## Defining parameters

 Level: $$N$$ = $$270 = 2 \cdot 3^{3} \cdot 5$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$9$$ Newform subspaces: $$23$$ Sturm bound: $$7776$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(270))$$.

Total New Old
Modular forms 2184 468 1716
Cusp forms 1705 468 1237
Eisenstein series 479 0 479

## Trace form

 $$468 q - 2 q^{2} - 2 q^{4} + 6 q^{5} + 12 q^{6} + 8 q^{7} + 10 q^{8} + 24 q^{9} + O(q^{10})$$ $$468 q - 2 q^{2} - 2 q^{4} + 6 q^{5} + 12 q^{6} + 8 q^{7} + 10 q^{8} + 24 q^{9} + 14 q^{10} + 52 q^{11} + 6 q^{12} + 36 q^{13} + 24 q^{14} + 18 q^{15} - 2 q^{16} + 36 q^{17} - 12 q^{18} + 12 q^{19} - 4 q^{20} - 48 q^{21} + 8 q^{22} - 12 q^{23} + 26 q^{25} - 64 q^{26} - 54 q^{27} - 4 q^{28} - 48 q^{29} - 36 q^{30} + 24 q^{31} - 2 q^{32} - 78 q^{33} - 20 q^{34} - 102 q^{35} - 36 q^{36} - 24 q^{37} - 94 q^{38} - 36 q^{39} - 26 q^{40} - 140 q^{41} - 48 q^{42} - 72 q^{43} - 40 q^{44} - 120 q^{45} - 48 q^{46} - 180 q^{47} - 12 q^{48} - 66 q^{49} - 114 q^{50} - 96 q^{51} - 12 q^{52} - 96 q^{53} - 14 q^{55} - 32 q^{56} - 54 q^{57} + 4 q^{58} - 26 q^{59} + 18 q^{60} + 52 q^{61} + 56 q^{62} + 12 q^{63} + 10 q^{64} + 50 q^{65} - 52 q^{67} + 54 q^{68} - 36 q^{69} - 10 q^{70} - 48 q^{72} + 28 q^{73} - 12 q^{74} - 24 q^{75} - 14 q^{76} - 168 q^{77} - 72 q^{78} - 240 q^{79} - 12 q^{80} - 96 q^{81} - 28 q^{82} - 168 q^{83} - 120 q^{85} + 4 q^{86} - 192 q^{87} - 46 q^{88} - 130 q^{89} - 18 q^{90} - 60 q^{91} + 24 q^{92} - 132 q^{93} - 28 q^{94} - 142 q^{95} - 12 q^{96} - 72 q^{97} + 66 q^{98} - 48 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(270))$$

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
270.2.a $$\chi_{270}(1, \cdot)$$ 270.2.a.a 1 1
270.2.a.b 1
270.2.a.c 1
270.2.a.d 1
270.2.c $$\chi_{270}(109, \cdot)$$ 270.2.c.a 2 1
270.2.c.b 2
270.2.c.c 4
270.2.e $$\chi_{270}(91, \cdot)$$ 270.2.e.a 2 2
270.2.e.b 2
270.2.e.c 4
270.2.f $$\chi_{270}(53, \cdot)$$ 270.2.f.a 8 2
270.2.f.b 8
270.2.i $$\chi_{270}(19, \cdot)$$ 270.2.i.a 4 2
270.2.i.b 8
270.2.k $$\chi_{270}(31, \cdot)$$ 270.2.k.a 6 6
270.2.k.b 12
270.2.k.c 12
270.2.k.d 18
270.2.k.e 24
270.2.m $$\chi_{270}(17, \cdot)$$ 270.2.m.a 8 4
270.2.m.b 16
270.2.p $$\chi_{270}(49, \cdot)$$ 270.2.p.a 108 6
270.2.r $$\chi_{270}(23, \cdot)$$ 270.2.r.a 216 12

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(270))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(270)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(45))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(54))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(90))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(135))$$$$^{\oplus 2}$$