## Defining parameters

 Level: $$N$$ = $$196 = 2^{2} \cdot 7^{2}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$8$$ Newform subspaces: $$27$$ Sturm bound: $$9408$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(196))$$.

Total New Old
Modular forms 3678 2022 1656
Cusp forms 3378 1926 1452
Eisenstein series 300 96 204

## Trace form

 $$1926q - 15q^{2} + 12q^{3} - 15q^{4} - 54q^{5} - 21q^{6} - 24q^{7} - 129q^{8} - 114q^{9} + O(q^{10})$$ $$1926q - 15q^{2} + 12q^{3} - 15q^{4} - 54q^{5} - 21q^{6} - 24q^{7} - 129q^{8} - 114q^{9} + 3q^{10} + 168q^{11} + 315q^{12} + 258q^{13} + 138q^{14} + 384q^{15} + 153q^{16} - 162q^{17} - 63q^{18} - 636q^{19} - 21q^{20} - 684q^{21} - 291q^{22} - 168q^{23} - 717q^{24} + 1062q^{25} - 813q^{26} + 1224q^{27} - 666q^{28} + 1722q^{29} - 1821q^{30} - 672q^{31} - 1695q^{32} - 1842q^{33} - 21q^{34} - 498q^{35} + 1983q^{36} - 3474q^{37} + 3219q^{38} - 1050q^{39} + 3075q^{40} + 1782q^{41} + 1983q^{42} + 1644q^{43} + 4143q^{44} + 5766q^{45} + 2295q^{46} + 2508q^{47} + 3642q^{49} - 2958q^{50} + 2016q^{51} - 5205q^{52} - 1122q^{53} - 9165q^{54} - 3114q^{55} - 3558q^{56} - 4518q^{57} - 5049q^{58} - 4344q^{59} - 4437q^{60} - 4794q^{61} - 21q^{62} - 3588q^{63} + 3495q^{64} - 690q^{65} + 8523q^{66} + 1344q^{67} + 12147q^{68} - 978q^{69} + 7323q^{70} - 1008q^{71} + 11961q^{72} + 1374q^{73} + 3807q^{74} + 3636q^{75} - 21q^{76} + 3150q^{77} - 3399q^{78} + 2184q^{79} - 918q^{80} + 16002q^{81} - 390q^{82} + 5832q^{83} + 8946q^{84} + 1542q^{85} - 9576q^{86} + 3120q^{87} - 8778q^{88} + 8394q^{89} - 16380q^{90} - 2610q^{91} - 3156q^{92} - 7254q^{93} - 2526q^{94} - 13440q^{95} - 10368q^{96} - 10068q^{97} - 18780q^{98} - 11664q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(196))$$

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
196.4.a $$\chi_{196}(1, \cdot)$$ 196.4.a.a 1 1
196.4.a.b 1
196.4.a.c 1
196.4.a.d 1
196.4.a.e 2
196.4.a.f 2
196.4.a.g 2
196.4.d $$\chi_{196}(195, \cdot)$$ 196.4.d.a 4 1
196.4.d.b 20
196.4.d.c 32
196.4.e $$\chi_{196}(165, \cdot)$$ 196.4.e.a 2 2
196.4.e.b 2
196.4.e.c 2
196.4.e.d 2
196.4.e.e 2
196.4.e.f 2
196.4.e.g 4
196.4.e.h 4
196.4.f $$\chi_{196}(19, \cdot)$$ 196.4.f.a 4 2
196.4.f.b 8
196.4.f.c 16
196.4.f.d 20
196.4.f.e 64
196.4.i $$\chi_{196}(29, \cdot)$$ 196.4.i.a 84 6
196.4.j $$\chi_{196}(27, \cdot)$$ 196.4.j.a 492 6
196.4.m $$\chi_{196}(9, \cdot)$$ 196.4.m.a 168 12
196.4.p $$\chi_{196}(3, \cdot)$$ 196.4.p.a 984 12

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(196))$$ into lower level spaces

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