## Defining parameters

 Level: $$N$$ = $$56 = 2^{3} \cdot 7$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$6$$ Newform subspaces: $$10$$ Sturm bound: $$384$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(56))$$.

Total New Old
Modular forms 132 62 70
Cusp forms 61 42 19
Eisenstein series 71 20 51

## Trace form

 $$42 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 6 q^{6} - 6 q^{7} - 12 q^{8} - 18 q^{9} + O(q^{10})$$ $$42 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 6 q^{6} - 6 q^{7} - 12 q^{8} - 18 q^{9} - 6 q^{10} - 12 q^{11} - 6 q^{12} - 6 q^{13} - 6 q^{14} - 24 q^{15} - 6 q^{16} - 18 q^{17} + 12 q^{18} - 6 q^{19} + 12 q^{20} + 18 q^{22} + 12 q^{23} + 30 q^{24} + 6 q^{25} + 24 q^{26} + 24 q^{27} + 42 q^{28} + 42 q^{30} + 12 q^{31} + 24 q^{32} - 6 q^{33} + 30 q^{34} - 6 q^{35} + 18 q^{36} - 6 q^{37} + 12 q^{38} + 24 q^{40} - 36 q^{41} - 42 q^{42} - 12 q^{43} - 30 q^{44} - 6 q^{45} - 36 q^{46} + 12 q^{47} - 66 q^{48} - 6 q^{49} - 54 q^{50} + 36 q^{51} - 60 q^{52} + 6 q^{53} - 78 q^{54} + 24 q^{55} - 48 q^{56} - 36 q^{58} + 30 q^{59} - 72 q^{60} - 60 q^{62} + 30 q^{63} - 54 q^{64} - 12 q^{65} - 30 q^{66} + 12 q^{67} - 6 q^{68} + 12 q^{69} + 18 q^{70} + 24 q^{71} + 12 q^{72} - 30 q^{73} + 78 q^{74} + 54 q^{75} + 66 q^{76} + 18 q^{77} + 72 q^{78} + 12 q^{79} + 108 q^{80} + 12 q^{81} + 66 q^{82} + 30 q^{83} + 102 q^{84} + 102 q^{86} + 36 q^{87} + 66 q^{88} - 6 q^{89} + 72 q^{90} - 18 q^{91} + 66 q^{92} + 6 q^{93} + 30 q^{94} - 12 q^{95} + 18 q^{96} + 36 q^{97} + 18 q^{98} - 84 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(56))$$

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
56.2.a $$\chi_{56}(1, \cdot)$$ 56.2.a.a 1 1
56.2.a.b 1
56.2.b $$\chi_{56}(29, \cdot)$$ 56.2.b.a 2 1
56.2.b.b 4
56.2.e $$\chi_{56}(27, \cdot)$$ 56.2.e.a 2 1
56.2.e.b 4
56.2.f $$\chi_{56}(55, \cdot)$$ None 0 1
56.2.i $$\chi_{56}(9, \cdot)$$ 56.2.i.a 2 2
56.2.i.b 2
56.2.l $$\chi_{56}(31, \cdot)$$ None 0 2
56.2.m $$\chi_{56}(3, \cdot)$$ 56.2.m.a 12 2
56.2.p $$\chi_{56}(37, \cdot)$$ 56.2.p.a 12 2

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(56))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(56)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(28))$$$$^{\oplus 2}$$