# Properties

 Label 384.4.d Level $384$ Weight $4$ Character orbit 384.d Rep. character $\chi_{384}(193,\cdot)$ Character field $\Q$ Dimension $24$ Newform subspaces $6$ Sturm bound $256$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$256$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$5$$, $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(384, [\chi])$$.

Total New Old
Modular forms 208 24 184
Cusp forms 176 24 152
Eisenstein series 32 0 32

## Trace form

 $$24 q - 216 q^{9} + O(q^{10})$$ $$24 q - 216 q^{9} + 208 q^{17} - 424 q^{25} + 944 q^{41} - 264 q^{49} + 672 q^{57} + 832 q^{65} - 2640 q^{73} + 1944 q^{81} + 528 q^{89} - 3440 q^{97} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(384, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
384.4.d.a $2$ $22.657$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$-24$$ $$q-3iq^{3}+8iq^{5}-12q^{7}-9q^{9}-12iq^{11}+\cdots$$
384.4.d.b $2$ $22.657$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$24$$ $$q+3iq^{3}+8iq^{5}+12q^{7}-9q^{9}+12iq^{11}+\cdots$$
384.4.d.c $4$ $22.657$ $$\Q(i, \sqrt{13})$$ None $$0$$ $$0$$ $$0$$ $$-32$$ $$q-3\beta _{1}q^{3}+(-4\beta _{1}+\beta _{2})q^{5}+(-8+\cdots)q^{7}+\cdots$$
384.4.d.d $4$ $22.657$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3\zeta_{8}q^{3}+\zeta_{8}^{2}q^{5}-5\zeta_{8}^{3}q^{7}-9q^{9}+\cdots$$
384.4.d.e $4$ $22.657$ $$\Q(i, \sqrt{13})$$ None $$0$$ $$0$$ $$0$$ $$32$$ $$q+3\beta _{1}q^{3}+(-4\beta _{1}+\beta _{2})q^{5}+(8-\beta _{3})q^{7}+\cdots$$
384.4.d.f $8$ $22.657$ 8.0.1534132224.8 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}-\beta _{6}q^{5}-\beta _{2}q^{7}-9q^{9}+(-4\beta _{1}+\cdots)q^{11}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(384, [\chi])$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(384, [\chi]) \simeq$$ $$S_{4}^{\mathrm{new}}(8, [\chi])$$$$^{\oplus 10}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(16, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(24, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(32, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(48, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(64, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(96, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(128, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(192, [\chi])$$$$^{\oplus 2}$$