# Properties

 Label 675.2.l Level $675$ Weight $2$ Character orbit 675.l Rep. character $\chi_{675}(76,\cdot)$ Character field $\Q(\zeta_{9})$ Dimension $324$ Newform subspaces $8$ 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.l (of order $$9$$ and degree $$6$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$27$$ Character field: $$\Q(\zeta_{9})$$ Newform subspaces: $$8$$ 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 576 360 216
Cusp forms 504 324 180
Eisenstein series 72 36 36

## Trace form

 $$324 q + 6 q^{2} + 6 q^{3} + 6 q^{4} - 24 q^{6} + 6 q^{7} + 12 q^{8} + O(q^{10})$$ $$324 q + 6 q^{2} + 6 q^{3} + 6 q^{4} - 24 q^{6} + 6 q^{7} + 12 q^{8} - 21 q^{11} + 24 q^{12} + 6 q^{13} + 15 q^{14} - 24 q^{16} + 15 q^{17} - 21 q^{18} + 3 q^{19} + 12 q^{21} + 15 q^{22} - 24 q^{23} + 12 q^{24} + 18 q^{26} - 21 q^{27} + 12 q^{28} - 30 q^{29} - 27 q^{31} - 72 q^{32} + 6 q^{33} + 33 q^{34} + 18 q^{36} + 3 q^{37} + 6 q^{38} - 33 q^{39} + 3 q^{41} - 78 q^{42} + 15 q^{43} - 3 q^{44} - 9 q^{46} - 57 q^{47} - 99 q^{48} - 12 q^{49} - 108 q^{51} + 45 q^{52} - 54 q^{53} + 90 q^{54} + 69 q^{56} - 27 q^{57} + 33 q^{58} - 42 q^{59} - 36 q^{61} - 6 q^{62} - 21 q^{63} - 96 q^{64} - 15 q^{66} + 33 q^{67} - 75 q^{68} + 93 q^{69} - 63 q^{71} + 126 q^{72} + 12 q^{73} + 111 q^{74} - 36 q^{76} + 75 q^{77} - 36 q^{78} + 42 q^{79} - 108 q^{81} + 12 q^{82} - 9 q^{83} - 12 q^{84} - 111 q^{86} + 57 q^{87} - 78 q^{88} - 87 q^{89} - 9 q^{92} + 21 q^{93} + 69 q^{94} - 264 q^{96} + 51 q^{97} + 75 q^{98} - 99 q^{99} + O(q^{100})$$

## 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.l.a $6$ $5.390$ $$\Q(\zeta_{18})$$ None $$-6$$ $$0$$ $$0$$ $$-6$$ $$q+(-1-\zeta_{18}+\zeta_{18}^{4}+\zeta_{18}^{5})q^{2}+\cdots$$
675.2.l.b $6$ $5.390$ $$\Q(\zeta_{18})$$ None $$6$$ $$0$$ $$0$$ $$6$$ $$q+(1+\zeta_{18}-\zeta_{18}^{4}-\zeta_{18}^{5})q^{2}+(-\zeta_{18}^{2}+\cdots)q^{3}+\cdots$$
675.2.l.c $12$ $5.390$ 12.0.$$\cdots$$.1 None $$6$$ $$6$$ $$0$$ $$6$$ $$q+(1+\beta _{3}-\beta _{8})q^{2}+(1+\beta _{2}-\beta _{6}+\beta _{8}+\cdots)q^{3}+\cdots$$
675.2.l.d $30$ $5.390$ None $$0$$ $$-3$$ $$0$$ $$0$$
675.2.l.e $42$ $5.390$ None $$0$$ $$3$$ $$0$$ $$0$$
675.2.l.f $66$ $5.390$ None $$-6$$ $$0$$ $$0$$ $$-6$$
675.2.l.g $66$ $5.390$ None $$6$$ $$0$$ $$0$$ $$6$$
675.2.l.h $96$ $5.390$ None $$0$$ $$0$$ $$0$$ $$0$$

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

$$S_{2}^{\mathrm{old}}(675, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(27, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(135, [\chi])$$$$^{\oplus 2}$$