# Properties

 Label 306.2 Level 306 Weight 2 Dimension 680 Nonzero newspaces 10 Newform subspaces 47 Sturm bound 10368 Trace bound 5

## Defining parameters

 Level: $$N$$ = $$306 = 2 \cdot 3^{2} \cdot 17$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$10$$ Newform subspaces: $$47$$ Sturm bound: $$10368$$ Trace bound: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(306))$$.

Total New Old
Modular forms 2848 680 2168
Cusp forms 2337 680 1657
Eisenstein series 511 0 511

## Trace form

 $$680 q + 2 q^{2} + 6 q^{3} + 2 q^{4} - 6 q^{6} + 4 q^{7} - 4 q^{8} - 6 q^{9} + O(q^{10})$$ $$680 q + 2 q^{2} + 6 q^{3} + 2 q^{4} - 6 q^{6} + 4 q^{7} - 4 q^{8} - 6 q^{9} + 4 q^{10} + 10 q^{11} + 20 q^{13} + 20 q^{14} + 6 q^{16} + 22 q^{17} + 12 q^{18} + 20 q^{19} + 4 q^{20} - 12 q^{21} + 10 q^{22} + 4 q^{23} + 6 q^{24} + 26 q^{25} - 4 q^{26} - 8 q^{28} + 32 q^{29} + 24 q^{31} + 2 q^{32} + 18 q^{33} - 3 q^{34} + 32 q^{35} - 6 q^{36} + 48 q^{37} - 26 q^{38} - 64 q^{39} - 106 q^{41} - 96 q^{42} - 74 q^{43} - 84 q^{44} - 128 q^{45} - 72 q^{46} - 252 q^{47} - 6 q^{48} - 198 q^{49} - 178 q^{50} - 137 q^{51} - 92 q^{52} - 236 q^{53} - 130 q^{54} - 176 q^{55} + 4 q^{56} - 166 q^{57} - 84 q^{58} - 170 q^{59} - 64 q^{60} - 64 q^{61} - 128 q^{62} - 72 q^{63} - 4 q^{64} - 92 q^{65} - 16 q^{66} + 26 q^{67} + q^{68} + 32 q^{70} + 96 q^{71} - 6 q^{72} + 40 q^{73} + 28 q^{74} + 30 q^{75} - 2 q^{76} + 68 q^{77} + 12 q^{78} + 72 q^{79} + 16 q^{80} + 18 q^{81} + 32 q^{82} + 64 q^{83} + 12 q^{84} + 4 q^{85} + 30 q^{86} - 132 q^{87} + 26 q^{88} - 120 q^{89} - 112 q^{91} + 4 q^{92} - 128 q^{93} + 52 q^{94} - 176 q^{95} - 22 q^{97} + 48 q^{98} - 180 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(306))$$

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 available newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
306.2.a $$\chi_{306}(1, \cdot)$$ 306.2.a.a 1 1
306.2.a.b 1
306.2.a.c 1
306.2.a.d 1
306.2.a.e 2
306.2.a.f 2
306.2.b $$\chi_{306}(271, \cdot)$$ 306.2.b.a 2 1
306.2.b.b 2
306.2.b.c 2
306.2.b.d 2
306.2.e $$\chi_{306}(103, \cdot)$$ 306.2.e.a 2 2
306.2.e.b 4
306.2.e.c 6
306.2.e.d 6
306.2.e.e 6
306.2.e.f 8
306.2.g $$\chi_{306}(55, \cdot)$$ 306.2.g.a 2 2
306.2.g.b 2
306.2.g.c 2
306.2.g.d 2
306.2.g.e 2
306.2.g.f 2
306.2.g.g 4
306.2.j $$\chi_{306}(67, \cdot)$$ 306.2.j.a 4 2
306.2.j.b 16
306.2.j.c 16
306.2.l $$\chi_{306}(19, \cdot)$$ 306.2.l.a 4 4
306.2.l.b 4
306.2.l.c 4
306.2.l.d 8
306.2.l.e 8
306.2.n $$\chi_{306}(13, \cdot)$$ 306.2.n.a 4 4
306.2.n.b 4
306.2.n.c 4
306.2.n.d 4
306.2.n.e 24
306.2.n.f 32
306.2.o $$\chi_{306}(71, \cdot)$$ 306.2.o.a 8 8
306.2.o.b 8
306.2.o.c 8
306.2.o.d 8
306.2.o.e 8
306.2.o.f 8
306.2.r $$\chi_{306}(25, \cdot)$$ 306.2.r.a 64 8
306.2.r.b 80
306.2.s $$\chi_{306}(5, \cdot)$$ 306.2.s.a 144 16
306.2.s.b 144

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(306))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(306)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(3))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(17))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(34))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(51))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(102))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(153))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(306))$$$$^{\oplus 1}$$