Space of modular forms of level 64, weight 16, and trivial character

• Label: 64.16.a
• Level: $64$
• Weight: $16$
• Character orbit: 64.a
• Rep. character: $\chi_{64}(1,\cdot)$
• Character field: $\Q$
• Dimension: $29$
• Newform subspaces: $17$
• Sturm bound: $128$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 64 = 2^{6} \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	64.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 17 \)
Sturm bound:			\(128\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{16}(\Gamma_0(64))\).

	Total	New	Old
Modular forms	126	31	95
Cusp forms	114	29	85
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
\(+\)		\(64\)	\(16\)	\(48\)		\(58\)	\(15\)	\(43\)		\(6\)	\(1\)	\(5\)
\(-\)		\(62\)	\(15\)	\(47\)		\(56\)	\(14\)	\(42\)		\(6\)	\(1\)	\(5\)

Trace form

29 * q + 2 * q^5 + 129140161 * q^9 - 249054550 * q^13 + 1457109626 * q^17 - 5416494784 * q^21 + 182317487707 * q^25 + 132406679002 * q^29 + 300944700576 * q^33 - 542612215806 * q^37 - 1706183644526 * q^41 - 5596728658710 * q^45 + 15599130675525 * q^49 + 26260504559314 * q^53 + 2870955174240 * q^57 - 14841673253158 * q^61 + 33637323215076 * q^65 - 310348970932288 * q^69 + 29276453840546 * q^73 - 105257654517312 * q^77 + 314167235594005 * q^81 + 621986239739268 * q^85 - 210132158414126 * q^89 + 2864542625815808 * q^93 - 66463991539542 * q^97

