# Properties

 Label 384.9.b Level $384$ Weight $9$ Character orbit 384.b Rep. character $\chi_{384}(319,\cdot)$ Character field $\Q$ Dimension $64$ Newform subspaces $4$ Sturm bound $576$ Trace bound $17$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 384.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$8$$ Character field: $$\Q$$ Newform subspaces: $$4$$ Sturm bound: $$576$$ Trace bound: $$17$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{9}(384, [\chi])$$.

Total New Old
Modular forms 528 64 464
Cusp forms 496 64 432
Eisenstein series 32 0 32

## Trace form

 $$64 q + 139968 q^{9} + O(q^{10})$$ $$64 q + 139968 q^{9} + 309120 q^{17} - 6416576 q^{25} - 8749440 q^{41} - 76975296 q^{49} + 6200064 q^{57} + 29299200 q^{65} - 80055680 q^{73} + 306110016 q^{81} + 365339520 q^{89} - 121481856 q^{97} + O(q^{100})$$

## Decomposition of $$S_{9}^{\mathrm{new}}(384, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
384.9.b.a $8$ $156.433$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{3}q^{3}-\beta _{5}q^{5}-7\beta _{4}q^{7}+3^{7}q^{9}+\cdots$$
384.9.b.b $8$ $156.433$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{1}q^{3}+\beta _{2}q^{5}-\beta _{4}q^{7}+3^{7}q^{9}+\cdots$$
384.9.b.c $16$ $156.433$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}-\beta _{2}q^{5}+\beta _{5}q^{7}+3^{7}q^{9}+\cdots$$
384.9.b.d $32$ $156.433$ None $$0$$ $$0$$ $$0$$ $$0$$

## Decomposition of $$S_{9}^{\mathrm{old}}(384, [\chi])$$ into lower level spaces

$$S_{9}^{\mathrm{old}}(384, [\chi]) \cong$$ $$S_{9}^{\mathrm{new}}(8, [\chi])$$$$^{\oplus 10}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(24, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(32, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(64, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(96, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(128, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(192, [\chi])$$$$^{\oplus 2}$$