## Defining parameters

 Level: $$N$$ = $$22 = 2 \cdot 11$$ Weight: $$k$$ = $$7$$ Nonzero newspaces: $$2$$ Newform subspaces: $$2$$ Sturm bound: $$210$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{7}(\Gamma_1(22))$$.

Total New Old
Modular forms 100 30 70
Cusp forms 80 30 50
Eisenstein series 20 0 20

## Trace form

 $$30q + 400q^{6} + 720q^{7} - 4160q^{9} + O(q^{10})$$ $$30q + 400q^{6} + 720q^{7} - 4160q^{9} + 3200q^{11} + 5120q^{12} + 3600q^{13} - 2080q^{14} - 8660q^{15} - 5880q^{17} + 18400q^{18} - 23850q^{19} - 37940q^{23} + 12800q^{24} + 129840q^{25} + 50400q^{26} - 53250q^{27} - 32640q^{28} - 204800q^{29} - 134160q^{30} - 6900q^{31} + 153450q^{33} + 117600q^{34} + 469520q^{35} + 36800q^{36} + 103200q^{37} + 28080q^{38} - 328640q^{39} - 107520q^{40} - 541660q^{41} - 521920q^{42} + 251200q^{44} + 1180380q^{45} + 213120q^{46} + 193640q^{47} - 645420q^{49} - 364480q^{50} - 1347330q^{51} - 291840q^{52} + 81120q^{53} + 714960q^{55} + 1416050q^{57} + 1061760q^{58} + 94170q^{59} + 414720q^{60} - 1184160q^{61} - 226640q^{62} - 921200q^{63} + 321920q^{66} + 22320q^{67} - 188160q^{68} + 1356080q^{69} + 304080q^{70} - 242800q^{71} - 896000q^{72} - 590340q^{73} - 610400q^{74} - 2419830q^{75} - 1972220q^{77} + 1361600q^{78} + 93240q^{79} + 573440q^{80} + 1519730q^{81} + 741600q^{82} + 4588350q^{83} + 2357760q^{84} + 4132920q^{85} + 2113440q^{86} - 806400q^{88} - 2260900q^{89} - 6535040q^{90} - 8984700q^{91} - 2419200q^{92} + 143240q^{93} - 2410800q^{94} + 661320q^{95} + 1009290q^{97} + 2419140q^{99} + O(q^{100})$$

## Decomposition of $$S_{7}^{\mathrm{new}}(\Gamma_1(22))$$

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
22.7.b $$\chi_{22}(21, \cdot)$$ 22.7.b.a 6 1
22.7.d $$\chi_{22}(7, \cdot)$$ 22.7.d.a 24 4

## Decomposition of $$S_{7}^{\mathrm{old}}(\Gamma_1(22))$$ into lower level spaces

$$S_{7}^{\mathrm{old}}(\Gamma_1(22)) \cong$$ $$S_{7}^{\mathrm{new}}(\Gamma_1(11))$$$$^{\oplus 2}$$