Decomposition of \(S_{16}^{\mathrm{new}}(\Gamma_0(64))\) 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
64.16.a.a	$64$	$16$	64.a	1.a	$1$	$1$	$1$	$91.324$	\(\Q\)	None	✓				2.16.a.a	$1$	$0$	\(0\)	\(-6252\)	\(-90510\)	\(56\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-6252q^{3}-90510q^{5}+56q^{7}+24738597q^{9}+\cdots\)
64.16.a.b	$64$	$16$	64.a	1.a	$1$	$1$	$1$	$91.324$	\(\Q\)	None	✓				8.16.a.a	$1$	$1$	\(0\)	\(-3444\)	\(-313358\)	\(2324616\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3444q^{3}-313358q^{5}+2324616q^{7}+\cdots\)
64.16.a.c	$64$	$16$	64.a	1.a	$1$	$1$	$1$	$91.324$	\(\Q\)	None	✓				1.16.a.a	$1$	$1$	\(0\)	\(-3348\)	\(-52110\)	\(-2822456\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3348q^{3}-52110q^{5}-2822456q^{7}+\cdots\)
64.16.a.d	$64$	$16$	64.a	1.a	$1$	$1$	$1$	$91.324$	\(\Q\)	None	✓				8.16.a.b	$1$	$0$	\(0\)	\(-2700\)	\(251890\)	\(1374072\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2700q^{3}+251890q^{5}+1374072q^{7}+\cdots\)
64.16.a.e	$64$	$16$	64.a	1.a	$1$	$1$	$1$	$91.324$	\(\Q\)	None	✓				4.16.a.a	$1$	$1$	\(0\)	\(-276\)	\(132210\)	\(3585736\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-276q^{3}+132210q^{5}+3585736q^{7}+\cdots\)
64.16.a.f	$64$	$16$	64.a	1.a	$1$	$1$	$1$	$91.324$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				32.16.a.a	$2$	$1$	\(0\)	\(0\)	\(-217382\)	\(0\)		$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-217382q^{5}-3^{15}q^{9}+372237858q^{13}+\cdots\)
64.16.a.g	$64$	$16$	64.a	1.a	$1$	$1$	$1$	$91.324$	\(\Q\)	None	✓				4.16.a.a	$1$	$0$	\(0\)	\(276\)	\(132210\)	\(-3585736\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+276q^{3}+132210q^{5}-3585736q^{7}+\cdots\)
64.16.a.h	$64$	$16$	64.a	1.a	$1$	$1$	$1$	$91.324$	\(\Q\)	None	✓				8.16.a.b	$1$	$1$	\(0\)	\(2700\)	\(251890\)	\(-1374072\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2700q^{3}+251890q^{5}-1374072q^{7}+\cdots\)
64.16.a.i	$64$	$16$	64.a	1.a	$1$	$1$	$1$	$91.324$	\(\Q\)	None	✓				1.16.a.a	$1$	$0$	\(0\)	\(3348\)	\(-52110\)	\(2822456\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3348q^{3}-52110q^{5}+2822456q^{7}+\cdots\)
64.16.a.j	$64$	$16$	64.a	1.a	$1$	$1$	$1$	$91.324$	\(\Q\)	None	✓				8.16.a.a	$1$	$0$	\(0\)	\(3444\)	\(-313358\)	\(-2324616\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3444q^{3}-313358q^{5}-2324616q^{7}+\cdots\)
64.16.a.k	$64$	$16$	64.a	1.a	$1$	$1$	$1$	$91.324$	\(\Q\)	None	✓				2.16.a.a	$1$	$1$	\(0\)	\(6252\)	\(-90510\)	\(-56\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+6252q^{3}-90510q^{5}-56q^{7}+24738597q^{9}+\cdots\)
64.16.a.l	$64$	$16$	64.a	1.a	$1$	$2$	$2$	$91.324$	\(\Q(\sqrt{58}) \)	None	✓				8.16.a.c	$1$	$1$	\(0\)	\(-4072\)	\(140260\)	\(-126192\)		$-$	$-$	$2^{7}\cdot 5$	$\mathrm{SU}(2)$	\(q+(-2036+\beta )q^{3}+(70130-6^{2}\beta )q^{5}+\cdots\)
64.16.a.m	$64$	$16$	64.a	1.a	$1$	$2$	$2$	$91.324$	\(\Q(\sqrt{2497}) \)	None	✓				32.16.a.b	$2$	$1$	\(0\)	\(0\)	\(262580\)	\(0\)		$-$	$-$	$2^{6}\cdot 3$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+131290q^{5}-14\beta q^{7}+8663445q^{9}+\cdots\)
64.16.a.n	$64$	$16$	64.a	1.a	$1$	$2$	$2$	$91.324$	\(\Q(\sqrt{58}) \)	None	✓				8.16.a.c	$1$	$0$	\(0\)	\(4072\)	\(140260\)	\(126192\)		$+$	$+$	$2^{7}\cdot 5$	$\mathrm{SU}(2)$	\(q+(2036+\beta )q^{3}+(70130+6^{2}\beta )q^{5}+\cdots\)
64.16.a.o	$64$	$16$	64.a	1.a	$1$	$4$	$4$	$91.324$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				32.16.a.c	$1$	$0$	\(0\)	\(-2912\)	\(98280\)	\(2755776\)		$+$	$+$	$2^{27}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(-728-\beta _{1})q^{3}+(24570-6\beta _{1}+\cdots)q^{5}+\cdots\)
64.16.a.p	$64$	$16$	64.a	1.a	$1$	$4$	$4$	$91.324$	\(\Q(\sqrt{1170 +120 \sqrt{78}})\)	None	✓		✓		32.16.a.d	$2$	$1$	\(0\)	\(0\)	\(-378520\)	\(0\)		$-$	$-$	$2^{27}\cdot 3^{5}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-94630+\beta _{2})q^{5}+(-313\beta _{1}+\cdots)q^{7}+\cdots\)
64.16.a.q	$64$	$16$	64.a	1.a	$1$	$4$	$4$	$91.324$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				32.16.a.c	$1$	$0$	\(0\)	\(2912\)	\(98280\)	\(-2755776\)		$+$	$+$	$2^{27}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(728+\beta _{1})q^{3}+(24570-6\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{16}^{\mathrm{old}}(\Gamma_0(64))\) into lower level spaces

\( S_{16}^{\mathrm{old}}(\Gamma_0(64)) \simeq \) \(S_{16}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 7}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 6}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 5}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 3}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 2}\)