# Properties

 Label 507.2.k Level $507$ Weight $2$ Character orbit 507.k Rep. character $\chi_{507}(80,\cdot)$ Character field $\Q(\zeta_{12})$ Dimension $164$ Newform subspaces $11$ Sturm bound $121$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 507.k (of order $$12$$ and degree $$4$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$39$$ Character field: $$\Q(\zeta_{12})$$ Newform subspaces: $$11$$ 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 300 244 56
Cusp forms 188 164 24
Eisenstein series 112 80 32

## Trace form

 $$164q + 2q^{3} + 12q^{4} + 2q^{6} + 14q^{7} + 2q^{9} + O(q^{10})$$ $$164q + 2q^{3} + 12q^{4} + 2q^{6} + 14q^{7} + 2q^{9} - 12q^{10} + 14q^{15} + 4q^{16} - 4q^{18} + 2q^{19} - 22q^{21} - 12q^{22} - 18q^{24} - 76q^{27} - 18q^{30} + 6q^{31} - 16q^{33} + 36q^{34} + 36q^{36} + 30q^{37} - 56q^{40} + 24q^{42} - 30q^{43} + 20q^{45} + 106q^{48} - 18q^{49} - 46q^{54} + 4q^{55} - 28q^{57} - 28q^{58} - 44q^{60} - 36q^{61} - 16q^{63} + 8q^{67} + 32q^{70} - 12q^{72} + 62q^{73} + 18q^{75} + 36q^{76} - 112q^{79} + 14q^{81} + 24q^{82} + 8q^{84} - 12q^{85} - 22q^{87} + 12q^{88} - 10q^{93} - 112q^{94} - 16q^{96} + 18q^{97} - 40q^{99} + O(q^{100})$$

## 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.k.a $$4$$ $$4.048$$ $$\Q(\zeta_{12})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$-8$$ $$q+(\zeta_{12}+\zeta_{12}^{3})q^{3}+2\zeta_{12}q^{4}+(-1+\cdots)q^{7}+\cdots$$
507.2.k.b $$4$$ $$4.048$$ $$\Q(\zeta_{12})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$8$$ $$q+(\zeta_{12}+\zeta_{12}^{3})q^{3}+2\zeta_{12}q^{4}+(1+\zeta_{12}+\cdots)q^{7}+\cdots$$
507.2.k.c $$4$$ $$4.048$$ $$\Q(\zeta_{12})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$10$$ $$q+(-\zeta_{12}-\zeta_{12}^{3})q^{3}+2\zeta_{12}q^{4}+(2+\cdots)q^{7}+\cdots$$
507.2.k.d $$8$$ $$4.048$$ 8.0.56070144.2 None $$0$$ $$-2$$ $$0$$ $$4$$ $$q+(-\beta _{1}-\beta _{2}+\beta _{3}-\beta _{4}+2\beta _{5}+\beta _{7})q^{2}+\cdots$$
507.2.k.e $$8$$ $$4.048$$ 8.0.56070144.2 None $$0$$ $$-2$$ $$0$$ $$-4$$ $$q+(\beta _{2}-\beta _{5}-\beta _{7})q^{2}+(-\beta _{2}+\beta _{4})q^{3}+\cdots$$
507.2.k.f $$8$$ $$4.048$$ 8.0.56070144.2 None $$0$$ $$-2$$ $$0$$ $$4$$ $$q+(\beta _{2}-\beta _{5}-\beta _{7})q^{2}+(\beta _{2}+\beta _{4}-\beta _{5}+\cdots)q^{3}+\cdots$$
507.2.k.g $$8$$ $$4.048$$ 8.0.56070144.2 $$\Q(\sqrt{-39})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{6}q^{2}+(-\beta _{2}-2\beta _{3})q^{3}+(-2+2\beta _{3}+\cdots)q^{4}+\cdots$$
507.2.k.h $$8$$ $$4.048$$ 8.0.56070144.2 $$\Q(\sqrt{-39})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{5}q^{2}+(2\beta _{2}+\beta _{3})q^{3}+(1+2\beta _{2}+\cdots)q^{4}+\cdots$$
507.2.k.i $$8$$ $$4.048$$ $$\Q(\zeta_{24})$$ None $$0$$ $$4$$ $$0$$ $$4$$ $$q+\zeta_{24}^{7}q^{2}+(\zeta_{24}+\zeta_{24}^{4}-\zeta_{24}^{7})q^{3}+\cdots$$
507.2.k.j $$8$$ $$4.048$$ $$\Q(\zeta_{24})$$ None $$0$$ $$4$$ $$0$$ $$-4$$ $$q+\zeta_{24}^{7}q^{2}+(-\zeta_{24}+\zeta_{24}^{4}+\zeta_{24}^{7})q^{3}+\cdots$$
507.2.k.k $$96$$ $$4.048$$ None $$0$$ $$0$$ $$0$$ $$0$$

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

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