# Properties

 Label 507.2.f Level $507$ Weight $2$ Character orbit 507.f Rep. character $\chi_{507}(239,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $84$ Newform subspaces $7$ Sturm bound $121$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 507.f (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$39$$ Character field: $$\Q(i)$$ Newform subspaces: $$7$$ Sturm bound: $$121$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$2$$, $$5$$, $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(507, [\chi])$$.

Total New Old
Modular forms 148 124 24
Cusp forms 92 84 8
Eisenstein series 56 40 16

## Trace form

 $$84q + 4q^{3} + 4q^{6} - 4q^{7} + 4q^{9} + O(q^{10})$$ $$84q + 4q^{3} + 4q^{6} - 4q^{7} + 4q^{9} - 8q^{15} - 12q^{16} - 8q^{18} - 4q^{19} + 4q^{21} + 24q^{22} - 12q^{24} - 56q^{27} - 4q^{28} + 20q^{31} + 16q^{33} - 4q^{37} - 16q^{40} + 16q^{45} - 24q^{46} - 88q^{48} + 4q^{54} - 40q^{55} + 4q^{57} - 8q^{58} + 8q^{60} - 24q^{61} + 4q^{63} - 36q^{66} + 20q^{67} - 8q^{70} + 24q^{72} - 4q^{73} + 4q^{76} - 8q^{79} + 28q^{81} + 4q^{84} + 40q^{87} - 20q^{93} + 64q^{94} - 20q^{96} - 28q^{97} - 32q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(507, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
507.2.f.a $$4$$ $$4.048$$ $$\Q(\zeta_{8})$$ None $$0$$ $$-4$$ $$0$$ $$-4$$ $$q+\zeta_{8}q^{2}+(-1-\zeta_{8}-\zeta_{8}^{3})q^{3}-\zeta_{8}^{2}q^{4}+\cdots$$
507.2.f.b $$4$$ $$4.048$$ $$\Q(\zeta_{12})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$-2$$ $$q+(\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{3}+2\zeta_{12}^{2}q^{4}+\cdots$$
507.2.f.c $$4$$ $$4.048$$ $$\Q(\zeta_{12})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$2$$ $$q+(\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{3}+2\zeta_{12}^{2}q^{4}+\cdots$$
507.2.f.d $$8$$ $$4.048$$ 8.0.56070144.2 $$\Q(\sqrt{-39})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{1}q^{2}+\beta _{4}q^{3}+(-2\beta _{2}-\beta _{5})q^{4}+\cdots$$
507.2.f.e $$8$$ $$4.048$$ 8.0.56070144.2 None $$0$$ $$4$$ $$0$$ $$-8$$ $$q+(\beta _{1}+\beta _{2}+\beta _{4})q^{2}+(\beta _{4}+\beta _{5})q^{3}+\cdots$$
507.2.f.f $$8$$ $$4.048$$ 8.0.56070144.2 None $$0$$ $$4$$ $$0$$ $$8$$ $$q+(\beta _{1}+\beta _{2}+\beta _{4})q^{2}+(1-\beta _{2}-\beta _{4}+\cdots)q^{3}+\cdots$$
507.2.f.g $$48$$ $$4.048$$ None $$0$$ $$0$$ $$0$$ $$0$$

## Decomposition of $$S_{2}^{\mathrm{old}}(507, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(507, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(39, [\chi])$$$$^{\oplus 2}$$