## Defining parameters

 Level: $$N$$ = $$196 = 2^{2} \cdot 7^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$8$$ Newform subspaces: $$16$$ Sturm bound: $$4704$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(196))$$.

Total New Old
Modular forms 1326 706 620
Cusp forms 1027 610 417
Eisenstein series 299 96 203

## Trace form

 $$610q - 15q^{2} + 2q^{3} - 15q^{4} - 24q^{5} - 21q^{6} + 4q^{7} - 33q^{8} - 34q^{9} + O(q^{10})$$ $$610q - 15q^{2} + 2q^{3} - 15q^{4} - 24q^{5} - 21q^{6} + 4q^{7} - 33q^{8} - 34q^{9} - 33q^{10} - 6q^{11} - 45q^{12} - 50q^{13} - 30q^{14} - 12q^{15} - 39q^{16} - 24q^{17} - 27q^{18} - 2q^{19} - 21q^{20} - 47q^{21} - 3q^{22} + 6q^{23} + 3q^{24} - 46q^{25} + 3q^{26} + 20q^{27} + 6q^{28} - 78q^{29} - 9q^{30} - 14q^{31} - 15q^{32} - 60q^{33} - 21q^{34} - 15q^{35} - 57q^{36} - 84q^{37} - 57q^{38} - 45q^{39} - 45q^{40} - 108q^{41} - 33q^{42} - 26q^{43} - 57q^{44} - 159q^{45} - 45q^{46} - 60q^{47} - 110q^{49} - 66q^{50} - 78q^{51} - 21q^{52} - 42q^{53} + 15q^{54} - 69q^{55} - 30q^{56} - 44q^{57} - 9q^{58} - 24q^{59} + 3q^{60} - 57q^{61} - 21q^{62} - 18q^{63} - 57q^{64} - 54q^{65} - 9q^{66} - 14q^{67} - 21q^{68} - 54q^{69} - 27q^{70} + 9q^{72} - 104q^{73} + 3q^{74} + 8q^{75} - 21q^{76} - 63q^{77} + 21q^{78} - 26q^{79} + 90q^{80} - 28q^{81} + 60q^{82} + 36q^{83} + 210q^{84} - 54q^{85} + 162q^{86} + 156q^{87} + 174q^{88} - 36q^{89} + 378q^{90} + 92q^{91} + 120q^{92} + 184q^{93} + 228q^{94} + 162q^{95} + 384q^{96} + 124q^{97} + 288q^{98} + 144q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(196))$$

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
196.2.a $$\chi_{196}(1, \cdot)$$ 196.2.a.a 1 1
196.2.a.b 1
196.2.a.c 2
196.2.d $$\chi_{196}(195, \cdot)$$ 196.2.d.a 4 1
196.2.d.b 4
196.2.d.c 8
196.2.e $$\chi_{196}(165, \cdot)$$ 196.2.e.a 2 2
196.2.e.b 4
196.2.f $$\chi_{196}(19, \cdot)$$ 196.2.f.a 4 2
196.2.f.b 4
196.2.f.c 8
196.2.f.d 16
196.2.i $$\chi_{196}(29, \cdot)$$ 196.2.i.a 24 6
196.2.j $$\chi_{196}(27, \cdot)$$ 196.2.j.a 156 6
196.2.m $$\chi_{196}(9, \cdot)$$ 196.2.m.a 60 12
196.2.p $$\chi_{196}(3, \cdot)$$ 196.2.p.a 312 12

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(196))$$ into lower level spaces

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