## Defining parameters

 Level: $$N$$ = $$392 = 2^{3} \cdot 7^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$12$$ Newform subspaces: $$52$$ Sturm bound: $$18816$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(392))$$.

Total New Old
Modular forms 5064 2636 2428
Cusp forms 4345 2442 1903
Eisenstein series 719 194 525

## Trace form

 $$2442 q - 30 q^{2} - 30 q^{3} - 30 q^{4} - 30 q^{6} - 36 q^{7} - 54 q^{8} - 48 q^{9} + O(q^{10})$$ $$2442 q - 30 q^{2} - 30 q^{3} - 30 q^{4} - 30 q^{6} - 36 q^{7} - 54 q^{8} - 48 q^{9} - 30 q^{10} - 18 q^{11} - 30 q^{12} + 12 q^{13} - 36 q^{14} - 30 q^{15} - 30 q^{16} - 48 q^{17} - 66 q^{18} - 30 q^{19} - 66 q^{20} - 114 q^{22} - 66 q^{23} - 102 q^{24} - 96 q^{25} - 90 q^{26} - 90 q^{27} - 84 q^{28} - 126 q^{30} - 66 q^{31} - 90 q^{32} - 72 q^{33} - 102 q^{34} - 36 q^{35} - 114 q^{36} + 12 q^{37} - 66 q^{38} - 42 q^{39} - 90 q^{40} - 12 q^{41} - 54 q^{43} + 18 q^{44} + 12 q^{45} + 30 q^{46} - 66 q^{47} + 90 q^{48} - 78 q^{49} - 12 q^{50} - 114 q^{51} + 78 q^{52} - 12 q^{53} + 114 q^{54} - 90 q^{55} + 6 q^{56} - 156 q^{57} + 30 q^{58} - 102 q^{59} + 102 q^{60} + 78 q^{62} - 72 q^{63} + 30 q^{64} - 60 q^{65} + 18 q^{66} - 66 q^{67} - 30 q^{68} - 24 q^{69} - 60 q^{70} - 126 q^{71} - 66 q^{72} - 24 q^{73} - 198 q^{74} - 150 q^{75} - 174 q^{76} - 18 q^{77} - 222 q^{78} - 66 q^{79} - 258 q^{80} - 192 q^{81} - 174 q^{82} - 186 q^{83} - 144 q^{84} - 84 q^{85} - 246 q^{86} - 282 q^{87} - 174 q^{88} - 156 q^{89} - 186 q^{90} - 108 q^{91} - 210 q^{92} - 264 q^{93} - 102 q^{94} - 186 q^{95} - 78 q^{96} - 240 q^{97} - 60 q^{98} - 120 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(392))$$

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
392.2.a $$\chi_{392}(1, \cdot)$$ 392.2.a.a 1 1
392.2.a.b 1
392.2.a.c 1
392.2.a.d 1
392.2.a.e 1
392.2.a.f 1
392.2.a.g 2
392.2.a.h 2
392.2.b $$\chi_{392}(197, \cdot)$$ 392.2.b.a 2 1
392.2.b.b 2
392.2.b.c 4
392.2.b.d 4
392.2.b.e 6
392.2.b.f 6
392.2.b.g 12
392.2.e $$\chi_{392}(195, \cdot)$$ 392.2.e.a 4 1
392.2.e.b 4
392.2.e.c 8
392.2.e.d 8
392.2.e.e 12
392.2.f $$\chi_{392}(391, \cdot)$$ None 0 1
392.2.i $$\chi_{392}(177, \cdot)$$ 392.2.i.a 2 2
392.2.i.b 2
392.2.i.c 2
392.2.i.d 2
392.2.i.e 2
392.2.i.f 2
392.2.i.g 4
392.2.i.h 4
392.2.l $$\chi_{392}(31, \cdot)$$ None 0 2
392.2.m $$\chi_{392}(19, \cdot)$$ 392.2.m.a 4 2
392.2.m.b 8
392.2.m.c 8
392.2.m.d 8
392.2.m.e 8
392.2.m.f 8
392.2.m.g 12
392.2.m.h 16
392.2.p $$\chi_{392}(165, \cdot)$$ 392.2.p.a 4 2
392.2.p.b 4
392.2.p.c 4
392.2.p.d 8
392.2.p.e 8
392.2.p.f 8
392.2.p.g 12
392.2.p.h 24
392.2.q $$\chi_{392}(57, \cdot)$$ 392.2.q.a 42 6
392.2.q.b 42
392.2.t $$\chi_{392}(55, \cdot)$$ None 0 6
392.2.u $$\chi_{392}(27, \cdot)$$ 392.2.u.a 324 6
392.2.x $$\chi_{392}(29, \cdot)$$ 392.2.x.a 324 6
392.2.y $$\chi_{392}(9, \cdot)$$ 392.2.y.a 84 12
392.2.y.b 84
392.2.z $$\chi_{392}(37, \cdot)$$ 392.2.z.a 648 12
392.2.bc $$\chi_{392}(3, \cdot)$$ 392.2.bc.a 648 12
392.2.bd $$\chi_{392}(47, \cdot)$$ None 0 12

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(392))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(392)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(28))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(49))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(56))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(98))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(196))$$$$^{\oplus 2}$$