# Properties

 Label 504.2.s Level $504$ Weight $2$ Character orbit 504.s Rep. character $\chi_{504}(289,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $20$ Newform subspaces $9$ Sturm bound $192$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$504 = 2^{3} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 504.s (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$9$$ Sturm bound: $$192$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$5$$, $$11$$, $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(504, [\chi])$$.

Total New Old
Modular forms 224 20 204
Cusp forms 160 20 140
Eisenstein series 64 0 64

## Trace form

 $$20 q - 2 q^{5} + O(q^{10})$$ $$20 q - 2 q^{5} - 2 q^{11} - 6 q^{17} - 10 q^{19} + 2 q^{23} - 20 q^{25} + 16 q^{29} + 6 q^{31} + 30 q^{35} - 2 q^{37} + 8 q^{43} + 6 q^{47} + 36 q^{49} + 18 q^{53} + 4 q^{55} + 18 q^{59} + 10 q^{61} + 20 q^{65} + 2 q^{67} - 48 q^{71} + 22 q^{73} - 50 q^{77} - 14 q^{79} - 56 q^{83} - 60 q^{85} - 34 q^{89} - 44 q^{91} - 38 q^{95} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(504, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
504.2.s.a $2$ $4.024$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-4$$ $$5$$ $$q+(-4+4\zeta_{6})q^{5}+(3-\zeta_{6})q^{7}-3q^{13}+\cdots$$
504.2.s.b $2$ $4.024$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-1$$ $$-5$$ $$q+(-1+\zeta_{6})q^{5}+(-2-\zeta_{6})q^{7}+5\zeta_{6}q^{11}+\cdots$$
504.2.s.c $2$ $4.024$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-1$$ $$-4$$ $$q+(-1+\zeta_{6})q^{5}+(-3+2\zeta_{6})q^{7}+3\zeta_{6}q^{11}+\cdots$$
504.2.s.d $2$ $4.024$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-1$$ $$1$$ $$q+(-1+\zeta_{6})q^{5}+(2-3\zeta_{6})q^{7}+3\zeta_{6}q^{11}+\cdots$$
504.2.s.e $2$ $4.024$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-1$$ $$4$$ $$q+(-1+\zeta_{6})q^{5}+(1+2\zeta_{6})q^{7}-\zeta_{6}q^{11}+\cdots$$
504.2.s.f $2$ $4.024$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$1$$ $$-5$$ $$q+(1-\zeta_{6})q^{5}+(-2-\zeta_{6})q^{7}-5\zeta_{6}q^{11}+\cdots$$
504.2.s.g $2$ $4.024$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$2$$ $$5$$ $$q+(2-2\zeta_{6})q^{5}+(3-\zeta_{6})q^{7}-6\zeta_{6}q^{11}+\cdots$$
504.2.s.h $2$ $4.024$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$4$$ $$5$$ $$q+(4-4\zeta_{6})q^{5}+(3-\zeta_{6})q^{7}-3q^{13}+\cdots$$
504.2.s.i $4$ $4.024$ $$\Q(\sqrt{-3}, \sqrt{-19})$$ None $$0$$ $$0$$ $$-1$$ $$-6$$ $$q+(-1+2\beta _{1}-\beta _{3})q^{5}+(-1-\beta _{1}+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(504, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(504, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(21, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(28, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(42, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(56, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(63, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(84, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(126, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(168, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(252, [\chi])$$$$^{\oplus 2}$$