# Properties

 Label 825.2.k Level $825$ Weight $2$ Character orbit 825.k Rep. character $\chi_{825}(518,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $120$ Newform subspaces $12$ Sturm bound $240$ Trace bound $14$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 825.k (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$15$$ Character field: $$\Q(i)$$ Newform subspaces: $$12$$ Sturm bound: $$240$$ Trace bound: $$14$$ Distinguishing $$T_p$$: $$2$$, $$7$$, $$13$$, $$29$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(825, [\chi])$$.

Total New Old
Modular forms 264 120 144
Cusp forms 216 120 96
Eisenstein series 48 0 48

## Trace form

 $$120 q + 2 q^{3} + 16 q^{6} + O(q^{10})$$ $$120 q + 2 q^{3} + 16 q^{6} - 8 q^{12} + 16 q^{13} - 136 q^{16} + 12 q^{18} + 32 q^{21} - 4 q^{27} - 32 q^{28} + 24 q^{31} - 2 q^{33} - 48 q^{36} + 36 q^{37} + 64 q^{42} + 4 q^{48} - 32 q^{51} - 88 q^{52} - 40 q^{57} - 8 q^{58} + 40 q^{61} - 4 q^{63} + 28 q^{67} - 48 q^{72} - 32 q^{76} + 40 q^{78} + 164 q^{81} - 8 q^{82} - 16 q^{87} + 24 q^{88} - 24 q^{91} - 58 q^{93} - 256 q^{96} - 76 q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(825, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
825.2.k.a $$4$$ $$6.588$$ $$\Q(\zeta_{8})$$ None $$-4$$ $$-4$$ $$0$$ $$-8$$ $$q+(-1+\zeta_{8}^{2}-\zeta_{8}^{3})q^{2}+(-1-\zeta_{8}^{2}+\cdots)q^{3}+\cdots$$
825.2.k.b $$4$$ $$6.588$$ $$\Q(\zeta_{8})$$ None $$-4$$ $$-4$$ $$0$$ $$8$$ $$q+(-1+\zeta_{8}^{2}-\zeta_{8}^{3})q^{2}+(-1-\zeta_{8}^{2}+\cdots)q^{3}+\cdots$$
825.2.k.c $$4$$ $$6.588$$ $$\Q(\zeta_{8})$$ None $$-4$$ $$0$$ $$0$$ $$8$$ $$q+(-1+\zeta_{8}^{2}-\zeta_{8}^{3})q^{2}+(-\zeta_{8}-\zeta_{8}^{2}+\cdots)q^{3}+\cdots$$
825.2.k.d $$4$$ $$6.588$$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}-\beta _{1}q^{3}+\beta _{2}q^{4}-3\beta _{2}q^{6}+\cdots$$
825.2.k.e $$4$$ $$6.588$$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+\beta _{1}q^{3}+\beta _{2}q^{4}+3\beta _{2}q^{6}+\cdots$$
825.2.k.f $$4$$ $$6.588$$ $$\Q(\zeta_{8})$$ None $$4$$ $$4$$ $$0$$ $$8$$ $$q+(1-\zeta_{8}^{2}-\zeta_{8}^{3})q^{2}+(1-\zeta_{8}-\zeta_{8}^{3})q^{3}+\cdots$$
825.2.k.g $$4$$ $$6.588$$ $$\Q(\zeta_{8})$$ None $$4$$ $$4$$ $$0$$ $$-8$$ $$q+(1-\zeta_{8}^{2}-\zeta_{8}^{3})q^{2}+(1+\zeta_{8}^{2}-\zeta_{8}^{3})q^{3}+\cdots$$
825.2.k.h $$4$$ $$6.588$$ $$\Q(\zeta_{8})$$ None $$4$$ $$4$$ $$0$$ $$8$$ $$q+(1-\zeta_{8}^{2}-\zeta_{8}^{3})q^{2}+(1+\zeta_{8}^{2}+\zeta_{8}^{3})q^{3}+\cdots$$
825.2.k.i $$16$$ $$6.588$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$-4$$ $$-4$$ $$0$$ $$-8$$ $$q+\beta _{3}q^{2}-\beta _{4}q^{3}+(-\beta _{1}-\beta _{3}-\beta _{13}+\cdots)q^{4}+\cdots$$
825.2.k.j $$16$$ $$6.588$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$4$$ $$2$$ $$0$$ $$-8$$ $$q+\beta _{1}q^{2}+\beta _{14}q^{3}+(\beta _{1}+\beta _{3}+\beta _{13}+\cdots)q^{4}+\cdots$$
825.2.k.k $$28$$ $$6.588$$ None $$0$$ $$0$$ $$0$$ $$0$$
825.2.k.l $$28$$ $$6.588$$ None $$0$$ $$0$$ $$0$$ $$0$$

## Decomposition of $$S_{2}^{\mathrm{old}}(825, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(825, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(75, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(165, [\chi])$$$$^{\oplus 2}$$