## Defining parameters

 Level: $$N$$ = $$27 = 3^{3}$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$3$$ Newform subspaces: $$4$$ Sturm bound: $$162$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(27))$$.

Total New Old
Modular forms 69 51 18
Cusp forms 39 35 4
Eisenstein series 30 16 14

## Trace form

 $$35q - 3q^{2} - 6q^{3} - 13q^{4} - 21q^{5} - 18q^{6} - 11q^{7} - 9q^{8} + O(q^{10})$$ $$35q - 3q^{2} - 6q^{3} - 13q^{4} - 21q^{5} - 18q^{6} - 11q^{7} - 9q^{8} + 3q^{10} - 3q^{11} - 15q^{12} - 23q^{13} - 21q^{14} - 9q^{15} - 13q^{16} - 9q^{17} + 63q^{18} - 2q^{19} + 219q^{20} + 132q^{21} + 51q^{22} + 168q^{23} + 144q^{24} + 29q^{25} - 90q^{27} - 110q^{28} - 246q^{29} - 243q^{30} - 41q^{31} - 387q^{32} - 207q^{33} - 81q^{34} - 252q^{35} - 360q^{36} + 16q^{37} - 51q^{38} + 15q^{39} - 21q^{40} + 249q^{41} + 486q^{42} + 43q^{43} + 639q^{44} + 477q^{45} + 165q^{46} + 483q^{47} + 453q^{48} + 39q^{49} + 264q^{50} + 36q^{51} + 91q^{52} - 54q^{54} - 114q^{55} - 363q^{56} - 192q^{57} - 129q^{58} - 561q^{59} - 846q^{60} - 191q^{61} - 900q^{62} - 585q^{63} + 53q^{64} - 435q^{65} - 423q^{66} + 256q^{67} + 126q^{68} + 99q^{69} + 591q^{70} + 315q^{71} + 720q^{72} + 97q^{73} + 219q^{74} + 255q^{75} + 451q^{76} + 195q^{77} + 180q^{78} + 151q^{79} + 36q^{81} - 330q^{82} + 51q^{83} - 588q^{84} - 207q^{85} - 75q^{86} - 279q^{87} - 717q^{88} + 72q^{89} + 288q^{90} - 7q^{91} - 51q^{92} + 591q^{93} - 741q^{94} + 615q^{95} + 270q^{96} - 470q^{97} + 882q^{98} + 513q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(27))$$

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
27.3.b $$\chi_{27}(26, \cdot)$$ 27.3.b.a 1 1
27.3.b.b 2
27.3.d $$\chi_{27}(8, \cdot)$$ 27.3.d.a 2 2
27.3.f $$\chi_{27}(2, \cdot)$$ 27.3.f.a 30 6

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(27))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(27)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 2}$$