## Defining parameters

 Level: $$N$$ = $$392 = 2^{3} \cdot 7^{2}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$12$$ Sturm bound: $$37632$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(392))$$.

Total New Old
Modular forms 14472 7715 6757
Cusp forms 13752 7521 6231
Eisenstein series 720 194 526

## Trace form

 $$7521q - 32q^{2} - 34q^{3} - 42q^{4} - 2q^{5} - 2q^{6} - 36q^{7} - 14q^{8} + 11q^{9} + O(q^{10})$$ $$7521q - 32q^{2} - 34q^{3} - 42q^{4} - 2q^{5} - 2q^{6} - 36q^{7} - 14q^{8} + 11q^{9} - 86q^{10} - 158q^{11} - 86q^{12} - 134q^{13} - 36q^{14} - 198q^{15} - 14q^{16} - 50q^{17} - 352q^{18} - 322q^{19} - 962q^{20} + 168q^{21} - 318q^{22} - 138q^{23} + 938q^{24} + 601q^{25} + 1750q^{26} + 1406q^{27} + 1260q^{28} + 198q^{29} + 1874q^{30} + 1806q^{31} + 38q^{32} + 920q^{33} - 1082q^{34} - 540q^{35} - 2742q^{36} - 1422q^{37} - 3214q^{38} - 3522q^{39} - 1306q^{40} - 2366q^{41} + 1260q^{42} - 3362q^{43} + 3834q^{44} - 3350q^{45} + 2374q^{46} - 450q^{47} + 762q^{48} + 342q^{49} - 2222q^{50} + 5734q^{51} - 5234q^{52} + 3706q^{53} - 8302q^{54} + 8398q^{55} - 2346q^{56} + 6012q^{57} - 5082q^{58} + 4414q^{59} - 6906q^{60} + 1558q^{61} - 1882q^{62} - 744q^{63} + 1950q^{64} - 2008q^{65} + 3810q^{66} - 10678q^{67} - 3222q^{68} - 12232q^{69} - 1068q^{70} - 10774q^{71} - 394q^{72} - 3554q^{73} + 82q^{74} - 1730q^{75} + 4250q^{76} + 1242q^{77} + 9410q^{78} + 4686q^{79} + 14814q^{80} - 7989q^{81} + 11774q^{82} + 5090q^{83} + 8928q^{84} + 7712q^{85} + 14278q^{86} + 9486q^{87} + 9378q^{88} + 4594q^{89} + 11966q^{90} + 7116q^{91} + 6846q^{92} + 11224q^{93} + 9450q^{94} + 16318q^{95} + 2738q^{96} - 4610q^{97} - 5436q^{98} + 1324q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(392))$$

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
392.4.a $$\chi_{392}(1, \cdot)$$ 392.4.a.a 1 1
392.4.a.b 1
392.4.a.c 1
392.4.a.d 1
392.4.a.e 1
392.4.a.f 2
392.4.a.g 2
392.4.a.h 2
392.4.a.i 3
392.4.a.j 3
392.4.a.k 3
392.4.a.l 3
392.4.a.m 4
392.4.a.n 4
392.4.b $$\chi_{392}(197, \cdot)$$ n/a 118 1
392.4.e $$\chi_{392}(195, \cdot)$$ n/a 116 1
392.4.f $$\chi_{392}(391, \cdot)$$ None 0 1
392.4.i $$\chi_{392}(177, \cdot)$$ 392.4.i.a 2 2
392.4.i.b 2
392.4.i.c 2
392.4.i.d 2
392.4.i.e 2
392.4.i.f 2
392.4.i.g 2
392.4.i.h 2
392.4.i.i 4
392.4.i.j 4
392.4.i.k 4
392.4.i.l 4
392.4.i.m 6
392.4.i.n 6
392.4.i.o 8
392.4.i.p 8
392.4.l $$\chi_{392}(31, \cdot)$$ None 0 2
392.4.m $$\chi_{392}(19, \cdot)$$ n/a 232 2
392.4.p $$\chi_{392}(165, \cdot)$$ n/a 232 2
392.4.q $$\chi_{392}(57, \cdot)$$ n/a 252 6
392.4.t $$\chi_{392}(55, \cdot)$$ None 0 6
392.4.u $$\chi_{392}(27, \cdot)$$ n/a 996 6
392.4.x $$\chi_{392}(29, \cdot)$$ n/a 996 6
392.4.y $$\chi_{392}(9, \cdot)$$ n/a 504 12
392.4.z $$\chi_{392}(37, \cdot)$$ n/a 1992 12
392.4.bc $$\chi_{392}(3, \cdot)$$ n/a 1992 12
392.4.bd $$\chi_{392}(47, \cdot)$$ None 0 12

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(392))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(392)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(28))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(49))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(56))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(98))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(196))$$$$^{\oplus 2}$$