## Defining parameters

 Level: $$N$$ = $$57 = 3 \cdot 19$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$6$$ Newform subspaces: $$11$$ Sturm bound: $$720$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(57))$$.

Total New Old
Modular forms 276 194 82
Cusp forms 204 162 42
Eisenstein series 72 32 40

## Trace form

 $$162q - 9q^{3} - 18q^{4} - 9q^{6} - 18q^{7} - 9q^{9} + O(q^{10})$$ $$162q - 9q^{3} - 18q^{4} - 9q^{6} - 18q^{7} - 9q^{9} - 18q^{10} - 81q^{12} - 138q^{13} - 144q^{14} - 63q^{15} - 162q^{16} - 18q^{17} - 18q^{18} + 24q^{19} + 144q^{20} + 54q^{21} + 198q^{22} + 90q^{23} + 207q^{24} + 234q^{25} + 288q^{26} - 72q^{27} - 264q^{28} - 288q^{29} - 450q^{30} - 234q^{31} - 450q^{32} - 225q^{33} - 288q^{34} - 144q^{35} - 63q^{36} - 36q^{37} + 126q^{38} + 90q^{39} + 432q^{40} + 144q^{41} + 441q^{42} + 414q^{43} + 630q^{44} + 720q^{45} + 792q^{46} + 360q^{47} + 1188q^{48} + 414q^{49} + 702q^{50} + 342q^{51} - 18q^{52} + 99q^{54} - 18q^{55} - 54q^{57} - 36q^{58} - 630q^{60} - 858q^{61} - 1170q^{62} - 405q^{63} - 1602q^{64} - 1080q^{65} - 1377q^{66} - 1170q^{67} - 882q^{68} - 792q^{69} - 1098q^{70} - 288q^{71} - 747q^{72} - 342q^{73} - 144q^{74} - 18q^{75} + 162q^{76} + 54q^{77} - 279q^{78} + 114q^{79} - 792q^{80} - 477q^{81} + 1008q^{82} - 18q^{83} - 117q^{84} + 774q^{85} + 810q^{86} + 351q^{87} + 774q^{88} + 1242q^{89} + 126q^{90} + 1302q^{91} + 1242q^{92} + 459q^{93} + 648q^{94} + 54q^{95} + 1494q^{96} + 144q^{97} + 576q^{98} + 1233q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(57))$$

We only show spaces with odd 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
57.3.b $$\chi_{57}(20, \cdot)$$ 57.3.b.a 12 1
57.3.c $$\chi_{57}(37, \cdot)$$ 57.3.c.a 2 1
57.3.c.b 4
57.3.g $$\chi_{57}(31, \cdot)$$ 57.3.g.a 6 2
57.3.g.b 6
57.3.h $$\chi_{57}(11, \cdot)$$ 57.3.h.a 8 2
57.3.h.b 16
57.3.k $$\chi_{57}(10, \cdot)$$ 57.3.k.a 18 6
57.3.k.b 24
57.3.l $$\chi_{57}(5, \cdot)$$ 57.3.l.a 6 6
57.3.l.b 60

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(57))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(57)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(19))$$$$^{\oplus 2}$$