# Properties

 Label 324.3.f Level $324$ Weight $3$ Character orbit 324.f Rep. character $\chi_{324}(55,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $92$ Newform subspaces $19$ Sturm bound $162$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$324 = 2^{2} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 324.f (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$36$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$19$$ Sturm bound: $$162$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$5$$, $$7$$, $$11$$

## Dimensions

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

Total New Old
Modular forms 240 100 140
Cusp forms 192 92 100
Eisenstein series 48 8 40

## Trace form

 $$92q + 2q^{4} + O(q^{10})$$ $$92q + 2q^{4} - 20q^{10} + 4q^{13} + 2q^{16} - 6q^{22} - 186q^{25} + 60q^{28} + 40q^{34} - 8q^{37} - 98q^{40} + 432q^{46} + 242q^{49} + 52q^{52} - 80q^{58} + 4q^{61} - 412q^{64} - 54q^{70} - 56q^{73} - 420q^{76} - 416q^{82} + 104q^{85} - 186q^{88} + 60q^{94} + 124q^{97} + O(q^{100})$$

## 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.f.a $$2$$ $$8.828$$ $$\Q(\sqrt{-3})$$ None $$-4$$ $$0$$ $$-2$$ $$12$$ $$q-2q^{2}+4q^{4}+(-2+2\zeta_{6})q^{5}+(8+\cdots)q^{7}+\cdots$$
324.3.f.b $$2$$ $$8.828$$ $$\Q(\sqrt{-3})$$ None $$-4$$ $$0$$ $$7$$ $$-15$$ $$q-2q^{2}+4q^{4}+(7-7\zeta_{6})q^{5}+(-10+\cdots)q^{7}+\cdots$$
324.3.f.c $$2$$ $$8.828$$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$0$$ $$-7$$ $$15$$ $$q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+(-7+7\zeta_{6})q^{5}+\cdots$$
324.3.f.d $$2$$ $$8.828$$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$0$$ $$2$$ $$-12$$ $$q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+(2-2\zeta_{6})q^{5}+\cdots$$
324.3.f.e $$2$$ $$8.828$$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-1})$$ $$-2$$ $$0$$ $$8$$ $$0$$ $$q-2\zeta_{6}q^{2}+(-4+4\zeta_{6})q^{4}+(8-8\zeta_{6})q^{5}+\cdots$$
324.3.f.f $$2$$ $$8.828$$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-1})$$ $$2$$ $$0$$ $$-8$$ $$0$$ $$q+2\zeta_{6}q^{2}+(-4+4\zeta_{6})q^{4}+(-8+8\zeta_{6})q^{5}+\cdots$$
324.3.f.g $$2$$ $$8.828$$ $$\Q(\sqrt{-3})$$ None $$2$$ $$0$$ $$-2$$ $$-12$$ $$q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+(-2+2\zeta_{6})q^{5}+\cdots$$
324.3.f.h $$2$$ $$8.828$$ $$\Q(\sqrt{-3})$$ None $$2$$ $$0$$ $$7$$ $$15$$ $$q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+(7-7\zeta_{6})q^{5}+\cdots$$
324.3.f.i $$2$$ $$8.828$$ $$\Q(\sqrt{-3})$$ None $$4$$ $$0$$ $$-7$$ $$-15$$ $$q+2q^{2}+4q^{4}+(-7+7\zeta_{6})q^{5}+(-10+\cdots)q^{7}+\cdots$$
324.3.f.j $$2$$ $$8.828$$ $$\Q(\sqrt{-3})$$ None $$4$$ $$0$$ $$2$$ $$12$$ $$q+2q^{2}+4q^{4}+(2-2\zeta_{6})q^{5}+(8-4\zeta_{6})q^{7}+\cdots$$
324.3.f.k $$4$$ $$8.828$$ $$\Q(\zeta_{12})$$ $$\Q(\sqrt{-1})$$ $$-4$$ $$0$$ $$-8$$ $$0$$ $$q+(-2+2\zeta_{12})q^{2}-4\zeta_{12}q^{4}+(-4\zeta_{12}+\cdots)q^{5}+\cdots$$
324.3.f.l $$4$$ $$8.828$$ $$\Q(\sqrt{-3}, \sqrt{13})$$ None $$-3$$ $$0$$ $$0$$ $$0$$ $$q+(-1+\beta _{1})q^{2}+(-\beta _{1}+3\beta _{2}+\beta _{3})q^{4}+\cdots$$
324.3.f.m $$4$$ $$8.828$$ $$\Q(\sqrt{-3}, \sqrt{13})$$ None $$3$$ $$0$$ $$0$$ $$0$$ $$q+(1-\beta _{1})q^{2}+(-\beta _{1}+3\beta _{2}+\beta _{3})q^{4}+\cdots$$
324.3.f.n $$4$$ $$8.828$$ $$\Q(\zeta_{12})$$ $$\Q(\sqrt{-1})$$ $$4$$ $$0$$ $$8$$ $$0$$ $$q+(2-2\zeta_{12})q^{2}-4\zeta_{12}q^{4}+(4\zeta_{12}+\cdots)q^{5}+\cdots$$
324.3.f.o $$8$$ $$8.828$$ 8.0.207360000.1 None $$0$$ $$0$$ $$0$$ $$-12$$ $$q+(-\beta _{2}+\beta _{3})q^{2}+(-1+\beta _{1}+\beta _{7})q^{4}+\cdots$$
324.3.f.p $$8$$ $$8.828$$ 8.0.207360000.1 None $$0$$ $$0$$ $$0$$ $$12$$ $$q-\beta _{2}q^{2}+(1-\beta _{5}-\beta _{7})q^{4}+(\beta _{2}+\beta _{3}+\cdots)q^{5}+\cdots$$
324.3.f.q $$12$$ $$8.828$$ 12.0.$$\cdots$$.2 None $$-1$$ $$0$$ $$-2$$ $$0$$ $$q+(-\beta _{3}+\beta _{6})q^{2}+(1+\beta _{2}+\beta _{4})q^{4}+\cdots$$
324.3.f.r $$12$$ $$8.828$$ 12.0.$$\cdots$$.2 None $$1$$ $$0$$ $$2$$ $$0$$ $$q+(\beta _{3}-\beta _{6})q^{2}+(1+\beta _{2}+\beta _{4})q^{4}+(1+\cdots)q^{5}+\cdots$$
324.3.f.s $$16$$ $$8.828$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{5}q^{2}+(1-\beta _{2}-\beta _{3}-\beta _{7})q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots$$

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

$$S_{3}^{\mathrm{old}}(324, [\chi]) \cong$$ $$S_{3}^{\mathrm{new}}(36, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(108, [\chi])$$$$^{\oplus 2}$$