# Properties

 Label 675.2.e Level $675$ Weight $2$ Character orbit 675.e Rep. character $\chi_{675}(226,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $32$ Newform subspaces $5$ Sturm bound $180$ Trace bound $2$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$675 = 3^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 675.e (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$9$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$5$$ Sturm bound: $$180$$ Trace bound: $$2$$ 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 44 172
Cusp forms 144 32 112
Eisenstein series 72 12 60

## Trace form

 $$32 q - 2 q^{2} - 12 q^{4} + 2 q^{7} + O(q^{10})$$ $$32 q - 2 q^{2} - 12 q^{4} + 2 q^{7} + 6 q^{11} + 2 q^{13} - 6 q^{14} - 8 q^{16} + 4 q^{17} + 8 q^{19} - 6 q^{22} - 6 q^{23} + 24 q^{26} - 16 q^{28} + 6 q^{29} + 4 q^{31} - 22 q^{32} - 14 q^{34} - 4 q^{37} + 10 q^{38} + 6 q^{41} + 2 q^{43} - 72 q^{44} - 16 q^{46} - 20 q^{47} + 10 q^{49} - 10 q^{52} - 8 q^{53} - 48 q^{56} - 18 q^{58} + 42 q^{59} + 10 q^{61} + 84 q^{62} + 28 q^{64} + 8 q^{67} + 26 q^{68} - 24 q^{71} + 8 q^{73} + 48 q^{74} - 30 q^{76} - 6 q^{77} + 2 q^{79} + 48 q^{82} + 6 q^{83} - 12 q^{86} - 18 q^{88} + 60 q^{89} - 60 q^{91} - 36 q^{92} - 2 q^{94} - 16 q^{97} - 76 q^{98} + O(q^{100})$$

## 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.e.a $2$ $5.390$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$0$$ $$-3$$ $$q+(-1+\zeta_{6})q^{2}+\zeta_{6}q^{4}+(-3+3\zeta_{6})q^{7}+\cdots$$
675.2.e.b $6$ $5.390$ 6.0.954288.1 None $$-1$$ $$0$$ $$0$$ $$5$$ $$q+(\beta _{4}-\beta _{5})q^{2}+(-2-\beta _{1}-\beta _{2}+2\beta _{3}+\cdots)q^{4}+\cdots$$
675.2.e.c $8$ $5.390$ 8.0.1223810289.2 None $$-2$$ $$0$$ $$0$$ $$-1$$ $$q-\beta _{1}q^{2}+(\beta _{1}-\beta _{2}-\beta _{3}+\beta _{6})q^{4}+\cdots$$
675.2.e.d $8$ $5.390$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{24}^{7}q^{2}-\zeta_{24}^{2}q^{4}+(-2\zeta_{24}^{3}+\cdots)q^{7}+\cdots$$
675.2.e.e $8$ $5.390$ 8.0.1223810289.2 None $$2$$ $$0$$ $$0$$ $$1$$ $$q+\beta _{1}q^{2}+(\beta _{1}-\beta _{2}-\beta _{3}+\beta _{6})q^{4}+\cdots$$

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

$$S_{2}^{\mathrm{old}}(675, [\chi]) \cong$$