# Properties

 Label 330.2.s Level $330$ Weight $2$ Character orbit 330.s Rep. character $\chi_{330}(49,\cdot)$ Character field $\Q(\zeta_{10})$ Dimension $48$ Newform subspaces $3$ Sturm bound $144$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$330 = 2 \cdot 3 \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 330.s (of order $$10$$ and degree $$4$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$55$$ Character field: $$\Q(\zeta_{10})$$ Newform subspaces: $$3$$ Sturm bound: $$144$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(330, [\chi])$$.

Total New Old
Modular forms 320 48 272
Cusp forms 256 48 208
Eisenstein series 64 0 64

## Trace form

 $$48q + 12q^{4} + 8q^{5} - 4q^{6} + 12q^{9} + O(q^{10})$$ $$48q + 12q^{4} + 8q^{5} - 4q^{6} + 12q^{9} + 4q^{10} - 4q^{11} + 12q^{14} - 14q^{15} - 12q^{16} + 16q^{19} - 8q^{20} + 32q^{21} + 4q^{24} + 4q^{25} + 8q^{26} + 8q^{30} + 20q^{31} - 8q^{34} + 44q^{35} - 12q^{36} - 8q^{39} + 6q^{40} - 60q^{41} - 36q^{44} - 8q^{45} - 36q^{46} - 8q^{49} - 24q^{50} - 32q^{51} - 16q^{54} - 12q^{55} + 8q^{56} - 80q^{59} - 16q^{60} - 56q^{61} + 12q^{64} - 64q^{65} - 16q^{66} - 40q^{69} - 30q^{70} + 64q^{71} - 8q^{74} + 24q^{75} + 24q^{76} + 4q^{79} - 12q^{80} - 12q^{81} + 8q^{84} + 88q^{86} - 24q^{89} - 4q^{90} + 68q^{91} + 20q^{94} - 60q^{95} - 4q^{96} + 4q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(330, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
330.2.s.a $$8$$ $$2.635$$ $$\Q(\zeta_{20})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{20}q^{2}-\zeta_{20}^{3}q^{3}+\zeta_{20}^{2}q^{4}+(\zeta_{20}+\cdots)q^{5}+\cdots$$
330.2.s.b $$8$$ $$2.635$$ $$\Q(\zeta_{20})$$ None $$0$$ $$0$$ $$10$$ $$0$$ $$q+\zeta_{20}q^{2}-\zeta_{20}^{3}q^{3}+\zeta_{20}^{2}q^{4}+(2\zeta_{20}^{2}+\cdots)q^{5}+\cdots$$
330.2.s.c $$32$$ $$2.635$$ None $$0$$ $$0$$ $$-2$$ $$0$$

## Decomposition of $$S_{2}^{\mathrm{old}}(330, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(330, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(55, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(110, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(165, [\chi])$$$$^{\oplus 2}$$