## Defining parameters

 Level: $$N$$ = $$49 = 7^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$4$$ Newform subspaces: $$5$$ Sturm bound: $$392$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(49))$$.

Total New Old
Modular forms 128 118 10
Cusp forms 69 69 0
Eisenstein series 59 49 10

## Trace form

 $$69 q - 18 q^{2} - 17 q^{3} - 14 q^{4} - 15 q^{5} - 9 q^{6} - 14 q^{7} - 24 q^{8} - 8 q^{9} + O(q^{10})$$ $$69 q - 18 q^{2} - 17 q^{3} - 14 q^{4} - 15 q^{5} - 9 q^{6} - 14 q^{7} - 24 q^{8} - 8 q^{9} - 3 q^{10} - 9 q^{11} + 7 q^{12} - 7 q^{13} - 15 q^{15} + 10 q^{16} - 3 q^{17} + 18 q^{18} - q^{19} + 21 q^{20} + 7 q^{21} - 3 q^{22} + 3 q^{23} + 39 q^{24} + 10 q^{25} + 21 q^{26} + 19 q^{27} + 28 q^{28} - 9 q^{29} + 51 q^{30} + 11 q^{31} + 42 q^{32} + 27 q^{33} + 33 q^{34} + 21 q^{35} - 2 q^{36} - 11 q^{37} - 3 q^{38} - 14 q^{39} - 57 q^{40} - 21 q^{41} - 21 q^{42} - 37 q^{43} - 63 q^{44} - 48 q^{45} - 75 q^{46} - 15 q^{47} - 72 q^{48} - 56 q^{49} - 51 q^{50} - 33 q^{51} - 77 q^{52} - 9 q^{53} - 27 q^{54} - 54 q^{55} - 42 q^{56} - q^{57} - 15 q^{58} - 3 q^{59} + 21 q^{60} - 8 q^{61} + 33 q^{62} + 42 q^{63} + 16 q^{64} + 63 q^{65} + 123 q^{66} + 47 q^{67} + 105 q^{68} + 75 q^{69} + 105 q^{70} + 33 q^{71} + 174 q^{72} + 53 q^{73} + 93 q^{74} + 103 q^{75} + 119 q^{76} + 63 q^{77} + 21 q^{78} + 59 q^{79} + 102 q^{80} + 16 q^{81} - 21 q^{83} - 56 q^{84} - 15 q^{85} - 36 q^{86} - 69 q^{87} - 72 q^{88} - 15 q^{89} - 186 q^{90} - 7 q^{91} - 18 q^{92} - 145 q^{93} - 66 q^{94} - 69 q^{95} - 168 q^{96} - 28 q^{97} - 168 q^{98} - 72 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(49))$$

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
49.2.a $$\chi_{49}(1, \cdot)$$ 49.2.a.a 1 1
49.2.c $$\chi_{49}(18, \cdot)$$ 49.2.c.a 2 2
49.2.e $$\chi_{49}(8, \cdot)$$ 49.2.e.a 6 6
49.2.e.b 12
49.2.g $$\chi_{49}(2, \cdot)$$ 49.2.g.a 48 12