# Properties

 Label 507.2.j Level $507$ Weight $2$ Character orbit 507.j Rep. character $\chi_{507}(316,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $54$ Newform subspaces $9$ Sturm bound $121$ Trace bound $10$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 507.j (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$13$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$9$$ Sturm bound: $$121$$ Trace bound: $$10$$ Distinguishing $$T_p$$: $$2$$, $$5$$

## Dimensions

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

Total New Old
Modular forms 150 54 96
Cusp forms 94 54 40
Eisenstein series 56 0 56

## Trace form

 $$54q - q^{3} + 30q^{4} + 3q^{7} - 27q^{9} + O(q^{10})$$ $$54q - q^{3} + 30q^{4} + 3q^{7} - 27q^{9} - 4q^{10} + 6q^{11} + 12q^{12} - 8q^{14} - 6q^{15} - 20q^{16} - 4q^{17} - 6q^{19} - 12q^{20} + 8q^{22} - 10q^{23} - 34q^{25} + 2q^{27} - 6q^{28} + 14q^{29} - 4q^{30} + 6q^{33} + 10q^{35} + 30q^{36} + 8q^{38} - 24q^{40} + 12q^{41} - 13q^{43} - 6q^{45} + 4q^{48} + 24q^{49} + 8q^{51} - 56q^{53} - 12q^{56} + 6q^{59} + 17q^{61} - 16q^{62} - 3q^{63} - 32q^{64} - 8q^{66} - 15q^{67} + 24q^{68} + 2q^{69} + 18q^{71} + 12q^{74} - q^{75} + 12q^{76} + 20q^{77} - 2q^{79} + 24q^{80} - 27q^{81} - 20q^{82} - 6q^{84} - 2q^{87} - 24q^{88} - 12q^{89} + 8q^{90} - 32q^{92} - 3q^{93} + 12q^{94} - 32q^{95} + 9q^{97} + O(q^{100})$$

## 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.j.a $$2$$ $$4.048$$ $$\Q(\sqrt{-3})$$ None $$-3$$ $$1$$ $$0$$ $$-6$$ $$q+(-1-\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots$$
507.2.j.b $$2$$ $$4.048$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$1$$ $$0$$ $$3$$ $$q+(1-\zeta_{6})q^{3}-2\zeta_{6}q^{4}+(2-4\zeta_{6})q^{5}+\cdots$$
507.2.j.c $$2$$ $$4.048$$ $$\Q(\sqrt{-3})$$ None $$3$$ $$1$$ $$0$$ $$6$$ $$q+(1+\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots$$
507.2.j.d $$4$$ $$4.048$$ $$\Q(\zeta_{12})$$ None $$0$$ $$2$$ $$0$$ $$0$$ $$q+\zeta_{12}q^{2}+(1-\zeta_{12}^{2})q^{3}-\zeta_{12}^{2}q^{4}+\cdots$$
507.2.j.e $$4$$ $$4.048$$ $$\Q(\zeta_{12})$$ None $$0$$ $$2$$ $$0$$ $$0$$ $$q+\zeta_{12}q^{2}+(1-\zeta_{12}^{2})q^{3}-\zeta_{12}^{2}q^{4}+\cdots$$
507.2.j.f $$8$$ $$4.048$$ $$\Q(\zeta_{24})$$ None $$0$$ $$-4$$ $$0$$ $$0$$ $$q+\zeta_{24}^{3}q^{2}-\zeta_{24}q^{3}+(1-\zeta_{24}-\zeta_{24}^{4}+\cdots)q^{4}+\cdots$$
507.2.j.g $$8$$ $$4.048$$ 8.0.1731891456.1 None $$0$$ $$-4$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+\beta _{2}q^{3}+(2+3\beta _{2}-\beta _{4}+\beta _{5}+\cdots)q^{4}+\cdots$$
507.2.j.h $$12$$ $$4.048$$ 12.0.$$\cdots$$.1 None $$0$$ $$-6$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+(-1+\beta _{7})q^{3}+(1-\beta _{3}+\beta _{4}+\cdots)q^{4}+\cdots$$
507.2.j.i $$12$$ $$4.048$$ 12.0.$$\cdots$$.1 None $$0$$ $$6$$ $$0$$ $$0$$ $$q+(-\beta _{1}+\beta _{8}+2\beta _{11})q^{2}+\beta _{7}q^{3}+\cdots$$

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

$$S_{2}^{\mathrm{old}}(507, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(13, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(39, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(169, [\chi])$$$$^{\oplus 2}$$