## Defining parameters

 Level: $$N$$ = $$56 = 2^{3} \cdot 7$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$6$$ Newform subspaces: $$12$$ Sturm bound: $$768$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(56))$$.

Total New Old
Modular forms 324 164 160
Cusp forms 252 144 108
Eisenstein series 72 20 52

## Trace form

 $$144q - 2q^{2} + 2q^{3} + 18q^{4} + 4q^{5} - 62q^{6} - 14q^{7} - 92q^{8} - 28q^{9} + O(q^{10})$$ $$144q - 2q^{2} + 2q^{3} + 18q^{4} + 4q^{5} - 62q^{6} - 14q^{7} - 92q^{8} - 28q^{9} + 106q^{10} + 124q^{11} + 106q^{12} + 34q^{13} - 22q^{14} - 120q^{15} - 38q^{16} - 50q^{17} + 152q^{18} + 74q^{19} + 292q^{20} - 72q^{21} + 246q^{22} + 588q^{23} - 322q^{24} - 200q^{25} - 1316q^{26} - 952q^{27} - 1398q^{28} - 396q^{29} - 790q^{30} - 1356q^{31} + 488q^{32} - 1018q^{33} + 478q^{34} + 546q^{35} + 1314q^{36} + 954q^{37} + 1880q^{38} + 2712q^{39} + 296q^{40} + 1648q^{41} - 1078q^{42} + 1564q^{43} - 2190q^{44} + 1642q^{45} - 1664q^{46} - 1596q^{47} - 1410q^{48} + 928q^{49} + 1090q^{50} - 2588q^{51} + 3436q^{52} - 1490q^{53} + 3038q^{54} - 3848q^{55} + 2624q^{56} - 3408q^{57} + 3780q^{58} - 1226q^{59} + 4440q^{60} - 1604q^{61} + 1592q^{62} + 950q^{63} - 1878q^{64} - 652q^{65} - 1674q^{66} + 5036q^{67} + 1338q^{68} + 5780q^{69} + 578q^{70} + 4472q^{71} + 236q^{72} + 2386q^{73} - 1994q^{74} + 118q^{75} - 2734q^{76} - 186q^{77} - 5584q^{78} + 12q^{79} - 5412q^{80} - 666q^{81} - 6118q^{82} - 7666q^{83} - 9418q^{84} - 6184q^{85} - 3674q^{86} - 7188q^{87} - 5214q^{88} - 3158q^{89} - 6088q^{90} - 3066q^{91} - 6198q^{92} + 3130q^{93} - 3738q^{94} + 772q^{95} - 718q^{96} + 4072q^{97} + 4150q^{98} + 6532q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(56))$$

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
56.4.a $$\chi_{56}(1, \cdot)$$ 56.4.a.a 1 1
56.4.a.b 1
56.4.a.c 2
56.4.b $$\chi_{56}(29, \cdot)$$ 56.4.b.a 8 1
56.4.b.b 10
56.4.e $$\chi_{56}(27, \cdot)$$ 56.4.e.a 2 1
56.4.e.b 4
56.4.e.c 16
56.4.f $$\chi_{56}(55, \cdot)$$ None 0 1
56.4.i $$\chi_{56}(9, \cdot)$$ 56.4.i.a 6 2
56.4.i.b 6
56.4.l $$\chi_{56}(31, \cdot)$$ None 0 2
56.4.m $$\chi_{56}(3, \cdot)$$ 56.4.m.a 44 2
56.4.p $$\chi_{56}(37, \cdot)$$ 56.4.p.a 44 2

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(56))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(56)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(28))$$$$^{\oplus 2}$$