# Properties

 Label 507.2.b Level $507$ Weight $2$ Character orbit 507.b Rep. character $\chi_{507}(337,\cdot)$ Character field $\Q$ Dimension $26$ Newform subspaces $7$ Sturm bound $121$ Trace bound $10$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 507.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$13$$ Character field: $$\Q$$ Newform subspaces: $$7$$ 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 74 26 48
Cusp forms 46 26 20
Eisenstein series 28 0 28

## Trace form

 $$26q + 2q^{3} - 26q^{4} + 26q^{9} + O(q^{10})$$ $$26q + 2q^{3} - 26q^{4} + 26q^{9} + 4q^{10} + 2q^{12} - 16q^{14} + 34q^{16} + 16q^{17} - 20q^{22} + 4q^{23} - 34q^{25} + 2q^{27} - 20q^{29} + 4q^{30} - 4q^{35} - 26q^{36} + 16q^{38} - 12q^{40} + 12q^{42} + 20q^{43} - 18q^{48} - 10q^{49} - 8q^{51} - 28q^{53} - 12q^{55} - 12q^{61} + 4q^{62} - 6q^{64} + 8q^{66} - 36q^{68} + 4q^{69} + 12q^{74} + 18q^{75} + 40q^{77} + 4q^{79} + 26q^{81} - 4q^{82} + 8q^{87} + 12q^{88} + 4q^{90} - 28q^{92} + 20q^{95} + O(q^{100})$$

## 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.b.a $$2$$ $$4.048$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-2$$ $$0$$ $$0$$ $$q+iq^{2}-q^{3}+q^{4}+2iq^{5}-iq^{6}+\cdots$$
507.2.b.b $$2$$ $$4.048$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-2$$ $$0$$ $$0$$ $$q+iq^{2}-q^{3}+q^{4}-iq^{5}-iq^{6}-2iq^{7}+\cdots$$
507.2.b.c $$2$$ $$4.048$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-2$$ $$0$$ $$0$$ $$q-q^{3}+2q^{4}+2\zeta_{6}q^{5}-\zeta_{6}q^{7}+q^{9}+\cdots$$
507.2.b.d $$4$$ $$4.048$$ $$\Q(i, \sqrt{17})$$ None $$0$$ $$4$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+q^{3}+(-3+\beta _{3})q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots$$
507.2.b.e $$4$$ $$4.048$$ $$\Q(\zeta_{8})$$ None $$0$$ $$4$$ $$0$$ $$0$$ $$q+\zeta_{8}q^{2}+q^{3}+(-1-\zeta_{8}^{3})q^{4}+(-\zeta_{8}+\cdots)q^{5}+\cdots$$
507.2.b.f $$6$$ $$4.048$$ 6.0.153664.1 None $$0$$ $$-6$$ $$0$$ $$0$$ $$q+(\beta _{1}+\beta _{3}+\beta _{5})q^{2}-q^{3}+(-3-\beta _{2}+\cdots)q^{4}+\cdots$$
507.2.b.g $$6$$ $$4.048$$ 6.0.153664.1 None $$0$$ $$6$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+q^{3}+\beta _{2}q^{4}+(-\beta _{3}+\beta _{5})q^{5}+\cdots$$

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

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