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 $3$

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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 Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2
64.8.a.a 64.a 1.a $1$ $19.993$ \(\Q\) None \(0\) \(-84\) \(82\) \(456\) $-$ $\mathrm{SU}(2)$ \(q-84q^{3}+82q^{5}+456q^{7}+4869q^{9}+\cdots\)
64.8.a.b 64.a 1.a $1$ $19.993$ \(\Q\) None \(0\) \(-44\) \(-430\) \(-1224\) $+$ $\mathrm{SU}(2)$ \(q-44q^{3}-430q^{5}-1224q^{7}-251q^{9}+\cdots\)
64.8.a.c 64.a 1.a $1$ $19.993$ \(\Q\) None \(0\) \(-12\) \(210\) \(1016\) $+$ $\mathrm{SU}(2)$ \(q-12q^{3}+210q^{5}+1016q^{7}-2043q^{9}+\cdots\)
64.8.a.d 64.a 1.a $1$ $19.993$ \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(58\) \(0\) $-$ $N(\mathrm{U}(1))$ \(q+58q^{5}-3^{7}q^{9}+8898q^{13}-40094q^{17}+\cdots\)
64.8.a.e 64.a 1.a $1$ $19.993$ \(\Q\) None \(0\) \(12\) \(210\) \(-1016\) $-$ $\mathrm{SU}(2)$ \(q+12q^{3}+210q^{5}-1016q^{7}-2043q^{9}+\cdots\)
64.8.a.f 64.a 1.a $1$ $19.993$ \(\Q\) None \(0\) \(44\) \(-430\) \(1224\) $-$ $\mathrm{SU}(2)$ \(q+44q^{3}-430q^{5}+1224q^{7}-251q^{9}+\cdots\)
64.8.a.g 64.a 1.a $1$ $19.993$ \(\Q\) None \(0\) \(84\) \(82\) \(-456\) $+$ $\mathrm{SU}(2)$ \(q+84q^{3}+82q^{5}-456q^{7}+4869q^{9}+\cdots\)
64.8.a.h 64.a 1.a $2$ $19.993$ \(\Q(\sqrt{10}) \) None \(0\) \(-16\) \(180\) \(-1248\) $+$ $\mathrm{SU}(2)$ \(q+(-8+\beta )q^{3}+(90-8\beta )q^{5}+(-624+\cdots)q^{7}+\cdots\)
64.8.a.i 64.a 1.a $2$ $19.993$ \(\Q(\sqrt{15}) \) None \(0\) \(0\) \(-140\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-70q^{5}-18\beta q^{7}+1653q^{9}+\cdots\)
64.8.a.j 64.a 1.a $2$ $19.993$ \(\Q(\sqrt{10}) \) None \(0\) \(16\) \(180\) \(1248\) $+$ $\mathrm{SU}(2)$ \(q+(8+\beta )q^{3}+(90+8\beta )q^{5}+(624+14\beta )q^{7}+\cdots\)

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

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