# Properties

 Label 507.2.e Level $507$ Weight $2$ Character orbit 507.e Rep. character $\chi_{507}(22,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $50$ Newform subspaces $12$ Sturm bound $121$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 507.e (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$13$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$12$$ Sturm bound: $$121$$ Trace bound: $$5$$ 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 50 100
Cusp forms 94 50 44
Eisenstein series 56 0 56

## Trace form

 $$50 q + 2 q^{2} + q^{3} - 24 q^{4} + 8 q^{5} - q^{7} - 12 q^{8} - 25 q^{9} + O(q^{10})$$ $$50 q + 2 q^{2} + q^{3} - 24 q^{4} + 8 q^{5} - q^{7} - 12 q^{8} - 25 q^{9} + 10 q^{10} - 6 q^{11} - 20 q^{12} + 8 q^{14} - 2 q^{15} - 22 q^{16} - 2 q^{17} - 4 q^{18} + 2 q^{20} + 10 q^{21} + 4 q^{22} + 2 q^{23} + 12 q^{24} + 50 q^{25} - 2 q^{27} - 18 q^{28} + 8 q^{29} + 12 q^{30} - 10 q^{31} + 14 q^{32} - 2 q^{33} - 4 q^{34} - 2 q^{35} - 24 q^{36} + 12 q^{37} + 32 q^{38} - 20 q^{40} + 10 q^{41} - 12 q^{42} + 23 q^{43} + 16 q^{44} - 4 q^{45} - 4 q^{46} - 12 q^{47} + 12 q^{48} - 24 q^{49} - 28 q^{50} - 24 q^{51} - 36 q^{53} + 20 q^{55} - 16 q^{56} - 24 q^{57} - 26 q^{58} - 14 q^{59} + 33 q^{61} + 12 q^{62} - q^{63} - 12 q^{64} - 8 q^{66} + 3 q^{67} + 42 q^{68} + 14 q^{69} + 8 q^{70} + 34 q^{71} + 6 q^{72} + 2 q^{73} + 10 q^{74} + 15 q^{75} + 4 q^{76} - 36 q^{77} - 46 q^{79} + 22 q^{80} - 25 q^{81} + 38 q^{82} + 48 q^{83} - 14 q^{84} + 14 q^{85} + 6 q^{87} + 12 q^{88} + 4 q^{89} - 20 q^{90} - 88 q^{92} - 3 q^{93} - 16 q^{94} - 8 q^{96} - 15 q^{97} + 22 q^{98} + 12 q^{99} + O(q^{100})$$

## 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.e.a $2$ $4.048$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$1$$ $$4$$ $$4$$ $$q+(-1+\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots$$
507.2.e.b $2$ $4.048$ $$\Q(\sqrt{-3})$$ None $$1$$ $$1$$ $$-4$$ $$-4$$ $$q+(1-\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots$$
507.2.e.c $2$ $4.048$ $$\Q(\sqrt{-3})$$ None $$1$$ $$1$$ $$2$$ $$2$$ $$q+(1-\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots$$
507.2.e.d $4$ $4.048$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$-2$$ $$-2$$ $$0$$ $$0$$ $$q+(-1+\beta _{1}-\beta _{2})q^{2}+(-1-\beta _{2})q^{3}+\cdots$$
507.2.e.e $4$ $4.048$ $$\Q(\zeta_{12})$$ None $$0$$ $$2$$ $$0$$ $$0$$ $$q-\zeta_{12}^{2}q^{2}+\zeta_{12}q^{3}+(-1+\zeta_{12}+\cdots)q^{4}+\cdots$$
507.2.e.f $4$ $4.048$ $$\Q(\zeta_{12})$$ None $$0$$ $$2$$ $$0$$ $$0$$ $$q+\zeta_{12}q^{3}+(2-2\zeta_{12})q^{4}-2\zeta_{12}^{3}q^{5}+\cdots$$
507.2.e.g $4$ $4.048$ $$\Q(\sqrt{-3}, \sqrt{17})$$ None $$1$$ $$-2$$ $$6$$ $$-3$$ $$q+\beta _{1}q^{2}-\beta _{2}q^{3}+(-2+\beta _{1}+2\beta _{2}+\cdots)q^{4}+\cdots$$
507.2.e.h $4$ $4.048$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$2$$ $$-2$$ $$0$$ $$0$$ $$q+(1+\beta _{1}+\beta _{2})q^{2}+(-1-\beta _{2})q^{3}+\cdots$$
507.2.e.i $6$ $4.048$ 6.0.64827.1 None $$-3$$ $$3$$ $$12$$ $$-2$$ $$q+(-2+2\beta _{1}+\beta _{4}+2\beta _{5})q^{2}+(1-\beta _{5})q^{3}+\cdots$$
507.2.e.j $6$ $4.048$ 6.0.64827.1 None $$-1$$ $$-3$$ $$8$$ $$-10$$ $$q-\beta _{1}q^{2}+(-1+\beta _{5})q^{3}+(\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots$$
507.2.e.k $6$ $4.048$ 6.0.64827.1 None $$1$$ $$-3$$ $$-8$$ $$10$$ $$q+\beta _{1}q^{2}+(-1+\beta _{5})q^{3}+(\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots$$
507.2.e.l $6$ $4.048$ 6.0.64827.1 None $$3$$ $$3$$ $$-12$$ $$2$$ $$q+(2-2\beta _{1}-\beta _{4}-2\beta _{5})q^{2}+(1-\beta _{5})q^{3}+\cdots$$

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

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