# Properties

 Label 378.3 Level 378 Weight 3 Dimension 1964 Nonzero newspaces 16 Newform subspaces 28 Sturm bound 23328 Trace bound 11

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## Defining parameters

 Level: $$N$$ = $$378 = 2 \cdot 3^{3} \cdot 7$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$16$$ Newform subspaces: $$28$$ Sturm bound: $$23328$$ Trace bound: $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(378))$$.

Total New Old
Modular forms 8136 1964 6172
Cusp forms 7416 1964 5452
Eisenstein series 720 0 720

## Trace form

 $$1964 q - 72 q^{5} - 24 q^{6} - 12 q^{7} + 24 q^{9} + O(q^{10})$$ $$1964 q - 72 q^{5} - 24 q^{6} - 12 q^{7} + 24 q^{9} + 48 q^{10} + 72 q^{11} + 24 q^{12} - 4 q^{13} + 36 q^{15} - 16 q^{16} - 108 q^{17} + 96 q^{18} - 64 q^{19} + 72 q^{20} + 228 q^{21} - 48 q^{22} + 504 q^{23} + 284 q^{25} + 144 q^{26} - 108 q^{27} + 44 q^{28} - 36 q^{29} - 288 q^{30} + 68 q^{31} - 648 q^{33} - 48 q^{34} - 702 q^{35} - 336 q^{36} - 252 q^{37} - 504 q^{38} - 204 q^{39} - 96 q^{40} - 72 q^{41} - 48 q^{42} - 96 q^{43} + 756 q^{45} + 96 q^{46} + 1080 q^{47} + 48 q^{48} + 752 q^{49} + 1440 q^{50} + 1296 q^{51} + 392 q^{52} + 2988 q^{53} + 936 q^{54} + 1440 q^{55} + 144 q^{56} + 576 q^{57} + 576 q^{58} + 648 q^{59} + 144 q^{60} + 188 q^{61} - 570 q^{63} - 192 q^{64} - 1116 q^{65} + 288 q^{66} + 832 q^{67} - 576 q^{68} - 216 q^{69} - 468 q^{70} - 1152 q^{71} + 384 q^{72} - 1600 q^{73} - 1008 q^{74} - 1464 q^{75} - 160 q^{76} - 2592 q^{77} - 576 q^{78} - 1052 q^{79} - 648 q^{81} - 384 q^{82} - 1944 q^{83} - 216 q^{84} - 1752 q^{85} - 1512 q^{86} - 2124 q^{87} - 432 q^{88} - 2124 q^{89} - 1440 q^{90} - 604 q^{91} - 1008 q^{92} - 924 q^{93} - 1704 q^{94} - 576 q^{95} - 192 q^{96} - 1684 q^{97} - 1080 q^{98} - 252 q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(378))$$

We only show spaces with odd 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
378.3.b $$\chi_{378}(323, \cdot)$$ 378.3.b.a 4 1
378.3.b.b 4
378.3.b.c 8
378.3.c $$\chi_{378}(55, \cdot)$$ 378.3.c.a 4 1
378.3.c.b 4
378.3.c.c 12
378.3.i $$\chi_{378}(179, \cdot)$$ 378.3.i.a 32 2
378.3.j $$\chi_{378}(19, \cdot)$$ 378.3.j.a 32 2
378.3.n $$\chi_{378}(271, \cdot)$$ 378.3.n.a 4 2
378.3.n.b 4
378.3.n.c 12
378.3.n.d 12
378.3.n.e 12
378.3.o $$\chi_{378}(181, \cdot)$$ 378.3.o.a 32 2
378.3.p $$\chi_{378}(73, \cdot)$$ 378.3.p.a 32 2
378.3.q $$\chi_{378}(71, \cdot)$$ 378.3.q.a 24 2
378.3.r $$\chi_{378}(233, \cdot)$$ 378.3.r.a 32 2
378.3.s $$\chi_{378}(53, \cdot)$$ 378.3.s.a 4 2
378.3.s.b 4
378.3.s.c 4
378.3.s.d 8
378.3.s.e 24
378.3.x $$\chi_{378}(103, \cdot)$$ 378.3.x.a 288 6
378.3.y $$\chi_{378}(11, \cdot)$$ 378.3.y.a 288 6
378.3.bb $$\chi_{378}(29, \cdot)$$ 378.3.bb.a 216 6
378.3.bc $$\chi_{378}(65, \cdot)$$ 378.3.bc.a 288 6
378.3.bd $$\chi_{378}(13, \cdot)$$ 378.3.bd.a 288 6
378.3.be $$\chi_{378}(31, \cdot)$$ 378.3.be.a 288 6

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(378))$$ into lower level spaces

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