Space of modular forms of level 64, weight 8, and trivial character

• 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: $3$

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.

\(2\)		Total				Cusp				Eisenstein
		All	New	Old		All	New	Old		All	New	Old
\(+\)		\(32\)	\(8\)	\(24\)		\(26\)	\(7\)	\(19\)		\(6\)	\(1\)	\(5\)
\(-\)		\(30\)	\(7\)	\(23\)		\(24\)	\(6\)	\(18\)		\(6\)	\(1\)	\(5\)

Trace form

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

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(64))\) 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
64.8.a.a	$64$	$8$	64.a	1.a	$1$	$1$	$1$	$19.993$	\(\Q\)	None	✓				8.8.a.a	$1$	$1$	\(0\)	\(-84\)	\(82\)	\(456\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-84q^{3}+82q^{5}+456q^{7}+4869q^{9}+\cdots\)
64.8.a.b	$64$	$8$	64.a	1.a	$1$	$1$	$1$	$19.993$	\(\Q\)	None	✓				8.8.a.b	$1$	$0$	\(0\)	\(-44\)	\(-430\)	\(-1224\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-44q^{3}-430q^{5}-1224q^{7}-251q^{9}+\cdots\)
64.8.a.c	$64$	$8$	64.a	1.a	$1$	$1$	$1$	$19.993$	\(\Q\)	None	✓				2.8.a.a	$1$	$0$	\(0\)	\(-12\)	\(210\)	\(1016\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-12q^{3}+210q^{5}+1016q^{7}-2043q^{9}+\cdots\)
64.8.a.d	$64$	$8$	64.a	1.a	$1$	$1$	$1$	$19.993$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				32.8.a.a	$2$	$1$	\(0\)	\(0\)	\(58\)	\(0\)		$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+58q^{5}-3^{7}q^{9}+8898q^{13}-40094q^{17}+\cdots\)
64.8.a.e	$64$	$8$	64.a	1.a	$1$	$1$	$1$	$19.993$	\(\Q\)	None	✓				2.8.a.a	$1$	$1$	\(0\)	\(12\)	\(210\)	\(-1016\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+12q^{3}+210q^{5}-1016q^{7}-2043q^{9}+\cdots\)
64.8.a.f	$64$	$8$	64.a	1.a	$1$	$1$	$1$	$19.993$	\(\Q\)	None	✓				8.8.a.b	$1$	$1$	\(0\)	\(44\)	\(-430\)	\(1224\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+44q^{3}-430q^{5}+1224q^{7}-251q^{9}+\cdots\)
64.8.a.g	$64$	$8$	64.a	1.a	$1$	$1$	$1$	$19.993$	\(\Q\)	None	✓				8.8.a.a	$1$	$0$	\(0\)	\(84\)	\(82\)	\(-456\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+84q^{3}+82q^{5}-456q^{7}+4869q^{9}+\cdots\)
64.8.a.h	$64$	$8$	64.a	1.a	$1$	$2$	$2$	$19.993$	\(\Q(\sqrt{10}) \)	None	✓				32.8.a.b	$1$	$0$	\(0\)	\(-16\)	\(180\)	\(-1248\)		$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(-8+\beta )q^{3}+(90-8\beta )q^{5}+(-624+\cdots)q^{7}+\cdots\)
64.8.a.i	$64$	$8$	64.a	1.a	$1$	$2$	$2$	$19.993$	\(\Q(\sqrt{15}) \)	None	✓		✓		32.8.a.c	$2$	$1$	\(0\)	\(0\)	\(-140\)	\(0\)		$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-70q^{5}-18\beta q^{7}+1653q^{9}+\cdots\)
64.8.a.j	$64$	$8$	64.a	1.a	$1$	$2$	$2$	$19.993$	\(\Q(\sqrt{10}) \)	None	✓				32.8.a.b	$1$	$0$	\(0\)	\(16\)	\(180\)	\(1248\)		$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(8+\beta )q^{3}+(90+8\beta )q^{5}+(624+14\beta )q^{7}+\cdots\)

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

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