## Defining parameters

 Level: $$N$$ = $$322 = 2 \cdot 7 \cdot 23$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$8$$ Newform subspaces: $$28$$ Sturm bound: $$12672$$ Trace bound: $$6$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(322))$$.

Total New Old
Modular forms 3432 1011 2421
Cusp forms 2905 1011 1894
Eisenstein series 527 0 527

## Trace form

 $$1011 q + 3 q^{2} + 8 q^{3} - q^{4} + 6 q^{5} - q^{7} + 3 q^{8} + 11 q^{9} + O(q^{10})$$ $$1011 q + 3 q^{2} + 8 q^{3} - q^{4} + 6 q^{5} - q^{7} + 3 q^{8} + 11 q^{9} + 6 q^{10} + 12 q^{11} + 8 q^{12} + 22 q^{13} + 3 q^{14} - 20 q^{15} - q^{16} - 38 q^{17} - 73 q^{18} - 28 q^{19} - 38 q^{20} - 58 q^{21} - 32 q^{22} - 65 q^{23} - 47 q^{25} - 38 q^{26} - 100 q^{27} - 23 q^{28} - 2 q^{29} - 64 q^{30} - 4 q^{31} + 3 q^{32} + 4 q^{33} + 30 q^{34} - 16 q^{35} + 11 q^{36} - 54 q^{37} + 24 q^{38} - 48 q^{39} + 6 q^{40} - 14 q^{41} - 60 q^{43} + 12 q^{44} - 54 q^{45} + 23 q^{46} - 16 q^{47} + 8 q^{48} - 67 q^{49} + 21 q^{50} + 8 q^{51} + 22 q^{52} - 2 q^{53} + 4 q^{54} - 104 q^{55} - 19 q^{56} - 132 q^{57} - 70 q^{58} - 148 q^{59} - 20 q^{60} - 130 q^{61} - 108 q^{62} - 99 q^{63} - q^{64} - 224 q^{65} - 128 q^{66} - 12 q^{67} - 82 q^{68} - 128 q^{69} - 82 q^{70} - 148 q^{71} - 73 q^{72} - 18 q^{73} - 134 q^{74} - 248 q^{75} + 16 q^{76} - 98 q^{77} - 60 q^{78} - 156 q^{79} - 38 q^{80} - 121 q^{81} - 34 q^{82} - 80 q^{83} - 14 q^{84} - 24 q^{85} + 16 q^{86} - 36 q^{87} + 12 q^{88} + 58 q^{89} + 78 q^{90} - 22 q^{91} + 23 q^{92} + 112 q^{93} + 24 q^{94} - 12 q^{95} - 58 q^{97} - 41 q^{98} - 196 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(322))$$

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
322.2.a $$\chi_{322}(1, \cdot)$$ 322.2.a.a 1 1
322.2.a.b 1
322.2.a.c 1
322.2.a.d 1
322.2.a.e 2
322.2.a.f 2
322.2.a.g 3
322.2.c $$\chi_{322}(321, \cdot)$$ 322.2.c.a 4 1
322.2.c.b 4
322.2.c.c 4
322.2.c.d 4
322.2.e $$\chi_{322}(93, \cdot)$$ 322.2.e.a 8 2
322.2.e.b 8
322.2.e.c 8
322.2.e.d 8
322.2.g $$\chi_{322}(45, \cdot)$$ 322.2.g.a 16 2
322.2.g.b 16
322.2.i $$\chi_{322}(29, \cdot)$$ 322.2.i.a 10 10
322.2.i.b 10
322.2.i.c 20
322.2.i.d 40
322.2.i.e 40
322.2.k $$\chi_{322}(83, \cdot)$$ 322.2.k.a 80 10
322.2.k.b 80
322.2.m $$\chi_{322}(9, \cdot)$$ 322.2.m.a 160 20
322.2.m.b 160
322.2.o $$\chi_{322}(5, \cdot)$$ 322.2.o.a 160 20
322.2.o.b 160

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(322))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(322)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(23))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(46))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(161))$$$$^{\oplus 2}$$