# Properties

 Label 1380.2 Level 1380 Weight 2 Dimension 19996 Nonzero newspaces 24 Sturm bound 202752 Trace bound 13

## 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

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

## 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

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