76q^{50} + 112q^{51} + 476q^{52} + 80q^{53} + 134q^{54} + 104q^{55} + 348q^{56} + 70q^{57} + 288q^{58} + 104q^{59} + 117q^{60} - 192q^{61} + 112q^{62} + 94q^{63} + 68q^{64} - 40q^{65} - 4q^{66} + 8q^{67} + 32q^{68} + 34q^{69} - 116q^{70} - 46q^{72} - 184q^{73} - 44q^{74} + 95q^{75} - 416q^{76} + 256q^{77} - 274q^{78} + 40q^{79} - 206q^{80} + 168q^{81} - 420q^{82} + 88q^{83} - 442q^{84} + 64q^{85} - 568q^{86} + 92q^{87} - 604q^{88} + 224q^{89} - 384q^{90} + 128q^{91} - 452q^{92} + 392q^{93} - 560q^{94} + 154q^{95} - 612q^{96} + 372q^{97} - 400q^{98} + 160q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(1380))$$

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
1380.2.a $$\chi_{1380}(1, \cdot)$$ 1380.2.a.a 1 1
1380.2.a.b 1
1380.2.a.c 1
1380.2.a.d 1
1380.2.a.e 1
1380.2.a.f 2
1380.2.a.g 2
1380.2.a.h 2
1380.2.a.i 2
1380.2.a.j 3
1380.2.f $$\chi_{1380}(829, \cdot)$$ 1380.2.f.a 6 1
1380.2.f.b 14
1380.2.g $$\chi_{1380}(919, \cdot)$$ n/a 144 1
1380.2.h $$\chi_{1380}(1151, \cdot)$$ n/a 176 1
1380.2.i $$\chi_{1380}(1241, \cdot)$$ 1380.2.i.a 16 1
1380.2.i.b 16
1380.2.n $$\chi_{1380}(689, \cdot)$$ 1380.2.n.a 48 1
1380.2.o $$\chi_{1380}(599, \cdot)$$ n/a 264 1
1380.2.p $$\chi_{1380}(91, \cdot)$$ 1380.2.p.a 48 1
1380.2.p.b 48
1380.2.q $$\chi_{1380}(827, \cdot)$$ n/a 560 2
1380.2.r $$\chi_{1380}(737, \cdot)$$ 1380.2.r.a 4 2
1380.2.r.b 4
1380.2.r.c 80
1380.2.s $$\chi_{1380}(967, \cdot)$$ n/a 264 2
1380.2.t $$\chi_{1380}(1057, \cdot)$$ 1380.2.t.a 48 2
1380.2.y $$\chi_{1380}(121, \cdot)$$ n/a 160 10
1380.2.z $$\chi_{1380}(451, \cdot)$$ n/a 960 10
1380.2.ba $$\chi_{1380}(59, \cdot)$$ n/a 2800 10
1380.2.bb $$\chi_{1380}(89, \cdot)$$ n/a 480 10
1380.2.bg $$\chi_{1380}(221, \cdot)$$ n/a 320 10
1380.2.bh $$\chi_{1380}(71, \cdot)$$ n/a 1920 10
1380.2.bi $$\chi_{1380}(19, \cdot)$$ n/a 1440 10
1380.2.bj $$\chi_{1380}(49, \cdot)$$ n/a 240 10
1380.2.bs $$\chi_{1380}(37, \cdot)$$ n/a 480 20
1380.2.bt $$\chi_{1380}(127, \cdot)$$ n/a 2880 20
1380.2.bu $$\chi_{1380}(77, \cdot)$$ n/a 960 20
1380.2.bv $$\chi_{1380}(83, \cdot)$$ n/a 5600 20

"n/a" means that newforms for that character have not been added to the database yet

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(1380))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(1380)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(23))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(46))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(60))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(69))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(92))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(115))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(138))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(230))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(276))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(345))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(460))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(690))$$$$^{\oplus 2}$$