# Properties

 Label 324.3.c Level $324$ Weight $3$ Character orbit 324.c Rep. character $\chi_{324}(161,\cdot)$ Character field $\Q$ Dimension $8$ Newform subspaces $2$ Sturm bound $162$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$324 = 2^{2} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 324.c (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$3$$ Character field: $$\Q$$ Newform subspaces: $$2$$ Sturm bound: $$162$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(324, [\chi])$$.

Total New Old
Modular forms 126 8 118
Cusp forms 90 8 82
Eisenstein series 36 0 36

## Trace form

 $$8 q - 2 q^{7} + O(q^{10})$$ $$8 q - 2 q^{7} - 14 q^{13} - 26 q^{19} + 2 q^{25} + 46 q^{31} - 8 q^{37} + 40 q^{43} + 18 q^{49} + 126 q^{55} - 146 q^{61} - 212 q^{67} - 170 q^{73} + 142 q^{79} + 252 q^{85} - 82 q^{91} + 232 q^{97} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(324, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
324.3.c.a $$4$$ $$8.828$$ $$\Q(\sqrt{-2}, \sqrt{3})$$ None $$0$$ $$0$$ $$0$$ $$-4$$ $$q+\beta _{1}q^{5}+(-1+\beta _{2})q^{7}-\beta _{3}q^{11}+\cdots$$
324.3.c.b $$4$$ $$8.828$$ $$\Q(\sqrt{-3}, \sqrt{-11})$$ None $$0$$ $$0$$ $$0$$ $$2$$ $$q+\beta _{1}q^{5}+(1-\beta _{2})q^{7}+(-\beta _{1}+\beta _{3})q^{11}+\cdots$$

## Decomposition of $$S_{3}^{\mathrm{old}}(324, [\chi])$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(324, [\chi]) \cong$$ $$S_{3}^{\mathrm{new}}(12, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(18, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(27, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(54, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(81, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(108, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(162, [\chi])$$$$^{\oplus 2}$$