Space of modular forms of level 64, weight 10, and trivial character

• Label: 64.10.a
• Level: $64$
• Weight: $10$
• Character orbit: 64.a
• Rep. character: $\chi_{64}(1,\cdot)$
• Character field: $\Q$
• Dimension: $17$
• Newform subspaces: $13$
• Sturm bound: $80$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 64 = 2^{6} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	64.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(80\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(64))\).

	Total	New	Old
Modular forms	78	19	59
Cusp forms	66	17	49
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
\(+\)		\(38\)	\(9\)	\(29\)		\(32\)	\(8\)	\(24\)		\(6\)	\(1\)	\(5\)
\(-\)		\(40\)	\(10\)	\(30\)		\(34\)	\(9\)	\(25\)		\(6\)	\(1\)	\(5\)

Trace form

17 * q + 2 * q^5 + 98413 * q^9 + 194618 * q^13 - 203998 * q^17 - 1998592 * q^21 + 6799887 * q^25 - 633398 * q^29 - 77184 * q^33 + 18215938 * q^37 - 11923478 * q^41 - 18175686 * q^45 + 63412809 * q^49 + 236380786 * q^53 + 109664640 * q^57 + 299486666 * q^61 - 179110092 * q^65 - 989715712 * q^69 + 120854874 * q^73 - 564067584 * q^77 + 670025305 * q^81 + 1741382148 * q^85 - 1122883862 * q^89 - 3447808 * q^93 + 359465138 * q^97

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(64))\) 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
64.10.a.a	$64$	$10$	64.a	1.a	$1$	$1$	$1$	$32.962$	\(\Q\)	None	✓				4.10.a.a	$1$	$1$	\(0\)	\(-228\)	\(666\)	\(-6328\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-228q^{3}+666q^{5}-6328q^{7}+32301q^{9}+\cdots\)
64.10.a.b	$64$	$10$	64.a	1.a	$1$	$1$	$1$	$32.962$	\(\Q\)	None	✓				2.10.a.a	$1$	$0$	\(0\)	\(-156\)	\(-870\)	\(952\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-156q^{3}-870q^{5}+952q^{7}+4653q^{9}+\cdots\)
64.10.a.c	$64$	$10$	64.a	1.a	$1$	$1$	$1$	$32.962$	\(\Q\)	None	✓				8.10.a.b	$1$	$1$	\(0\)	\(-68\)	\(-1510\)	\(10248\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-68q^{3}-1510q^{5}+10248q^{7}-15059q^{9}+\cdots\)
64.10.a.d	$64$	$10$	64.a	1.a	$1$	$1$	$1$	$32.962$	\(\Q\)	None	✓				8.10.a.a	$1$	$0$	\(0\)	\(-60\)	\(2074\)	\(4344\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-60q^{3}+2074q^{5}+4344q^{7}-16083q^{9}+\cdots\)
64.10.a.e	$64$	$10$	64.a	1.a	$1$	$1$	$1$	$32.962$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				32.10.a.a	$2$	$0$	\(0\)	\(0\)	\(-2398\)	\(0\)		$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-2398q^{5}-3^{9}q^{9}-112806q^{13}+\cdots\)
64.10.a.f	$64$	$10$	64.a	1.a	$1$	$1$	$1$	$32.962$	\(\Q\)	None	✓				8.10.a.a	$1$	$1$	\(0\)	\(60\)	\(2074\)	\(-4344\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+60q^{3}+2074q^{5}-4344q^{7}-16083q^{9}+\cdots\)
64.10.a.g	$64$	$10$	64.a	1.a	$1$	$1$	$1$	$32.962$	\(\Q\)	None	✓				8.10.a.b	$1$	$0$	\(0\)	\(68\)	\(-1510\)	\(-10248\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+68q^{3}-1510q^{5}-10248q^{7}-15059q^{9}+\cdots\)
64.10.a.h	$64$	$10$	64.a	1.a	$1$	$1$	$1$	$32.962$	\(\Q\)	None	✓				2.10.a.a	$1$	$1$	\(0\)	\(156\)	\(-870\)	\(-952\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+156q^{3}-870q^{5}-952q^{7}+4653q^{9}+\cdots\)
64.10.a.i	$64$	$10$	64.a	1.a	$1$	$1$	$1$	$32.962$	\(\Q\)	None	✓				4.10.a.a	$1$	$0$	\(0\)	\(228\)	\(666\)	\(6328\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+228q^{3}+666q^{5}+6328q^{7}+32301q^{9}+\cdots\)
64.10.a.j	$64$	$10$	64.a	1.a	$1$	$2$	$2$	$32.962$	\(\Q(\sqrt{106}) \)	None	✓				32.10.a.b	$1$	$1$	\(0\)	\(-176\)	\(-1404\)	\(2784\)		$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(-88+\beta )q^{3}+(-702+8\beta )q^{5}+\cdots\)
64.10.a.k	$64$	$10$	64.a	1.a	$1$	$2$	$2$	$32.962$	\(\Q(\sqrt{7}) \)	None	✓				32.10.a.d	$2$	$0$	\(0\)	\(0\)	\(68\)	\(0\)		$-$	$-$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+34q^{5}+86\beta q^{7}-3555q^{9}+\cdots\)
64.10.a.l	$64$	$10$	64.a	1.a	$1$	$2$	$2$	$32.962$	\(\Q(\sqrt{5}) \)	None	✓				32.10.a.c	$2$	$0$	\(0\)	\(0\)	\(4420\)	\(0\)		$-$	$-$	$2^{6}\cdot 3$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+2210q^{5}+42\beta q^{7}+26397q^{9}+\cdots\)
64.10.a.m	$64$	$10$	64.a	1.a	$1$	$2$	$2$	$32.962$	\(\Q(\sqrt{106}) \)	None	✓				32.10.a.b	$1$	$1$	\(0\)	\(176\)	\(-1404\)	\(-2784\)		$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(88+\beta )q^{3}+(-702-8\beta )q^{5}+(-1392+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(64))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(64)) \simeq \) \(S_{10}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 5}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 2}\)