# Properties

 Label 36.4 Level 36 Weight 4 Dimension 45 Nonzero newspaces 4 Newform subspaces 6 Sturm bound 288 Trace bound 3

## 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

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

## 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 available 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

$$S_{4}^{\mathrm{old}}(\Gamma_1(36)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 9}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(3))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 1}$$