# Properties

 Label 378.4 Level 378 Weight 4 Dimension 2944 Nonzero newspaces 16 Sturm bound 31104 Trace bound 11

## Defining parameters

 Level: $$N$$ = $$378 = 2 \cdot 3^{3} \cdot 7$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$16$$ Sturm bound: $$31104$$ Trace bound: $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(378))$$.

Total New Old
Modular forms 12024 2944 9080
Cusp forms 11304 2944 8360
Eisenstein series 720 0 720

## Trace form

 $$2944 q + 4 q^{2} - 8 q^{4} + 84 q^{5} - 24 q^{6} - 56 q^{7} - 80 q^{8} - 192 q^{9} + O(q^{10})$$ $$2944 q + 4 q^{2} - 8 q^{4} + 84 q^{5} - 24 q^{6} - 56 q^{7} - 80 q^{8} - 192 q^{9} - 120 q^{10} + 180 q^{11} + 24 q^{12} + 428 q^{13} + 496 q^{14} + 144 q^{15} - 32 q^{16} - 852 q^{17} - 552 q^{18} - 544 q^{19} - 480 q^{20} - 204 q^{21} - 528 q^{22} + 288 q^{23} - 38 q^{25} + 1328 q^{26} + 2754 q^{27} + 64 q^{28} + 2076 q^{29} + 1008 q^{30} + 716 q^{31} + 64 q^{32} - 270 q^{33} + 1800 q^{34} + 120 q^{35} - 1200 q^{36} - 220 q^{37} - 700 q^{38} - 1392 q^{39} - 480 q^{40} - 6420 q^{41} - 912 q^{42} - 2992 q^{43} - 1152 q^{44} - 3348 q^{45} - 1344 q^{46} - 2040 q^{47} - 384 q^{48} - 4994 q^{49} - 8468 q^{50} - 1188 q^{51} - 976 q^{52} + 2016 q^{53} + 936 q^{54} + 3456 q^{55} + 928 q^{56} + 10890 q^{57} + 5952 q^{58} + 15270 q^{59} + 4032 q^{60} + 4796 q^{61} + 8576 q^{62} + 11688 q^{63} + 2176 q^{64} + 5580 q^{65} - 1440 q^{66} - 2188 q^{67} + 4008 q^{68} - 8640 q^{69} - 516 q^{70} - 1752 q^{71} + 384 q^{72} - 2776 q^{73} - 4720 q^{74} - 924 q^{75} - 760 q^{76} - 16608 q^{77} - 1440 q^{78} + 5948 q^{79} + 768 q^{80} + 12312 q^{81} + 4344 q^{82} + 6132 q^{83} + 2160 q^{84} + 12528 q^{85} + 9920 q^{86} + 4356 q^{87} + 2928 q^{88} - 5982 q^{89} - 2520 q^{90} - 1246 q^{91} - 5376 q^{92} - 19392 q^{93} - 13704 q^{94} - 31416 q^{95} - 1920 q^{96} - 22072 q^{97} - 3788 q^{98} - 12564 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(378))$$

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
378.4.a $$\chi_{378}(1, \cdot)$$ 378.4.a.a 1 1
378.4.a.b 1
378.4.a.c 1
378.4.a.d 1
378.4.a.e 1
378.4.a.f 1
378.4.a.g 1
378.4.a.h 1
378.4.a.i 1
378.4.a.j 1
378.4.a.k 1
378.4.a.l 1
378.4.a.m 2
378.4.a.n 2
378.4.a.o 2
378.4.a.p 2
378.4.a.q 2
378.4.a.r 2
378.4.d $$\chi_{378}(377, \cdot)$$ 378.4.d.a 4 1
378.4.d.b 4
378.4.d.c 12
378.4.d.d 12
378.4.e $$\chi_{378}(37, \cdot)$$ 378.4.e.a 24 2
378.4.e.b 24
378.4.f $$\chi_{378}(127, \cdot)$$ 378.4.f.a 6 2
378.4.f.b 8
378.4.f.c 10
378.4.f.d 12
378.4.g $$\chi_{378}(109, \cdot)$$ 378.4.g.a 6 2
378.4.g.b 6
378.4.g.c 8
378.4.g.d 8
378.4.g.e 8
378.4.g.f 8
378.4.g.g 10
378.4.g.h 10
378.4.h $$\chi_{378}(289, \cdot)$$ 378.4.h.a 24 2
378.4.h.b 24
378.4.k $$\chi_{378}(215, \cdot)$$ 378.4.k.a 12 2
378.4.k.b 16
378.4.k.c 16
378.4.k.d 20
378.4.l $$\chi_{378}(143, \cdot)$$ 378.4.l.a 48 2
378.4.m $$\chi_{378}(125, \cdot)$$ 378.4.m.a 48 2
378.4.t $$\chi_{378}(17, \cdot)$$ 378.4.t.a 48 2
378.4.u $$\chi_{378}(43, \cdot)$$ n/a 324 6
378.4.v $$\chi_{378}(67, \cdot)$$ n/a 432 6
378.4.w $$\chi_{378}(25, \cdot)$$ n/a 432 6
378.4.z $$\chi_{378}(41, \cdot)$$ n/a 432 6
378.4.ba $$\chi_{378}(47, \cdot)$$ n/a 432 6
378.4.bf $$\chi_{378}(5, \cdot)$$ n/a 432 6

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(378))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(378)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(42))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(54))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(63))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(126))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(189))$$$$^{\oplus 2}$$