# Properties

 Label 33.3 Level 33 Weight 3 Dimension 50 Nonzero newspaces 4 Newform subspaces 6 Sturm bound 240 Trace bound 1

## Defining parameters

 Level: $$N$$ = $$33 = 3 \cdot 11$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$4$$ Newform subspaces: $$6$$ Sturm bound: $$240$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(33))$$.

Total New Old
Modular forms 100 66 34
Cusp forms 60 50 10
Eisenstein series 40 16 24

## Trace form

 $$50 q - 5 q^{3} - 10 q^{4} - 25 q^{6} - 40 q^{7} - 40 q^{8} - 15 q^{9} + O(q^{10})$$ $$50 q - 5 q^{3} - 10 q^{4} - 25 q^{6} - 40 q^{7} - 40 q^{8} - 15 q^{9} - 20 q^{10} + 10 q^{11} + 30 q^{12} + 20 q^{13} + 10 q^{14} - 15 q^{15} - 90 q^{16} - 10 q^{17} + 5 q^{18} - 70 q^{19} - 50 q^{20} - 10 q^{21} + 50 q^{22} + 40 q^{23} + 225 q^{24} + 150 q^{25} + 250 q^{26} + 70 q^{27} + 340 q^{28} + 160 q^{29} + 190 q^{30} + 190 q^{31} - 180 q^{33} - 380 q^{34} - 320 q^{35} - 425 q^{36} - 250 q^{37} - 250 q^{38} - 270 q^{39} - 380 q^{40} - 120 q^{41} - 360 q^{42} - 240 q^{43} - 330 q^{44} + 25 q^{45} - 80 q^{46} - 50 q^{47} - 50 q^{48} + 100 q^{49} + 330 q^{50} + 325 q^{51} + 280 q^{52} + 370 q^{53} + 870 q^{54} + 510 q^{55} + 680 q^{56} + 585 q^{57} + 460 q^{58} + 150 q^{59} + 530 q^{60} + 120 q^{61} + 40 q^{62} - 30 q^{63} + 350 q^{64} - 230 q^{66} + 190 q^{67} + 80 q^{68} - 145 q^{69} - 540 q^{70} - 520 q^{71} - 920 q^{72} - 940 q^{73} - 730 q^{74} - 875 q^{75} - 900 q^{76} - 570 q^{77} - 1120 q^{78} - 720 q^{79} - 370 q^{80} + 45 q^{81} + 210 q^{82} - 190 q^{83} + 530 q^{84} + 500 q^{85} + 520 q^{86} + 540 q^{87} + 1470 q^{88} + 380 q^{89} + 630 q^{90} + 980 q^{91} + 190 q^{92} + 255 q^{93} + 380 q^{94} + 430 q^{95} + 60 q^{96} + 200 q^{97} - 305 q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(33))$$

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 available newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
33.3.b $$\chi_{33}(23, \cdot)$$ 33.3.b.a 2 1
33.3.b.b 4
33.3.c $$\chi_{33}(10, \cdot)$$ 33.3.c.a 4 1
33.3.g $$\chi_{33}(7, \cdot)$$ 33.3.g.a 16 4
33.3.h $$\chi_{33}(5, \cdot)$$ 33.3.h.a 8 4
33.3.h.b 16

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(33))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(33)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(11))$$$$^{\oplus 2}$$