# Properties

 Label 309.2 Level 309 Weight 2 Dimension 2549 Nonzero newspaces 8 Newform subspaces 18 Sturm bound 14144 Trace bound 1

## Defining parameters

 Level: $$N$$ = $$309 = 3 \cdot 103$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$8$$ Newform subspaces: $$18$$ Sturm bound: $$14144$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(309))$$.

Total New Old
Modular forms 3740 2753 987
Cusp forms 3333 2549 784
Eisenstein series 407 204 203

## Trace form

 $$2549q - 3q^{2} - 52q^{3} - 109q^{4} - 6q^{5} - 54q^{6} - 110q^{7} - 15q^{8} - 52q^{9} + O(q^{10})$$ $$2549q - 3q^{2} - 52q^{3} - 109q^{4} - 6q^{5} - 54q^{6} - 110q^{7} - 15q^{8} - 52q^{9} - 120q^{10} - 12q^{11} - 58q^{12} - 116q^{13} - 24q^{14} - 57q^{15} - 133q^{16} - 18q^{17} - 54q^{18} - 122q^{19} - 42q^{20} - 59q^{21} - 138q^{22} - 24q^{23} - 66q^{24} - 133q^{25} - 42q^{26} - 52q^{27} - 158q^{28} - 30q^{29} - 69q^{30} - 134q^{31} - 63q^{32} - 63q^{33} - 156q^{34} - 48q^{35} - 58q^{36} - 140q^{37} - 60q^{38} - 65q^{39} - 192q^{40} - 42q^{41} - 75q^{42} - 146q^{43} - 84q^{44} - 57q^{45} - 174q^{46} - 48q^{47} - 82q^{48} - 159q^{49} - 93q^{50} - 69q^{51} - 200q^{52} - 54q^{53} - 54q^{54} - 174q^{55} - 120q^{56} - 71q^{57} - 192q^{58} - 60q^{59} - 93q^{60} - 164q^{61} - 96q^{62} - 59q^{63} - 229q^{64} - 84q^{65} - 87q^{66} - 170q^{67} - 126q^{68} - 75q^{69} - 246q^{70} - 72q^{71} - 66q^{72} - 176q^{73} - 114q^{74} - 82q^{75} - 242q^{76} - 96q^{77} - 93q^{78} - 182q^{79} - 186q^{80} - 52q^{81} - 228q^{82} - 84q^{83} - 39q^{84} - 6q^{85} + 72q^{86} + 21q^{87} + 534q^{88} + 114q^{89} + 33q^{90} + 228q^{91} + 240q^{92} + 121q^{93} + 162q^{94} + 186q^{95} + 498q^{96} + 242q^{97} + 645q^{98} + 39q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(309))$$

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
309.2.a $$\chi_{309}(1, \cdot)$$ 309.2.a.a 1 1
309.2.a.b 3
309.2.a.c 5
309.2.a.d 8
309.2.c $$\chi_{309}(308, \cdot)$$ 309.2.c.a 32 1
309.2.e $$\chi_{309}(46, \cdot)$$ 309.2.e.a 2 2
309.2.e.b 16
309.2.e.c 16
309.2.g $$\chi_{309}(47, \cdot)$$ 309.2.g.a 2 2
309.2.g.b 8
309.2.g.c 56
309.2.i $$\chi_{309}(13, \cdot)$$ 309.2.i.a 128 16
309.2.i.b 160
309.2.k $$\chi_{309}(80, \cdot)$$ 309.2.k.a 512 16
309.2.m $$\chi_{309}(4, \cdot)$$ 309.2.m.a 256 32
309.2.m.b 288
309.2.o $$\chi_{309}(5, \cdot)$$ 309.2.o.a 32 32
309.2.o.b 1024

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(309))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(309)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(103))$$$$^{\oplus 2}$$