## Defining parameters

 Level: $$N$$ = $$255 = 3 \cdot 5 \cdot 17$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$18$$ Newforms: $$31$$ Sturm bound: $$9216$$ Trace bound: $$10$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(255))$$.

Total New Old
Modular forms 2560 1663 897
Cusp forms 2049 1487 562
Eisenstein series 511 176 335

## Trace form

 $$1487q + 5q^{2} - 13q^{3} - 23q^{4} - q^{5} - 47q^{6} - 24q^{7} + 9q^{8} - 17q^{9} + O(q^{10})$$ $$1487q + 5q^{2} - 13q^{3} - 23q^{4} - q^{5} - 47q^{6} - 24q^{7} + 9q^{8} - 17q^{9} - 51q^{10} - 12q^{11} - 59q^{12} - 46q^{13} - 40q^{14} - 45q^{15} - 207q^{16} - 17q^{17} - 91q^{18} - 52q^{19} - 47q^{20} - 88q^{21} - 68q^{22} - 8q^{23} - 75q^{24} - 105q^{25} - 106q^{26} - 13q^{27} - 168q^{28} - 46q^{29} - 87q^{30} - 192q^{31} - 87q^{32} - 92q^{33} - 235q^{34} - 56q^{35} - 103q^{36} - 102q^{37} - 92q^{38} - 38q^{39} - 167q^{40} - 58q^{41} + 8q^{42} - 92q^{43} - 52q^{44} - q^{45} - 88q^{46} + 32q^{47} + 125q^{48} + 39q^{49} + 5q^{50} + 35q^{51} + 30q^{52} + 26q^{53} + 17q^{54} - 92q^{55} - 40q^{56} - 20q^{57} - 170q^{58} - 60q^{59} - 59q^{60} - 222q^{61} - 128q^{62} - 152q^{63} + 17q^{64} - 102q^{65} - 164q^{66} - 148q^{67} - 183q^{68} - 168q^{69} - 40q^{70} + 24q^{71} - 103q^{72} - 58q^{73} + 46q^{74} - 21q^{75} + 84q^{76} + 32q^{77} - 50q^{78} + 48q^{79} + 401q^{80} - 161q^{81} + 226q^{82} + 28q^{83} + 248q^{84} + 231q^{85} + 268q^{86} + 202q^{87} + 556q^{88} + 198q^{89} + 85q^{90} + 208q^{91} + 488q^{92} + 208q^{93} + 384q^{94} + 204q^{95} + 341q^{96} + 190q^{97} + 301q^{98} + 148q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(255))$$

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
255.2.a $$\chi_{255}(1, \cdot)$$ 255.2.a.a 2 1
255.2.a.b 2
255.2.a.c 3
255.2.a.d 4
255.2.b $$\chi_{255}(154, \cdot)$$ 255.2.b.a 2 1
255.2.b.b 4
255.2.b.c 10
255.2.d $$\chi_{255}(169, \cdot)$$ 255.2.d.a 8 1
255.2.d.b 8
255.2.g $$\chi_{255}(16, \cdot)$$ 255.2.g.a 4 1
255.2.g.b 8
255.2.j $$\chi_{255}(106, \cdot)$$ 255.2.j.a 8 2
255.2.j.b 16
255.2.k $$\chi_{255}(98, \cdot)$$ 255.2.k.a 8 2
255.2.k.b 56
255.2.m $$\chi_{255}(137, \cdot)$$ 255.2.m.a 64 2
255.2.o $$\chi_{255}(152, \cdot)$$ 255.2.o.a 8 2
255.2.o.b 56
255.2.r $$\chi_{255}(38, \cdot)$$ 255.2.r.a 8 2
255.2.r.b 56
255.2.s $$\chi_{255}(4, \cdot)$$ 255.2.s.a 32 2
255.2.v $$\chi_{255}(53, \cdot)$$ 255.2.v.a 128 4
255.2.w $$\chi_{255}(76, \cdot)$$ 255.2.w.a 16 4
255.2.w.b 32
255.2.z $$\chi_{255}(19, \cdot)$$ 255.2.z.a 80 4
255.2.ba $$\chi_{255}(2, \cdot)$$ 255.2.ba.a 128 4
255.2.bd $$\chi_{255}(7, \cdot)$$ 255.2.bd.a 144 8
255.2.be $$\chi_{255}(14, \cdot)$$ 255.2.be.a 32 8
255.2.be.b 224
255.2.bg $$\chi_{255}(11, \cdot)$$ 255.2.bg.a 192 8
255.2.bi $$\chi_{255}(22, \cdot)$$ 255.2.bi.a 144 8

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(255))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(255)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(17))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(51))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(85))$$$$^{\oplus 2}$$