# Properties

 Label 64.8.a Level $64$ Weight $8$ Character orbit 64.a Rep. character $\chi_{64}(1,\cdot)$ Character field $\Q$ Dimension $13$ Newform subspaces $10$ Sturm bound $64$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$64 = 2^{6}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 64.a (trivial) Character field: $$\Q$$ Newform subspaces: $$10$$ Sturm bound: $$64$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{8}(\Gamma_0(64))$$.

Total New Old
Modular forms 62 15 47
Cusp forms 50 13 37
Eisenstein series 12 2 10

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
$$+$$$$7$$
$$-$$$$6$$

## Trace form

 $$13 q + 2 q^{5} + 8017 q^{9} + O(q^{10})$$ $$13 q + 2 q^{5} + 8017 q^{9} - 7062 q^{13} + 2906 q^{17} + 31808 q^{21} + 156747 q^{25} - 51686 q^{29} - 103392 q^{33} + 1049602 q^{37} + 159922 q^{41} + 1266090 q^{45} + 823541 q^{49} - 4773806 q^{53} - 690720 q^{57} - 1972454 q^{61} - 489564 q^{65} + 2920640 q^{69} - 1342654 q^{73} + 3377856 q^{77} - 10190651 q^{81} - 2925692 q^{85} + 5223922 q^{89} - 947968 q^{93} - 9203702 q^{97} + O(q^{100})$$

## Decomposition of $$S_{8}^{\mathrm{new}}(\Gamma_0(64))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2
64.8.a.a $1$ $19.993$ $$\Q$$ None $$0$$ $$-84$$ $$82$$ $$456$$ $-$ $$q-84q^{3}+82q^{5}+456q^{7}+4869q^{9}+\cdots$$
64.8.a.b $1$ $19.993$ $$\Q$$ None $$0$$ $$-44$$ $$-430$$ $$-1224$$ $+$ $$q-44q^{3}-430q^{5}-1224q^{7}-251q^{9}+\cdots$$
64.8.a.c $1$ $19.993$ $$\Q$$ None $$0$$ $$-12$$ $$210$$ $$1016$$ $+$ $$q-12q^{3}+210q^{5}+1016q^{7}-2043q^{9}+\cdots$$
64.8.a.d $1$ $19.993$ $$\Q$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$58$$ $$0$$ $-$ $$q+58q^{5}-3^{7}q^{9}+8898q^{13}-40094q^{17}+\cdots$$
64.8.a.e $1$ $19.993$ $$\Q$$ None $$0$$ $$12$$ $$210$$ $$-1016$$ $-$ $$q+12q^{3}+210q^{5}-1016q^{7}-2043q^{9}+\cdots$$
64.8.a.f $1$ $19.993$ $$\Q$$ None $$0$$ $$44$$ $$-430$$ $$1224$$ $-$ $$q+44q^{3}-430q^{5}+1224q^{7}-251q^{9}+\cdots$$
64.8.a.g $1$ $19.993$ $$\Q$$ None $$0$$ $$84$$ $$82$$ $$-456$$ $+$ $$q+84q^{3}+82q^{5}-456q^{7}+4869q^{9}+\cdots$$
64.8.a.h $2$ $19.993$ $$\Q(\sqrt{10})$$ None $$0$$ $$-16$$ $$180$$ $$-1248$$ $+$ $$q+(-8+\beta )q^{3}+(90-8\beta )q^{5}+(-624+\cdots)q^{7}+\cdots$$
64.8.a.i $2$ $19.993$ $$\Q(\sqrt{15})$$ None $$0$$ $$0$$ $$-140$$ $$0$$ $-$ $$q+\beta q^{3}-70q^{5}-18\beta q^{7}+1653q^{9}+\cdots$$
64.8.a.j $2$ $19.993$ $$\Q(\sqrt{10})$$ None $$0$$ $$16$$ $$180$$ $$1248$$ $+$ $$q+(8+\beta )q^{3}+(90+8\beta )q^{5}+(624+14\beta )q^{7}+\cdots$$

## Decomposition of $$S_{8}^{\mathrm{old}}(\Gamma_0(64))$$ into lower level spaces

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