Space of modular forms of level 32, weight 4, and trivial character

• Label: 32.4.a
• Level: $32$
• Weight: $4$
• Character orbit: 32.a
• Rep. character: $\chi_{32}(1,\cdot)$
• Character field: $\Q$
• Dimension: $3$
• Newform subspaces: $3$
• Sturm bound: $16$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 32 = 2^{5} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	32.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(16\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(32))\).

	Total	New	Old
Modular forms	16	3	13
Cusp forms	8	3	5
Eisenstein series	8	0	8

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
\(+\)		\(9\)	\(2\)	\(7\)		\(5\)	\(2\)	\(3\)		\(4\)	\(0\)	\(4\)
\(-\)		\(7\)	\(1\)	\(6\)		\(3\)	\(1\)	\(2\)		\(4\)	\(0\)	\(4\)

Trace form

3 * q + 2 * q^5 + 47 * q^9 - 118 * q^13 - 154 * q^17 + 256 * q^21 + 309 * q^25 - 198 * q^29 - 640 * q^33 + 834 * q^37 + 590 * q^41 - 1334 * q^45 - 517 * q^49 - 302 * q^53 + 640 * q^57 + 890 * q^61 + 604 * q^65 + 768 * q^69 - 162 * q^73 - 1280 * q^77 + 11 * q^81 - 1468 * q^85 - 2322 * q^89 + 5120 * q^93 + 374 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(32))\) 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
32.4.a.a	$32$	$4$	32.a	1.a	$1$	$1$	$1$	$1.888$	\(\Q\)	None	✓	✓	✓	✓	32.4.a.a	$1$	$1$	\(0\)	\(-8\)	\(-10\)	\(-16\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{3}-10q^{5}-2^{4}q^{7}+37q^{9}+40q^{11}+\cdots\)
32.4.a.b	$32$	$4$	32.a	1.a	$1$	$1$	$1$	$1.888$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓	✓			32.4.a.b	$2$	$0$	\(0\)	\(0\)	\(22\)	\(0\)		$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+22q^{5}-3^{3}q^{9}-18q^{13}-94q^{17}+\cdots\)
32.4.a.c	$32$	$4$	32.a	1.a	$1$	$1$	$1$	$1.888$	\(\Q\)	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 \(S_{4}^{\mathrm{old}}(\Gamma_0(32))\) into lower level spaces

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