## Defining parameters

 Level: $$N$$ = $$33 = 3 \cdot 11$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$4$$ Newform subspaces: $$10$$ Sturm bound: $$320$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(33))$$.

Total New Old
Modular forms 140 100 40
Cusp forms 100 80 20
Eisenstein series 40 20 20

## Trace form

 $$80 q - 5 q^{3} - 10 q^{4} + 45 q^{6} + 10 q^{7} - 80 q^{8} - 85 q^{9} + O(q^{10})$$ $$80 q - 5 q^{3} - 10 q^{4} + 45 q^{6} + 10 q^{7} - 80 q^{8} - 85 q^{9} - 200 q^{10} - 100 q^{11} - 170 q^{12} - 50 q^{13} + 390 q^{14} + 305 q^{15} + 750 q^{16} + 300 q^{17} + 135 q^{18} - 460 q^{19} - 950 q^{20} - 420 q^{21} - 1270 q^{22} - 380 q^{23} - 1015 q^{24} - 370 q^{25} - 50 q^{26} + 25 q^{27} + 1040 q^{28} + 700 q^{29} + 1930 q^{30} + 1370 q^{31} + 1780 q^{32} + 1800 q^{33} + 1860 q^{34} + 1160 q^{35} + 1215 q^{36} - 50 q^{37} - 650 q^{38} - 1075 q^{39} - 1540 q^{40} - 2220 q^{41} - 2440 q^{42} - 160 q^{43} - 770 q^{44} - 2680 q^{45} - 1800 q^{46} - 1180 q^{47} - 2070 q^{48} - 1990 q^{49} - 1430 q^{50} - 1050 q^{51} - 2400 q^{52} + 320 q^{53} - 540 q^{54} + 1390 q^{55} + 360 q^{56} + 840 q^{57} - 1700 q^{58} - 1020 q^{59} + 3190 q^{60} + 1150 q^{61} + 2760 q^{62} + 4125 q^{63} + 5990 q^{64} + 3360 q^{65} + 7140 q^{66} + 3320 q^{67} + 4880 q^{68} + 3310 q^{69} + 5100 q^{70} + 1860 q^{71} - 110 q^{72} + 5450 q^{73} + 1950 q^{74} - 1080 q^{75} - 4000 q^{76} - 900 q^{77} - 7340 q^{78} - 3610 q^{79} - 9810 q^{80} - 7825 q^{81} - 12770 q^{82} - 7720 q^{83} - 11580 q^{84} - 12410 q^{85} - 5960 q^{86} - 5220 q^{87} - 11290 q^{88} - 3500 q^{89} + 7600 q^{90} + 2190 q^{91} + 11050 q^{92} + 6105 q^{93} + 13040 q^{94} + 3600 q^{95} + 14140 q^{96} + 5960 q^{97} + 5060 q^{98} + 4865 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(33))$$

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
33.4.a $$\chi_{33}(1, \cdot)$$ 33.4.a.a 1 1
33.4.a.b 1
33.4.a.c 2
33.4.a.d 2
33.4.d $$\chi_{33}(32, \cdot)$$ 33.4.d.a 2 1
33.4.d.b 8
33.4.e $$\chi_{33}(4, \cdot)$$ 33.4.e.a 4 4
33.4.e.b 8
33.4.e.c 12
33.4.f $$\chi_{33}(2, \cdot)$$ 33.4.f.a 40 4

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(33))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(33)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(11))$$$$^{\oplus 2}$$