## Defining parameters

 Level: $$N$$ = $$49 = 7^{2}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$4$$ Newform subspaces: $$12$$ Sturm bound: $$784$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(49))$$.

Total New Old
Modular forms 324 307 17
Cusp forms 264 258 6
Eisenstein series 60 49 11

## Trace form

 $$258q - 15q^{2} - 3q^{3} - 15q^{4} - 39q^{5} - 81q^{6} - 42q^{7} + 27q^{8} + 69q^{9} + O(q^{10})$$ $$258q - 15q^{2} - 3q^{3} - 15q^{4} - 39q^{5} - 81q^{6} - 42q^{7} + 27q^{8} + 69q^{9} + 39q^{10} - 15q^{11} - 105q^{12} - 21q^{13} - 42q^{14} - 171q^{15} - 135q^{16} - 171q^{17} + 21q^{18} + 297q^{19} + 315q^{20} + 210q^{21} - 15q^{22} - 435q^{23} - 297q^{24} - 435q^{25} - 21q^{26} - 81q^{27} - 210q^{28} - 51q^{29} + 111q^{30} + 249q^{31} + 621q^{32} - 123q^{33} + 255q^{34} + 42q^{35} - 1953q^{36} - 1443q^{37} - 2649q^{38} - 2163q^{39} - 3861q^{40} - 2121q^{41} + 105q^{42} + 957q^{43} + 3435q^{44} + 4803q^{45} + 4731q^{46} + 2313q^{47} + 7962q^{48} + 3864q^{49} + 4674q^{50} + 3429q^{51} + 6027q^{52} + 2085q^{53} + 2307q^{54} + 1005q^{55} - 252q^{56} - 2859q^{57} - 3537q^{58} - 4455q^{59} - 11445q^{60} - 5163q^{61} - 6801q^{62} - 4872q^{63} - 6105q^{64} - 1113q^{65} - 129q^{66} + 321q^{67} + 903q^{68} + 4623q^{69} + 357q^{70} + 3225q^{71} - 387q^{72} - 843q^{73} + 363q^{74} - 561q^{75} - 777q^{76} - 252q^{77} - 3255q^{78} - 1107q^{79} - 14724q^{80} - 17241q^{81} - 13272q^{82} - 11277q^{83} - 19362q^{84} - 6135q^{85} - 5406q^{86} - 1833q^{87} + 2076q^{88} - 39q^{89} + 14370q^{90} + 5187q^{91} + 7938q^{92} + 21489q^{93} + 18792q^{94} + 19473q^{95} + 35322q^{96} + 11046q^{97} + 27468q^{98} + 15972q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(49))$$

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
49.4.a $$\chi_{49}(1, \cdot)$$ 49.4.a.a 1 1
49.4.a.b 1
49.4.a.c 1
49.4.a.d 1
49.4.a.e 4
49.4.c $$\chi_{49}(18, \cdot)$$ 49.4.c.a 2 2
49.4.c.b 2
49.4.c.c 2
49.4.c.d 2
49.4.c.e 8
49.4.e $$\chi_{49}(8, \cdot)$$ 49.4.e.a 78 6
49.4.g $$\chi_{49}(2, \cdot)$$ 49.4.g.a 156 12

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(49))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(49)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 2}$$