## Defining parameters

 Level: $$N$$ = $$27 = 3^{3}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$3$$ Newform subspaces: $$5$$ Sturm bound: $$216$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(27))$$.

Total New Old
Modular forms 96 72 24
Cusp forms 66 56 10
Eisenstein series 30 16 14

## Trace form

 $$56q - 3q^{2} - 6q^{3} + 11q^{4} + 21q^{5} - 18q^{6} - 41q^{7} - 141q^{8} - 54q^{9} + O(q^{10})$$ $$56q - 3q^{2} - 6q^{3} + 11q^{4} + 21q^{5} - 18q^{6} - 41q^{7} - 141q^{8} - 54q^{9} - 45q^{10} + 123q^{11} + 147q^{12} + 103q^{13} + 111q^{14} - 36q^{15} - 205q^{16} - 405q^{17} - 639q^{18} - 95q^{19} - 609q^{20} - 138q^{21} + 27q^{22} + 435q^{23} + 1170q^{24} + 425q^{25} + 2442q^{26} + 1125q^{27} + 490q^{28} + 429q^{29} + 459q^{30} - 437q^{31} - 1071q^{32} - 639q^{33} - 9q^{34} - 1263q^{35} - 2088q^{36} - 797q^{37} - 2085q^{38} - 768q^{39} - 621q^{40} - 1599q^{41} - 3078q^{42} + 175q^{43} - 1749q^{44} - 360q^{45} + 873q^{46} + 1383q^{47} + 2289q^{48} + 33q^{49} + 3930q^{50} + 2655q^{51} - 41q^{52} + 2628q^{53} + 5454q^{54} + 216q^{55} + 5973q^{56} + 3426q^{57} + 1647q^{58} + 3036q^{59} + 1314q^{60} + 391q^{61} - 2346q^{62} - 2610q^{63} - 2239q^{64} - 6825q^{65} - 11115q^{66} - 3191q^{67} - 10476q^{68} - 6138q^{69} - 4185q^{70} - 5841q^{71} - 36q^{72} + 715q^{73} + 4359q^{74} + 2604q^{75} + 5539q^{76} + 4557q^{77} + 6066q^{78} + 3631q^{79} + 9678q^{80} + 3438q^{81} + 1134q^{82} + 4281q^{83} + 7674q^{84} + 5823q^{85} + 3657q^{86} + 2880q^{87} + 2547q^{88} - 4410q^{89} - 12510q^{90} - 1657q^{91} - 19311q^{92} - 11802q^{93} - 9045q^{94} - 10245q^{95} - 14094q^{96} - 4685q^{97} - 3546q^{98} - 1242q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(27))$$

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
27.4.a $$\chi_{27}(1, \cdot)$$ 27.4.a.a 1 1
27.4.a.b 1
27.4.a.c 2
27.4.c $$\chi_{27}(10, \cdot)$$ 27.4.c.a 4 2
27.4.e $$\chi_{27}(4, \cdot)$$ 27.4.e.a 48 6

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(27))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(27)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 2}$$