# Properties

 Label 768.2.j Level $768$ Weight $2$ Character orbit 768.j Rep. character $\chi_{768}(193,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $32$ Newform subspaces $6$ Sturm bound $256$ Trace bound $13$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 768.j (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$16$$ Character field: $$\Q(i)$$ Newform subspaces: $$6$$ Sturm bound: $$256$$ Trace bound: $$13$$ Distinguishing $$T_p$$: $$5$$, $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(768, [\chi])$$.

Total New Old
Modular forms 304 32 272
Cusp forms 208 32 176
Eisenstein series 96 0 96

## Trace form

 $$32 q + O(q^{10})$$ $$32 q - 32 q^{49} + 192 q^{65} - 32 q^{81} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(768, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
768.2.j.a $4$ $6.133$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$-8$$ $$0$$ $$q-\zeta_{8}q^{3}+(-2+2\zeta_{8}^{2})q^{5}+(3\zeta_{8}+3\zeta_{8}^{3})q^{7}+\cdots$$
768.2.j.b $4$ $6.133$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{8}^{3}q^{3}+(\zeta_{8}+\zeta_{8}^{3})q^{7}-\zeta_{8}^{2}q^{9}+\cdots$$
768.2.j.c $4$ $6.133$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{8}^{3}q^{3}+(-\zeta_{8}-\zeta_{8}^{3})q^{7}-\zeta_{8}^{2}q^{9}+\cdots$$
768.2.j.d $4$ $6.133$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$8$$ $$0$$ $$q+\zeta_{8}q^{3}+(2-2\zeta_{8}^{2})q^{5}+(3\zeta_{8}+3\zeta_{8}^{3})q^{7}+\cdots$$
768.2.j.e $8$ $6.133$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$-8$$ $$0$$ $$q-\zeta_{24}q^{3}+(-1+\zeta_{24}^{2}-\zeta_{24}^{6})q^{5}+\cdots$$
768.2.j.f $8$ $6.133$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$8$$ $$0$$ $$q-\zeta_{24}q^{3}+(1-\zeta_{24}^{2}+\zeta_{24}^{6})q^{5}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(768, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(768, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(16, [\chi])$$$$^{\oplus 10}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(48, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(64, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(128, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(192, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(256, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(384, [\chi])$$$$^{\oplus 2}$$