Space of modular forms of level 32, weight 10, and trivial character

• Label: 32.10.a
• Level: $32$
• Weight: $10$
• Character orbit: 32.a
• Rep. character: $\chi_{32}(1,\cdot)$
• Character field: $\Q$
• Dimension: $9$
• Newform subspaces: $5$
• Sturm bound: $40$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 32 = 2^{5} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	32.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(40\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(32))\).

	Total	New	Old
Modular forms	40	9	31
Cusp forms	32	9	23
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.

\(2\)	Dim
\(+\)	\(4\)
\(-\)	\(5\)

Trace form

\( 9 q + 718 q^{5} + 86789 q^{9} + O(q^{10}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(32))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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
32.10.a.a	$32$	$10$	32.a	1.a	$1$	$1$	$1$	$16.481$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(2398\)	\(0\)		$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+2398q^{5}-3^{9}q^{9}+112806q^{13}+\cdots\)
32.10.a.b	$32$	$10$	32.a	1.a	$1$	$2$	$2$	$16.481$	\(\Q(\sqrt{106}) \)	None	✓	$1$	$1$	\(0\)	\(-176\)	\(1404\)	\(-2784\)		$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(-88+\beta )q^{3}+(702-8\beta )q^{5}+(-1392+\cdots)q^{7}+\cdots\)
32.10.a.c	$32$	$10$	32.a	1.a	$1$	$2$	$2$	$16.481$	\(\Q(\sqrt{5}) \)	None	✓	$2$	$0$	\(0\)	\(0\)	\(-4420\)	\(0\)		$-$	$-$	$2^{6}\cdot 3$	$\mathrm{SU}(2)$	\(q-\beta q^{3}-2210q^{5}-42\beta q^{7}+26397q^{9}+\cdots\)
32.10.a.d	$32$	$10$	32.a	1.a	$1$	$2$	$2$	$16.481$	\(\Q(\sqrt{7}) \)	None	✓	$2$	$1$	\(0\)	\(0\)	\(-68\)	\(0\)		$+$	$+$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-34q^{5}-86\beta q^{7}-3555q^{9}+\cdots\)
32.10.a.e	$32$	$10$	32.a	1.a	$1$	$2$	$2$	$16.481$	\(\Q(\sqrt{106}) \)	None	✓	$1$	$0$	\(0\)	\(176\)	\(1404\)	\(2784\)		$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(88+\beta )q^{3}+(702+8\beta )q^{5}+(1392+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(32))\) into lower level spaces

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