## Defining parameters

 Level: $$N$$ = $$52 = 2^{2} \cdot 13$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$5$$ Newform subspaces: $$8$$ Sturm bound: $$336$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(52))$$.

Total New Old
Modular forms 114 61 53
Cusp forms 55 37 18
Eisenstein series 59 24 35

## Trace form

 $$37 q - 6 q^{2} - 6 q^{4} - 12 q^{5} - 6 q^{6} - 2 q^{7} - 6 q^{8} - 20 q^{9} + O(q^{10})$$ $$37 q - 6 q^{2} - 6 q^{4} - 12 q^{5} - 6 q^{6} - 2 q^{7} - 6 q^{8} - 20 q^{9} - 6 q^{10} - 6 q^{11} - 24 q^{13} - 12 q^{14} - 12 q^{15} - 6 q^{16} - 15 q^{17} + 12 q^{18} + 4 q^{19} + 18 q^{20} + 10 q^{21} + 24 q^{22} + 12 q^{23} + 42 q^{24} + 15 q^{25} + 24 q^{26} + 36 q^{27} + 24 q^{28} - 9 q^{29} + 42 q^{30} - 2 q^{31} + 24 q^{32} - 6 q^{33} + 18 q^{34} - 18 q^{35} - 6 q^{36} - 23 q^{37} - 20 q^{39} - 36 q^{40} - 3 q^{41} - 24 q^{42} + 2 q^{43} - 36 q^{44} + 3 q^{45} - 66 q^{46} - 6 q^{47} - 54 q^{48} + 50 q^{49} - 48 q^{50} - 60 q^{52} - 48 q^{54} + 6 q^{55} - 60 q^{56} + 34 q^{57} - 54 q^{58} - 6 q^{59} - 36 q^{60} + 21 q^{61} - 18 q^{62} - 2 q^{63} + 33 q^{65} + 24 q^{66} + 4 q^{67} + 72 q^{68} - 18 q^{69} + 36 q^{70} + 36 q^{71} + 72 q^{72} + 16 q^{73} + 78 q^{74} + 16 q^{75} + 54 q^{76} + 12 q^{77} + 108 q^{78} - 12 q^{79} + 108 q^{80} + 34 q^{81} + 54 q^{82} + 42 q^{83} + 36 q^{84} - 3 q^{85} + 48 q^{86} + 12 q^{87} + 36 q^{88} - 12 q^{89} - 2 q^{91} + 12 q^{92} - 26 q^{93} - 18 q^{94} - 96 q^{96} - 68 q^{97} - 18 q^{98} - 54 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(52))$$

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
52.2.a $$\chi_{52}(1, \cdot)$$ 52.2.a.a 1 1
52.2.d $$\chi_{52}(25, \cdot)$$ None 0 1
52.2.e $$\chi_{52}(9, \cdot)$$ 52.2.e.a 2 2
52.2.e.b 2
52.2.f $$\chi_{52}(31, \cdot)$$ 52.2.f.a 2 2
52.2.f.b 8
52.2.h $$\chi_{52}(17, \cdot)$$ 52.2.h.a 2 2
52.2.l $$\chi_{52}(7, \cdot)$$ 52.2.l.a 4 4
52.2.l.b 16

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(52))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(52)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(13))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(26))$$$$^{\oplus 2}$$