Space of modular forms of level 64, weight 4, and trivial character

• Label: 64.4.a
• Level: $64$
• Weight: $4$
• Character orbit: 64.a
• Rep. character: $\chi_{64}(1,\cdot)$
• Character field: $\Q$
• Dimension: $5$
• Newform subspaces: $5$
• Sturm bound: $32$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 64 = 2^{6} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	64.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(32\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(64))\).

	Total	New	Old
Modular forms	30	7	23
Cusp forms	18	5	13
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
\(+\)	\(3\)
\(-\)	\(2\)

Trace form

5 * q + 2 * q^5 + 25 * q^9 + 74 * q^13 - 54 * q^17 - 64 * q^21 + 67 * q^25 - 198 * q^29 - 288 * q^33 - 510 * q^37 + 194 * q^41 + 1290 * q^45 - 51 * q^49 + 786 * q^53 + 288 * q^57 - 1990 * q^61 + 516 * q^65 - 1216 * q^69 + 146 * q^73 + 3392 * q^77 - 611 * q^81 + 1668 * q^85 - 894 * q^89 - 6400 * q^93 - 582 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(64))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces					A-L signs	Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2
64.4.a.a	$64$	$4$	64.a	1.a	$1$	$1$	$1$	$3.776$	\(\Q\)	None	✓				32.4.a.a	$1$	$0$	\(0\)	\(-8\)	\(10\)	\(16\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{3}+10q^{5}+2^{4}q^{7}+37q^{9}+40q^{11}+\cdots\)
64.4.a.b	$64$	$4$	64.a	1.a	$1$	$1$	$1$	$3.776$	\(\Q\)	None	✓				8.4.a.a	$1$	$1$	\(0\)	\(-4\)	\(2\)	\(-24\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{3}+2q^{5}-24q^{7}-11q^{9}-44q^{11}+\cdots\)
64.4.a.c	$64$	$4$	64.a	1.a	$1$	$1$	$1$	$3.776$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				32.4.a.b	$2$	$1$	\(0\)	\(0\)	\(-22\)	\(0\)		$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-22q^{5}-3^{3}q^{9}+18q^{13}-94q^{17}+\cdots\)
64.4.a.d	$64$	$4$	64.a	1.a	$1$	$1$	$1$	$3.776$	\(\Q\)	None	✓				8.4.a.a	$1$	$0$	\(0\)	\(4\)	\(2\)	\(24\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{3}+2q^{5}+24q^{7}-11q^{9}+44q^{11}+\cdots\)
64.4.a.e	$64$	$4$	64.a	1.a	$1$	$1$	$1$	$3.776$	\(\Q\)	None	✓				32.4.a.a	$1$	$0$	\(0\)	\(8\)	\(10\)	\(-16\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{3}+10q^{5}-2^{4}q^{7}+37q^{9}-40q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(64))\) into lower level spaces

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