# Properties

 Label 384.8.d Level $384$ Weight $8$ Character orbit 384.d Rep. character $\chi_{384}(193,\cdot)$ Character field $\Q$ Dimension $56$ Newform subspaces $6$ Sturm bound $512$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 384.d (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$8$$ Character field: $$\Q$$ Newform subspaces: $$6$$ Sturm bound: $$512$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$5$$, $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{8}(384, [\chi])$$.

Total New Old
Modular forms 464 56 408
Cusp forms 432 56 376
Eisenstein series 32 0 32

## Trace form

 $$56 q - 40824 q^{9} + O(q^{10})$$ $$56 q - 40824 q^{9} - 11632 q^{17} - 810504 q^{25} - 1765136 q^{41} + 4544152 q^{49} - 6204384 q^{57} + 12748352 q^{65} + 11043568 q^{73} + 29760696 q^{81} + 28686288 q^{89} - 60259504 q^{97} + O(q^{100})$$

## Decomposition of $$S_{8}^{\mathrm{new}}(384, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
384.8.d.a $6$ $119.956$ $$\mathbb{Q}[x]/(x^{6} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$-136$$ $$q+3^{3}\beta _{1}q^{3}+(-37\beta _{1}+\beta _{3})q^{5}+(-23+\cdots)q^{7}+\cdots$$
384.8.d.b $6$ $119.956$ $$\mathbb{Q}[x]/(x^{6} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$136$$ $$q-3^{3}\beta _{1}q^{3}+(-37\beta _{1}+\beta _{3})q^{5}+(23+\cdots)q^{7}+\cdots$$
384.8.d.c $8$ $119.956$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$-2880$$ $$q-3^{3}\beta _{1}q^{3}+(28\beta _{1}+\beta _{5})q^{5}+(-360+\cdots)q^{7}+\cdots$$
384.8.d.d $8$ $119.956$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$2880$$ $$q+3^{3}\beta _{1}q^{3}+(28\beta _{1}+\beta _{5})q^{5}+(360+\cdots)q^{7}+\cdots$$
384.8.d.e $12$ $119.956$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3\beta _{1}q^{3}-\beta _{8}q^{5}+(\beta _{6}-\beta _{7})q^{7}-3^{6}q^{9}+\cdots$$
384.8.d.f $16$ $119.956$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{8}q^{3}-\beta _{1}q^{5}+\beta _{10}q^{7}-3^{6}q^{9}+\cdots$$

## Decomposition of $$S_{8}^{\mathrm{old}}(384, [\chi])$$ into lower level spaces

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