# Properties

 Label 1323.2.f Level $1323$ Weight $2$ Character orbit 1323.f Rep. character $\chi_{1323}(442,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $72$ Newform subspaces $8$ Sturm bound $336$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1323 = 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1323.f (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$9$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$8$$ Sturm bound: $$336$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$2$$, $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(1323, [\chi])$$.

Total New Old
Modular forms 384 92 292
Cusp forms 288 72 216
Eisenstein series 96 20 76

## Trace form

 $$72 q - 2 q^{2} - 30 q^{4} + 2 q^{5} + 24 q^{8} + O(q^{10})$$ $$72 q - 2 q^{2} - 30 q^{4} + 2 q^{5} + 24 q^{8} - 6 q^{11} - 18 q^{16} - 12 q^{17} + 12 q^{19} + 22 q^{20} - 12 q^{22} - 16 q^{23} - 18 q^{25} - 8 q^{26} - 6 q^{31} - 26 q^{32} + 6 q^{34} + 12 q^{37} - 14 q^{38} + 12 q^{40} + 22 q^{41} - 12 q^{43} + 44 q^{44} + 6 q^{47} + 18 q^{50} - 18 q^{52} - 16 q^{53} + 12 q^{55} - 18 q^{58} + 12 q^{59} - 96 q^{62} - 24 q^{64} - 24 q^{65} - 12 q^{67} - 12 q^{68} + 60 q^{71} + 36 q^{73} + 22 q^{74} - 6 q^{76} - 8 q^{80} + 30 q^{83} + 18 q^{85} - 58 q^{86} - 6 q^{88} + 20 q^{89} + 4 q^{92} + 6 q^{94} + 26 q^{95} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(1323, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1323.2.f.a $2$ $10.564$ $$\Q(\sqrt{-3})$$ None $$1$$ $$0$$ $$-1$$ $$0$$ $$q+(1-\zeta_{6})q^{2}+\zeta_{6}q^{4}-\zeta_{6}q^{5}+3q^{8}+\cdots$$
1323.2.f.b $2$ $10.564$ $$\Q(\sqrt{-3})$$ None $$1$$ $$0$$ $$1$$ $$0$$ $$q+(1-\zeta_{6})q^{2}+\zeta_{6}q^{4}+\zeta_{6}q^{5}+3q^{8}+\cdots$$
1323.2.f.c $6$ $10.564$ 6.0.309123.1 None $$-1$$ $$0$$ $$5$$ $$0$$ $$q+(\beta _{1}-\beta _{5})q^{2}+(-1+\beta _{2}+\beta _{4}+\beta _{5})q^{4}+\cdots$$
1323.2.f.d $6$ $10.564$ $$\Q(\zeta_{18})$$ None $$3$$ $$0$$ $$-3$$ $$0$$ $$q+(\zeta_{18}-\zeta_{18}^{3}-\zeta_{18}^{4}+\zeta_{18}^{5})q^{2}+\cdots$$
1323.2.f.e $10$ $10.564$ 10.0.$$\cdots$$.1 None $$-2$$ $$0$$ $$-4$$ $$0$$ $$q-\beta _{1}q^{2}+(-\beta _{3}+\beta _{6}+\beta _{7})q^{4}+(\beta _{6}+\cdots)q^{5}+\cdots$$
1323.2.f.f $10$ $10.564$ 10.0.$$\cdots$$.1 None $$-2$$ $$0$$ $$4$$ $$0$$ $$q-\beta _{1}q^{2}+(-\beta _{3}+\beta _{6}+\beta _{7})q^{4}+(-\beta _{6}+\cdots)q^{5}+\cdots$$
1323.2.f.g $12$ $10.564$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$2$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{6}q^{2}+(-\beta _{1}+\beta _{4}-\beta _{5}+\beta _{6}-\beta _{7}+\cdots)q^{4}+\cdots$$
1323.2.f.h $24$ $10.564$ None $$-4$$ $$0$$ $$0$$ $$0$$

## Decomposition of $$S_{2}^{\mathrm{old}}(1323, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(1323, [\chi]) \simeq$$ $$S_{2}^{\mathrm{new}}(63, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(189, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(441, [\chi])$$$$^{\oplus 2}$$