## Defining parameters

 Level: $$N$$ = $$33 = 3 \cdot 11$$ Weight: $$k$$ = $$5$$ Nonzero newspaces: $$4$$ Newform subspaces: $$4$$ Sturm bound: $$400$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{5}(\Gamma_1(33))$$.

Total New Old
Modular forms 180 126 54
Cusp forms 140 110 30
Eisenstein series 40 16 24

## Trace form

 $$110q - 5q^{3} - 10q^{4} - 85q^{6} + 140q^{7} + 480q^{8} + 165q^{9} + O(q^{10})$$ $$110q - 5q^{3} - 10q^{4} - 85q^{6} + 140q^{7} + 480q^{8} + 165q^{9} - 20q^{10} - 210q^{11} - 810q^{12} - 520q^{13} - 2430q^{14} - 615q^{15} + 1870q^{16} + 2490q^{17} + 1925q^{18} + 2030q^{19} + 1950q^{20} - 10q^{21} - 2590q^{22} - 1680q^{23} - 1515q^{24} - 5450q^{25} - 6750q^{26} - 2180q^{27} - 2460q^{28} + 960q^{29} + 10q^{30} + 7590q^{31} - 3480q^{33} - 4700q^{34} + 1920q^{35} + 14275q^{36} + 9110q^{37} + 12750q^{38} + 12960q^{39} + 12340q^{40} + 9360q^{41} - 5760q^{42} - 7720q^{43} - 10890q^{44} - 13745q^{45} - 8320q^{46} - 3030q^{47} - 6410q^{48} - 4080q^{49} - 11550q^{50} - 5q^{51} + 5640q^{52} + 750q^{53} + 11430q^{54} + 11950q^{55} + 12360q^{56} - 5085q^{57} - 15660q^{58} - 13950q^{59} - 56770q^{60} - 35900q^{61} - 39360q^{62} - 22200q^{63} - 29930q^{64} + 24190q^{66} + 23170q^{67} + 68160q^{68} + 48545q^{69} + 50540q^{70} - 13080q^{71} + 68440q^{72} - 29680q^{73} - 2130q^{74} + 26335q^{75} + 27260q^{76} + 27810q^{77} + 19880q^{78} + 61780q^{79} + 27870q^{80} - 35895q^{81} + 82930q^{82} + 35430q^{83} - 18910q^{84} + 7400q^{85} - 29520q^{86} - 29160q^{87} - 96650q^{88} - 23220q^{89} - 80430q^{90} - 167320q^{91} - 106770q^{92} - 81465q^{93} - 147940q^{94} - 71670q^{95} - 38220q^{96} - 54400q^{97} + 43915q^{99} + O(q^{100})$$

## Decomposition of $$S_{5}^{\mathrm{new}}(\Gamma_1(33))$$

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
33.5.b $$\chi_{33}(23, \cdot)$$ 33.5.b.a 14 1
33.5.c $$\chi_{33}(10, \cdot)$$ 33.5.c.a 8 1
33.5.g $$\chi_{33}(7, \cdot)$$ 33.5.g.a 32 4
33.5.h $$\chi_{33}(5, \cdot)$$ 33.5.h.a 56 4

## Decomposition of $$S_{5}^{\mathrm{old}}(\Gamma_1(33))$$ into lower level spaces

$$S_{5}^{\mathrm{old}}(\Gamma_1(33)) \cong$$ $$S_{5}^{\mathrm{new}}(\Gamma_1(11))$$$$^{\oplus 2}$$