## 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

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

## 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

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