# Properties

 Label 768.4.c Level $768$ Weight $4$ Character orbit 768.c Rep. character $\chi_{768}(767,\cdot)$ Character field $\Q$ Dimension $92$ Newform subspaces $22$ Sturm bound $512$ Trace bound $25$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 768.c (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$12$$ Character field: $$\Q$$ Newform subspaces: $$22$$ Sturm bound: $$512$$ Trace bound: $$25$$ Distinguishing $$T_p$$: $$5$$, $$7$$, $$11$$, $$13$$, $$23$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(768, [\chi])$$.

Total New Old
Modular forms 408 100 308
Cusp forms 360 92 268
Eisenstein series 48 8 40

## Trace form

 $$92 q + 4 q^{9} + O(q^{10})$$ $$92 q + 4 q^{9} - 1892 q^{25} + 104 q^{33} - 1900 q^{49} + 112 q^{57} + 872 q^{73} - 4 q^{81} + 3160 q^{97} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(768, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
768.4.c.a $2$ $45.313$ $$\Q(\sqrt{-2})$$ $$\Q(\sqrt{-2})$$ $$0$$ $$-10$$ $$0$$ $$0$$ $$q+(-5-\beta )q^{3}+(23+10\beta )q^{9}-18q^{11}+\cdots$$
768.4.c.b $2$ $45.313$ $$\Q(\sqrt{-2})$$ None $$0$$ $$-6$$ $$0$$ $$0$$ $$q+(-3+3\beta )q^{3}+4\beta q^{5}-12\beta q^{7}+\cdots$$
768.4.c.c $2$ $45.313$ $$\Q(\sqrt{-2})$$ None $$0$$ $$-6$$ $$0$$ $$0$$ $$q+(-3-3\beta )q^{3}+4\beta q^{5}-12\beta q^{7}+\cdots$$
768.4.c.d $2$ $45.313$ $$\Q(\sqrt{-26})$$ None $$0$$ $$-2$$ $$0$$ $$0$$ $$q+(-1-\beta )q^{3}+2\beta q^{5}-2\beta q^{7}+(-5^{2}+\cdots)q^{9}+\cdots$$
768.4.c.e $2$ $45.313$ $$\Q(\sqrt{-26})$$ None $$0$$ $$-2$$ $$0$$ $$0$$ $$q+(-1+\beta )q^{3}+2\beta q^{5}-2\beta q^{7}+(-5^{2}+\cdots)q^{9}+\cdots$$
768.4.c.f $2$ $45.313$ $$\Q(\sqrt{-26})$$ None $$0$$ $$2$$ $$0$$ $$0$$ $$q+(1+\beta )q^{3}+2\beta q^{5}+2\beta q^{7}+(-5^{2}+\cdots)q^{9}+\cdots$$
768.4.c.g $2$ $45.313$ $$\Q(\sqrt{-26})$$ None $$0$$ $$2$$ $$0$$ $$0$$ $$q+(1-\beta )q^{3}+2\beta q^{5}+2\beta q^{7}+(-5^{2}+\cdots)q^{9}+\cdots$$
768.4.c.h $2$ $45.313$ $$\Q(\sqrt{-2})$$ None $$0$$ $$6$$ $$0$$ $$0$$ $$q+(3-3\beta )q^{3}+4\beta q^{5}+12\beta q^{7}+(-9+\cdots)q^{9}+\cdots$$
768.4.c.i $2$ $45.313$ $$\Q(\sqrt{-2})$$ None $$0$$ $$6$$ $$0$$ $$0$$ $$q+(3+3\beta )q^{3}+4\beta q^{5}+12\beta q^{7}+(-9+\cdots)q^{9}+\cdots$$
768.4.c.j $2$ $45.313$ $$\Q(\sqrt{-2})$$ $$\Q(\sqrt{-2})$$ $$0$$ $$10$$ $$0$$ $$0$$ $$q+(5+\beta )q^{3}+(23+10\beta )q^{9}+18q^{11}+\cdots$$
768.4.c.k $4$ $45.313$ $$\Q(\zeta_{12})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{12}^{2}q^{3}-5\zeta_{12}q^{7}-3^{3}q^{9}-3\zeta_{12}^{3}q^{13}+\cdots$$
768.4.c.l $4$ $45.313$ $$\Q(\zeta_{8})$$ $$\Q(\sqrt{-2})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+(2\zeta_{8}-3\zeta_{8}^{2})q^{3}+(-23+5\zeta_{8}^{3})q^{9}+\cdots$$
768.4.c.m $4$ $45.313$ $$\Q(\sqrt{-6}, \sqrt{-14})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}+\beta _{3})q^{3}+(-2\beta _{1}-\beta _{2})q^{5}+(3\beta _{1}+\cdots)q^{7}+\cdots$$
768.4.c.n $4$ $45.313$ $$\Q(\sqrt{-6}, \sqrt{-14})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{3}q^{3}+(-2\beta _{1}-\beta _{2})q^{5}+(3\beta _{1}-2\beta _{2}+\cdots)q^{7}+\cdots$$
768.4.c.o $4$ $45.313$ $$\Q(\sqrt{-6}, \sqrt{-14})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(-\beta _{1}-\beta _{3})q^{3}+(-2\beta _{1}-\beta _{2})q^{5}+\cdots$$
768.4.c.p $4$ $45.313$ $$\Q(\sqrt{-6}, \sqrt{-14})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{3}q^{3}+(-2\beta _{1}-\beta _{2})q^{5}+(-3\beta _{1}+\cdots)q^{7}+\cdots$$
768.4.c.q $4$ $45.313$ $$\Q(\sqrt{-2}, \sqrt{3})$$ $$\Q(\sqrt{-6})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{3}q^{3}-7\beta _{1}q^{5}+\beta _{2}q^{7}+3^{3}q^{9}+\cdots$$
768.4.c.r $4$ $45.313$ $$\Q(\zeta_{12})$$ $$\Q(\sqrt{-6})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{12}^{2}q^{3}-\zeta_{12}^{3}q^{5}+17\zeta_{12}q^{7}+\cdots$$
768.4.c.s $8$ $45.313$ 8.0.$$\cdots$$.4 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(-\beta _{1}-\beta _{3})q^{3}+\beta _{4}q^{5}-\beta _{6}q^{7}+\cdots$$
768.4.c.t $8$ $45.313$ 8.0.1731891456.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{4}q^{3}+(3\beta _{2}+\beta _{4}+\beta _{6})q^{5}+\beta _{7}q^{7}+\cdots$$
768.4.c.u $8$ $45.313$ 8.0.1731891456.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{4}q^{3}+(3\beta _{2}+\beta _{4}+\beta _{6})q^{5}-\beta _{7}q^{7}+\cdots$$
768.4.c.v $16$ $45.313$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{4}q^{3}+\beta _{9}q^{5}+\beta _{6}q^{7}+(6-3\beta _{7}+\cdots)q^{9}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(768, [\chi])$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(768, [\chi]) \simeq$$ $$S_{4}^{\mathrm{new}}(12, [\chi])$$$$^{\oplus 7}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(24, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(48, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(96, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(192, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(384, [\chi])$$$$^{\oplus 2}$$