# Properties

 Label 1380.4 Level 1380 Weight 4 Dimension 60504 Nonzero newspaces 24 Sturm bound 405504 Trace bound 9

## Defining parameters

 Level: $$N$$ = $$1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$24$$ Sturm bound: $$405504$$ Trace bound: $$9$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(1380))$$.

Total New Old
Modular forms 153824 61016 92808
Cusp forms 150304 60504 89800
Eisenstein series 3520 512 3008

## Trace form

 $$60504 q + 20 q^{3} - 60 q^{4} + 40 q^{5} - 130 q^{6} - 32 q^{7} - 168 q^{8} - 76 q^{9} + O(q^{10})$$ $$60504 q + 20 q^{3} - 60 q^{4} + 40 q^{5} - 130 q^{6} - 32 q^{7} - 168 q^{8} - 76 q^{9} - 186 q^{10} + 32 q^{11} + 306 q^{12} + 64 q^{13} - 474 q^{15} - 260 q^{16} - 920 q^{17} - 534 q^{18} - 360 q^{19} + 760 q^{20} + 1444 q^{21} + 1552 q^{22} + 2200 q^{23} + 1060 q^{24} + 2708 q^{25} - 736 q^{26} + 824 q^{27} - 2780 q^{28} - 1120 q^{29} - 665 q^{30} - 2488 q^{31} - 680 q^{32} - 2252 q^{33} + 9484 q^{34} + 2700 q^{35} + 2330 q^{36} + 3744 q^{37} + 6484 q^{38} - 384 q^{39} + 2702 q^{40} - 456 q^{41} - 5506 q^{42} - 8144 q^{43} - 13244 q^{44} - 2124 q^{45} - 19180 q^{46} - 8240 q^{47} - 10366 q^{48} - 7552 q^{49} - 7468 q^{50} - 3872 q^{51} - 7364 q^{52} + 880 q^{53} + 1326 q^{54} + 7000 q^{55} + 17244 q^{56} + 12540 q^{57} + 24200 q^{58} + 13576 q^{59} + 1165 q^{60} - 936 q^{61} + 3344 q^{62} + 1900 q^{63} + 1860 q^{64} - 5240 q^{65} - 1108 q^{66} - 3296 q^{67} - 8864 q^{68} - 8236 q^{69} - 9236 q^{70} - 4800 q^{71} - 2158 q^{72} + 2336 q^{73} - 9460 q^{74} - 6560 q^{75} - 28048 q^{76} + 45248 q^{77} - 14314 q^{78} + 5160 q^{79} + 1154 q^{80} + 8188 q^{81} + 4236 q^{82} + 1016 q^{83} + 19286 q^{84} + 5936 q^{85} + 27736 q^{86} + 1100 q^{87} + 26148 q^{88} - 21800 q^{89} + 26040 q^{90} - 5520 q^{91} + 38924 q^{92} - 27816 q^{93} + 44064 q^{94} + 14204 q^{95} + 40108 q^{96} - 7832 q^{97} + 23824 q^{98} + 204 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(1380))$$

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
1380.4.a $$\chi_{1380}(1, \cdot)$$ 1380.4.a.a 1 1
1380.4.a.b 2
1380.4.a.c 3
1380.4.a.d 4
1380.4.a.e 5
1380.4.a.f 5
1380.4.a.g 5
1380.4.a.h 5
1380.4.a.i 7
1380.4.a.j 7
1380.4.f $$\chi_{1380}(829, \cdot)$$ 1380.4.f.a 30 1
1380.4.f.b 38
1380.4.g $$\chi_{1380}(919, \cdot)$$ n/a 432 1
1380.4.h $$\chi_{1380}(1151, \cdot)$$ n/a 528 1
1380.4.i $$\chi_{1380}(1241, \cdot)$$ 1380.4.i.a 48 1
1380.4.i.b 48
1380.4.n $$\chi_{1380}(689, \cdot)$$ n/a 144 1
1380.4.o $$\chi_{1380}(599, \cdot)$$ n/a 792 1
1380.4.p $$\chi_{1380}(91, \cdot)$$ n/a 288 1
1380.4.q $$\chi_{1380}(827, \cdot)$$ n/a 1712 2
1380.4.r $$\chi_{1380}(737, \cdot)$$ n/a 264 2
1380.4.s $$\chi_{1380}(967, \cdot)$$ n/a 792 2
1380.4.t $$\chi_{1380}(1057, \cdot)$$ n/a 144 2
1380.4.y $$\chi_{1380}(121, \cdot)$$ n/a 480 10
1380.4.z $$\chi_{1380}(451, \cdot)$$ n/a 2880 10
1380.4.ba $$\chi_{1380}(59, \cdot)$$ n/a 8560 10
1380.4.bb $$\chi_{1380}(89, \cdot)$$ n/a 1440 10
1380.4.bg $$\chi_{1380}(221, \cdot)$$ n/a 960 10
1380.4.bh $$\chi_{1380}(71, \cdot)$$ n/a 5760 10
1380.4.bi $$\chi_{1380}(19, \cdot)$$ n/a 4320 10
1380.4.bj $$\chi_{1380}(49, \cdot)$$ n/a 720 10
1380.4.bs $$\chi_{1380}(37, \cdot)$$ n/a 1440 20
1380.4.bt $$\chi_{1380}(127, \cdot)$$ n/a 8640 20
1380.4.bu $$\chi_{1380}(77, \cdot)$$ n/a 2880 20
1380.4.bv $$\chi_{1380}(83, \cdot)$$ n/a 17120 20

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(1380))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(1380)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(23))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(46))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(60))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(69))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(92))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(115))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(138))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(230))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(276))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(345))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(460))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(690))$$$$^{\oplus 2}$$