## Defining parameters

 Level: $$N$$ = $$22 = 2 \cdot 11$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$2$$ Newform subspaces: $$5$$ Sturm bound: $$120$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(22))$$.

Total New Old
Modular forms 55 15 40
Cusp forms 35 15 20
Eisenstein series 20 0 20

## Trace form

 $$15 q + 50 q^{6} + 20 q^{7} - 160 q^{9} + O(q^{10})$$ $$15 q + 50 q^{6} + 20 q^{7} - 160 q^{9} - 100 q^{10} - 100 q^{11} - 80 q^{12} - 40 q^{13} + 20 q^{14} + 410 q^{15} + 310 q^{17} + 230 q^{18} - 225 q^{19} - 420 q^{21} + 390 q^{23} + 200 q^{24} + 300 q^{25} + 200 q^{26} + 75 q^{27} - 120 q^{28} - 250 q^{29} - 780 q^{30} + 90 q^{31} - 160 q^{32} - 1265 q^{33} - 600 q^{34} - 1010 q^{35} - 100 q^{36} - 240 q^{37} - 60 q^{38} - 160 q^{39} + 240 q^{40} + 830 q^{41} + 2180 q^{42} + 2690 q^{43} + 1060 q^{44} + 1910 q^{45} + 360 q^{46} - 250 q^{47} - 1330 q^{49} - 680 q^{50} - 695 q^{51} - 960 q^{52} + 260 q^{53} - 2160 q^{54} - 370 q^{55} - 945 q^{57} - 1680 q^{58} - 1825 q^{59} - 480 q^{60} + 260 q^{61} + 1380 q^{62} + 1720 q^{63} + 760 q^{65} + 2920 q^{66} + 820 q^{67} + 1240 q^{68} + 840 q^{70} - 3240 q^{71} - 1120 q^{72} - 870 q^{73} - 1400 q^{74} - 5045 q^{75} - 760 q^{76} + 630 q^{77} - 2960 q^{78} + 240 q^{79} - 640 q^{80} + 2505 q^{81} + 1690 q^{82} + 4445 q^{83} + 1840 q^{84} + 2420 q^{85} + 2450 q^{86} + 3400 q^{87} + 560 q^{88} + 350 q^{89} + 2940 q^{90} + 3830 q^{91} - 1680 q^{92} + 6230 q^{93} - 1600 q^{94} - 2200 q^{95} - 3345 q^{97} - 4410 q^{98} - 7790 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(22))$$

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
22.4.a $$\chi_{22}(1, \cdot)$$ 22.4.a.a 1 1
22.4.a.b 1
22.4.a.c 1
22.4.c $$\chi_{22}(3, \cdot)$$ 22.4.c.a 4 4
22.4.c.b 8

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(22))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(22)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(11))$$$$^{\oplus 2}$$