## Defining parameters

 Level: $$N$$ = $$51 = 3 \cdot 17$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$5$$ Newform subspaces: $$7$$ Sturm bound: $$384$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(51))$$.

Total New Old
Modular forms 128 87 41
Cusp forms 65 55 10
Eisenstein series 63 32 31

## Trace form

 $$55q - 3q^{2} - 9q^{3} - 23q^{4} - 6q^{5} - 11q^{6} - 24q^{7} - 15q^{8} - 9q^{9} + O(q^{10})$$ $$55q - 3q^{2} - 9q^{3} - 23q^{4} - 6q^{5} - 11q^{6} - 24q^{7} - 15q^{8} - 9q^{9} - 26q^{10} + 4q^{11} + 9q^{12} - 14q^{13} + 8q^{14} + 10q^{15} + 25q^{16} - q^{17} + 13q^{18} - 20q^{19} + 14q^{20} + 8q^{21} - 20q^{22} - 8q^{23} + 17q^{24} + 9q^{25} + 30q^{26} - 9q^{27} + 24q^{28} + 10q^{29} + 38q^{30} + 16q^{31} + 17q^{32} + 20q^{33} + 61q^{34} + 16q^{35} + 17q^{36} + 10q^{37} + 20q^{38} - 6q^{39} + 22q^{40} - 2q^{41} - 32q^{42} - 12q^{43} - 20q^{44} - 38q^{45} - 56q^{46} - 48q^{47} - 95q^{48} - 73q^{49} - 93q^{50} - 57q^{51} - 130q^{52} - 30q^{53} - 27q^{54} - 24q^{55} - 40q^{56} - 12q^{57} + 6q^{58} + 4q^{59} - 10q^{60} + 18q^{61} + 16q^{62} + 56q^{63} + 17q^{64} + 36q^{65} + 36q^{66} - 4q^{67} + 81q^{68} + 72q^{69} + 112q^{70} + 24q^{71} + 73q^{72} + 110q^{73} + 54q^{74} + 89q^{75} + 84q^{76} + 64q^{77} + 118q^{78} + 32q^{79} + 86q^{80} + 47q^{81} + 74q^{82} + 28q^{83} + 40q^{84} + 2q^{85} - 68q^{86} - 38q^{87} - 4q^{88} - 42q^{89} - 98q^{90} - 32q^{91} - 8q^{92} - 72q^{93} - 32q^{94} - 56q^{95} - 111q^{96} - 50q^{97} - 99q^{98} - 60q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(51))$$

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
51.2.a $$\chi_{51}(1, \cdot)$$ 51.2.a.a 1 1
51.2.a.b 2
51.2.d $$\chi_{51}(16, \cdot)$$ 51.2.d.a 2 1
51.2.d.b 2
51.2.e $$\chi_{51}(4, \cdot)$$ 51.2.e.a 8 2
51.2.h $$\chi_{51}(19, \cdot)$$ 51.2.h.a 8 4
51.2.i $$\chi_{51}(5, \cdot)$$ 51.2.i.a 32 8

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(51))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(51)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(17))$$$$^{\oplus 2}$$