# Properties

 Label 675.2.k Level $675$ Weight $2$ Character orbit 675.k Rep. character $\chi_{675}(199,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $32$ Newform subspaces $3$ Sturm bound $180$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$675 = 3^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 675.k (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$45$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$3$$ Sturm bound: $$180$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(675, [\chi])$$.

Total New Old
Modular forms 216 40 176
Cusp forms 144 32 112
Eisenstein series 72 8 64

## Trace form

 $$32 q + 16 q^{4} + O(q^{10})$$ $$32 q + 16 q^{4} - 10 q^{11} + 18 q^{14} - 16 q^{16} + 8 q^{19} + 56 q^{26} + 14 q^{29} - 8 q^{31} + 18 q^{34} - 26 q^{41} + 8 q^{44} - 6 q^{49} - 60 q^{56} - 2 q^{59} + 10 q^{61} - 44 q^{64} + 64 q^{71} - 40 q^{74} - 14 q^{76} - 22 q^{79} - 28 q^{86} - 132 q^{89} - 4 q^{91} - 42 q^{94} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(675, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
675.2.k.a $4$ $5.390$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{12}q^{2}-\zeta_{12}^{2}q^{4}+3\zeta_{12}q^{7}-3\zeta_{12}^{3}q^{8}+\cdots$$
675.2.k.b $12$ $5.390$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{8}q^{2}+(2\beta _{6}-\beta _{9})q^{4}+(-\beta _{3}+\beta _{7}+\cdots)q^{7}+\cdots$$
675.2.k.c $16$ $5.390$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+(1+\beta _{3}+\beta _{7}+\beta _{11})q^{4}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(675, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(675, [\chi]) \simeq$$ $$S_{2}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(135, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(225, [\chi])$$$$^{\oplus 2}$$