# Properties

 Label 378.2 Level 378 Weight 2 Dimension 980 Nonzero newspaces 16 Newform subspaces 50 Sturm bound 15552 Trace bound 11

## Defining parameters

 Level: $$N$$ = $$378 = 2 \cdot 3^{3} \cdot 7$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$16$$ Newform subspaces: $$50$$ Sturm bound: $$15552$$ Trace bound: $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(378))$$.

Total New Old
Modular forms 4248 980 3268
Cusp forms 3529 980 2549
Eisenstein series 719 0 719

## Trace form

 $$980 q - 2 q^{2} - 2 q^{4} + 12 q^{5} + 12 q^{6} + 4 q^{7} + 10 q^{8} + 24 q^{9} + O(q^{10})$$ $$980 q - 2 q^{2} - 2 q^{4} + 12 q^{5} + 12 q^{6} + 4 q^{7} + 10 q^{8} + 24 q^{9} + 12 q^{10} + 36 q^{11} + 6 q^{12} + 24 q^{13} + 16 q^{14} + 36 q^{15} + 2 q^{16} + 48 q^{17} - 12 q^{18} + 24 q^{19} - 12 q^{20} - 24 q^{21} + 24 q^{22} + 38 q^{25} - 40 q^{26} - 54 q^{27} + 2 q^{28} - 48 q^{29} - 72 q^{30} + 24 q^{31} - 2 q^{32} - 54 q^{33} + 12 q^{34} + 30 q^{35} - 12 q^{36} + 32 q^{37} + 26 q^{38} + 84 q^{39} + 12 q^{40} + 60 q^{41} + 24 q^{42} + 80 q^{43} + 48 q^{47} + 12 q^{48} + 14 q^{49} - 38 q^{50} - 108 q^{51} - 24 q^{52} - 180 q^{53} - 36 q^{54} - 144 q^{55} + 4 q^{56} - 126 q^{57} - 96 q^{58} - 318 q^{59} - 72 q^{60} - 96 q^{61} - 208 q^{62} - 246 q^{63} + 10 q^{64} - 432 q^{65} - 144 q^{66} - 212 q^{67} - 150 q^{68} - 216 q^{69} - 174 q^{70} - 312 q^{71} - 48 q^{72} - 168 q^{73} - 232 q^{74} - 204 q^{75} - 90 q^{76} - 192 q^{77} - 144 q^{78} - 200 q^{79} - 24 q^{80} - 60 q^{82} + 12 q^{83} - 120 q^{85} - 40 q^{86} - 72 q^{87} - 42 q^{88} + 102 q^{89} - 36 q^{90} + 78 q^{91} + 24 q^{92} + 12 q^{93} - 12 q^{94} + 120 q^{95} - 12 q^{96} + 36 q^{97} + 46 q^{98} + 72 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(378))$$

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 available newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
378.2.a $$\chi_{378}(1, \cdot)$$ 378.2.a.a 1 1
378.2.a.b 1
378.2.a.c 1
378.2.a.d 1
378.2.a.e 1
378.2.a.f 1
378.2.a.g 1
378.2.a.h 1
378.2.d $$\chi_{378}(377, \cdot)$$ 378.2.d.a 4 1
378.2.d.b 4
378.2.d.c 4
378.2.e $$\chi_{378}(37, \cdot)$$ 378.2.e.a 2 2
378.2.e.b 2
378.2.e.c 6
378.2.e.d 6
378.2.f $$\chi_{378}(127, \cdot)$$ 378.2.f.a 2 2
378.2.f.b 2
378.2.f.c 4
378.2.f.d 4
378.2.g $$\chi_{378}(109, \cdot)$$ 378.2.g.a 2 2
378.2.g.b 2
378.2.g.c 2
378.2.g.d 2
378.2.g.e 2
378.2.g.f 2
378.2.g.g 4
378.2.g.h 4
378.2.h $$\chi_{378}(289, \cdot)$$ 378.2.h.a 2 2
378.2.h.b 2
378.2.h.c 6
378.2.h.d 6
378.2.k $$\chi_{378}(215, \cdot)$$ 378.2.k.a 4 2
378.2.k.b 4
378.2.k.c 4
378.2.k.d 8
378.2.l $$\chi_{378}(143, \cdot)$$ 378.2.l.a 16 2
378.2.m $$\chi_{378}(125, \cdot)$$ 378.2.m.a 16 2
378.2.t $$\chi_{378}(17, \cdot)$$ 378.2.t.a 16 2
378.2.u $$\chi_{378}(43, \cdot)$$ 378.2.u.a 6 6
378.2.u.b 12
378.2.u.c 24
378.2.u.d 30
378.2.u.e 36
378.2.v $$\chi_{378}(67, \cdot)$$ 378.2.v.a 72 6
378.2.v.b 72
378.2.w $$\chi_{378}(25, \cdot)$$ 378.2.w.a 72 6
378.2.w.b 72
378.2.z $$\chi_{378}(41, \cdot)$$ 378.2.z.a 144 6
378.2.ba $$\chi_{378}(47, \cdot)$$ 378.2.ba.a 144 6
378.2.bf $$\chi_{378}(5, \cdot)$$ 378.2.bf.a 144 6

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(378))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(378)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 16}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(3))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(42))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(54))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(63))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(126))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(189))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(378))$$$$^{\oplus 1}$$