# Properties

 Label 256.4.b Level $256$ Weight $4$ Character orbit 256.b Rep. character $\chi_{256}(129,\cdot)$ Character field $\Q$ Dimension $22$ Newform subspaces $9$ Sturm bound $128$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$256 = 2^{8}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 256.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$8$$ Character field: $$\Q$$ Newform subspaces: $$9$$ Sturm bound: $$128$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$3$$, $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(256, [\chi])$$.

Total New Old
Modular forms 108 26 82
Cusp forms 84 22 62
Eisenstein series 24 4 20

## Trace form

 $$22 q - 158 q^{9} + O(q^{10})$$ $$22 q - 158 q^{9} - 4 q^{17} - 346 q^{25} - 112 q^{33} + 4 q^{41} - 954 q^{49} + 784 q^{57} - 1480 q^{65} - 268 q^{73} + 374 q^{81} - 172 q^{89} + 3164 q^{97} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(256, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
256.4.b.a $2$ $15.104$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$-48$$ $$q+2iq^{3}-iq^{5}-24q^{7}+11q^{9}-22iq^{11}+\cdots$$
256.4.b.b $2$ $15.104$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$-40$$ $$q+iq^{3}-3iq^{5}-20q^{7}+23q^{9}-7iq^{11}+\cdots$$
256.4.b.c $2$ $15.104$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$-32$$ $$q+4iq^{3}+5iq^{5}-2^{4}q^{7}-37q^{9}+\cdots$$
256.4.b.d $2$ $15.104$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+11iq^{5}+3^{3}q^{9}+9iq^{13}-94q^{17}+\cdots$$
256.4.b.e $2$ $15.104$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$32$$ $$q+4iq^{3}-5iq^{5}+2^{4}q^{7}-37q^{9}+\cdots$$
256.4.b.f $2$ $15.104$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$40$$ $$q+iq^{3}+3iq^{5}+20q^{7}+23q^{9}-7iq^{11}+\cdots$$
256.4.b.g $2$ $15.104$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$48$$ $$q+2iq^{3}+iq^{5}+24q^{7}+11q^{9}-22iq^{11}+\cdots$$
256.4.b.h $4$ $15.104$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$-16$$ $$q+(\zeta_{12}-\zeta_{12}^{2})q^{3}+(\zeta_{12}+2\zeta_{12}^{2}+\cdots)q^{5}+\cdots$$
256.4.b.i $4$ $15.104$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$16$$ $$q+(\zeta_{12}-\zeta_{12}^{2})q^{3}+(-\zeta_{12}-2\zeta_{12}^{2}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(256, [\chi])$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(256, [\chi]) \cong$$ $$S_{4}^{\mathrm{new}}(64, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(8, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(32, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(128, [\chi])$$$$^{\oplus 2}$$