Properties

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

Defining parameters

 Level: $$N$$ = $$378 = 2 \cdot 3^{3} \cdot 7$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$16$$ Newforms: $$50$$ 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

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

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 the 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

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