## Defining parameters

 Level: $$N$$ = $$36 = 2^{2} \cdot 3^{2}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$4$$ Newform subspaces: $$6$$ Sturm bound: $$288$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(36))$$.

Total New Old
Modular forms 128 53 75
Cusp forms 88 45 43
Eisenstein series 40 8 32

## Trace form

 $$45q - 3q^{2} - 3q^{3} + 11q^{4} + 18q^{5} + 21q^{6} + 2q^{7} + 45q^{9} + O(q^{10})$$ $$45q - 3q^{2} - 3q^{3} + 11q^{4} + 18q^{5} + 21q^{6} + 2q^{7} + 45q^{9} - 152q^{10} + 15q^{11} - 6q^{12} + 72q^{13} - 78q^{14} - 180q^{15} + 263q^{16} - 240q^{17} - 120q^{18} - 70q^{19} - 234q^{20} - 246q^{21} - 465q^{22} + 138q^{23} + 3q^{24} + 76q^{25} + 648q^{27} + 348q^{28} + 768q^{29} + 438q^{30} + 32q^{31} + 687q^{32} - 999q^{33} - 11q^{34} - 960q^{35} + 1023q^{36} + 306q^{37} + 891q^{38} - 36q^{39} + 250q^{40} + 651q^{41} + 732q^{42} + 581q^{43} + 1188q^{45} + 528q^{46} + 90q^{47} - 1371q^{48} - 2092q^{49} - 1977q^{50} - 1647q^{51} - 2588q^{52} - 2622q^{53} - 3021q^{54} - 864q^{55} - 3006q^{56} - 243q^{57} - 86q^{58} + 1137q^{59} - 1302q^{60} + 3078q^{61} + 1896q^{63} - 502q^{64} + 1620q^{65} + 3114q^{66} + 119q^{67} + 4845q^{68} + 12q^{69} + 3444q^{70} + 480q^{71} + 5475q^{72} - 1692q^{73} + 5874q^{74} + 921q^{75} - 495q^{76} + 1818q^{77} + 2484q^{78} + 1064q^{79} + 1833q^{81} + 358q^{82} + 576q^{83} - 5850q^{84} + 3332q^{85} - 8331q^{86} - 1386q^{87} - 2349q^{88} - 294q^{89} - 11202q^{90} - 344q^{91} - 8724q^{92} - 1776q^{93} - 3192q^{94} - 3984q^{95} - 3900q^{96} - 7041q^{97} - 1854q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(36))$$

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
36.4.a $$\chi_{36}(1, \cdot)$$ 36.4.a.a 1 1
36.4.b $$\chi_{36}(35, \cdot)$$ 36.4.b.a 2 1
36.4.b.b 4
36.4.e $$\chi_{36}(13, \cdot)$$ 36.4.e.a 6 2
36.4.h $$\chi_{36}(11, \cdot)$$ 36.4.h.a 8 2
36.4.h.b 24

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(36))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(36)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 2}$$