# Properties

 Label 256.8.a Level $256$ Weight $8$ Character orbit 256.a Rep. character $\chi_{256}(1,\cdot)$ Character field $\Q$ Dimension $54$ Newform subspaces $18$ Sturm bound $256$ Trace bound $9$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$256 = 2^{8}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 256.a (trivial) Character field: $$\Q$$ Newform subspaces: $$18$$ Sturm bound: $$256$$ Trace bound: $$9$$ Distinguishing $$T_p$$: $$3$$, $$5$$, $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{8}(\Gamma_0(256))$$.

Total New Old
Modular forms 236 58 178
Cusp forms 212 54 158
Eisenstein series 24 4 20

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

$$2$$Dim.
$$+$$$$28$$
$$-$$$$26$$

## Trace form

 $$54 q + 36454 q^{9} + O(q^{10})$$ $$54 q + 36454 q^{9} - 4 q^{17} + 718754 q^{25} + 8744 q^{33} + 4 q^{41} + 6985446 q^{49} - 6213128 q^{57} - 2541280 q^{65} + 302356 q^{73} + 20203502 q^{81} + 9562100 q^{89} + 30397212 q^{97} + O(q^{100})$$

## Decomposition of $$S_{8}^{\mathrm{new}}(\Gamma_0(256))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2
256.8.a.a $1$ $79.971$ $$\Q$$ $$\Q(\sqrt{-2})$$ $$0$$ $$-86$$ $$0$$ $$0$$ $+$ $$q-86q^{3}+5209q^{9}+8814q^{11}-22182q^{17}+\cdots$$
256.8.a.b $1$ $79.971$ $$\Q$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$-556$$ $$0$$ $-$ $$q-556q^{5}-3^{7}q^{9}+13108q^{13}+40094q^{17}+\cdots$$
256.8.a.c $1$ $79.971$ $$\Q$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$556$$ $$0$$ $+$ $$q+556q^{5}-3^{7}q^{9}-13108q^{13}+40094q^{17}+\cdots$$
256.8.a.d $1$ $79.971$ $$\Q$$ $$\Q(\sqrt{-2})$$ $$0$$ $$86$$ $$0$$ $$0$$ $-$ $$q+86q^{3}+5209q^{9}-8814q^{11}-22182q^{17}+\cdots$$
256.8.a.e $2$ $79.971$ $$\Q(\sqrt{435})$$ None $$0$$ $$-60$$ $$0$$ $$0$$ $-$ $$q-30q^{3}+\beta q^{5}-2\beta q^{7}-1287q^{9}+\cdots$$
256.8.a.f $2$ $79.971$ $$\Q(\sqrt{46})$$ None $$0$$ $$0$$ $$-360$$ $$0$$ $+$ $$q+\beta q^{3}-180q^{5}+4\beta q^{7}+757q^{9}+\cdots$$
256.8.a.g $2$ $79.971$ $$\Q(\sqrt{2})$$ $$\Q(\sqrt{-2})$$ $$0$$ $$0$$ $$0$$ $$0$$ $-$ $$q+13\beta q^{3}-835q^{9}-181\beta q^{11}+22182q^{17}+\cdots$$
256.8.a.h $2$ $79.971$ $$\Q(\sqrt{46})$$ None $$0$$ $$0$$ $$360$$ $$0$$ $-$ $$q+\beta q^{3}+180q^{5}-4\beta q^{7}+757q^{9}+\cdots$$
256.8.a.i $2$ $79.971$ $$\Q(\sqrt{435})$$ None $$0$$ $$60$$ $$0$$ $$0$$ $+$ $$q+30q^{3}+\beta q^{5}+2\beta q^{7}-1287q^{9}+\cdots$$
256.8.a.j $4$ $79.971$ $$\Q(\sqrt{3}, \sqrt{601})$$ None $$0$$ $$-80$$ $$0$$ $$0$$ $-$ $$q+(-20-\beta _{1})q^{3}+(\beta _{2}+\beta _{3})q^{5}+(7\beta _{2}+\cdots)q^{7}+\cdots$$
256.8.a.k $4$ $79.971$ 4.4.3484860.2 None $$0$$ $$0$$ $$-304$$ $$0$$ $-$ $$q-\beta _{2}q^{3}+(-76+\beta _{1})q^{5}+(-\beta _{2}+\beta _{3})q^{7}+\cdots$$
256.8.a.l $4$ $79.971$ $$\Q(\sqrt{2}, \sqrt{5})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $+$ $$q-3\beta _{1}q^{3}-17\beta _{3}q^{5}+29\beta _{2}q^{7}-1827q^{9}+\cdots$$
256.8.a.m $4$ $79.971$ $$\Q(\sqrt{2}, \sqrt{85})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $-$ $$q+21\beta _{1}q^{3}-\beta _{2}q^{5}+\beta _{3}q^{7}+1341q^{9}+\cdots$$
256.8.a.n $4$ $79.971$ $$\Q(\sqrt{2}, \sqrt{5})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $+$ $$q+\beta _{2}q^{3}-\beta _{3}q^{5}-71\beta _{1}q^{7}+4573q^{9}+\cdots$$
256.8.a.o $4$ $79.971$ 4.4.3484860.2 None $$0$$ $$0$$ $$304$$ $$0$$ $+$ $$q-\beta _{2}q^{3}+(76-\beta _{1})q^{5}+(\beta _{2}-\beta _{3})q^{7}+\cdots$$
256.8.a.p $4$ $79.971$ $$\Q(\sqrt{3}, \sqrt{601})$$ None $$0$$ $$80$$ $$0$$ $$0$$ $+$ $$q+(20+\beta _{1})q^{3}+(-\beta _{2}-\beta _{3})q^{5}+(7\beta _{2}+\cdots)q^{7}+\cdots$$
256.8.a.q $6$ $79.971$ $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$-688$$ $-$ $$q-\beta _{1}q^{3}+(\beta _{1}+\beta _{2})q^{5}+(-115+\beta _{4}+\cdots)q^{7}+\cdots$$
256.8.a.r $6$ $79.971$ $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$688$$ $+$ $$q-\beta _{1}q^{3}+(-\beta _{1}-\beta _{2})q^{5}+(115-\beta _{4}+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{8}^{\mathrm{old}}(\Gamma_0(256))$$ into lower level spaces

$$S_{8}^{\mathrm{old}}(\Gamma_0(256)) \cong$$ $$S_{8}^{\mathrm{new}}(\Gamma_0(2))$$$$^{\oplus 8}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_0(8))$$$$^{\oplus 6}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_0(16))$$$$^{\oplus 5}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_0(32))$$$$^{\oplus 4}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_0(64))$$$$^{\oplus 3}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_0(128))$$$$^{\oplus 2}$$