Space of modular forms of level 16, weight 32, and trivial character

• Label: 16.32.a
• Level: $16$
• Weight: $32$
• Character orbit: 16.a
• Rep. character: $\chi_{16}(1,\cdot)$
• Character field: $\Q$
• Dimension: $15$
• Newform subspaces: $6$
• Sturm bound: $64$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 32 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(64\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{32}(\Gamma_0(16))\).

	Total	New	Old
Modular forms	65	16	49
Cusp forms	59	15	44
Eisenstein series	6	1	5

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
\(+\)	\(8\)
\(-\)	\(7\)

Trace form

\( 15 q + 14348908 q^{3} + 7974687378 q^{5} - 2805416096456 q^{7} + 3076621645160683 q^{9} + O(q^{10}) \)

Decomposition of \(S_{32}^{\mathrm{new}}(\Gamma_0(16))\) 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
16.32.a.a	$16$	$32$	16.a	1.a	$1$	$1$	$1$	$97.403$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(19984212\)	\(42951708750\)	\(16\!\cdots\!76\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+19984212q^{3}+42951708750q^{5}+\cdots\)
16.32.a.b	$16$	$32$	16.a	1.a	$1$	$2$	$2$	$97.403$	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-17363160\)	\(-19391218020\)	\(-30\!\cdots\!00\)		$-$	$-$	$2^{7}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-8681580-3^{3}\beta )q^{3}+(-9695609010+\cdots)q^{5}+\cdots\)
16.32.a.c	$16$	$32$	16.a	1.a	$1$	$2$	$2$	$97.403$	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-16716504\)	\(-26763065700\)	\(23\!\cdots\!08\)		$-$	$-$	$2^{7}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(-8358252-\beta )q^{3}+(-13381532850+\cdots)q^{5}+\cdots\)
16.32.a.d	$16$	$32$	16.a	1.a	$1$	$2$	$2$	$97.403$	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(31205160\)	\(1872305820\)	\(-60\!\cdots\!80\)		$-$	$-$	$2^{7}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(15602580-\beta )q^{3}+(936152910+\cdots)q^{5}+\cdots\)
16.32.a.e	$16$	$32$	16.a	1.a	$1$	$4$	$4$	$97.403$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-11055408\)	\(-3872402632\)	\(11\!\cdots\!56\)		$+$	$+$	$2^{43}\cdot 3^{5}\cdot 5$	$\mathrm{SU}(2)$	\(q+(-2763852+\beta _{1})q^{3}+(-968100658+\cdots)q^{5}+\cdots\)
16.32.a.f	$16$	$32$	16.a	1.a	$1$	$4$	$4$	$97.403$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(8294608\)	\(13177359160\)	\(-18\!\cdots\!16\)		$+$	$+$	$2^{44}\cdot 3^{4}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(2073652+\beta _{1})q^{3}+(3294339790+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{32}^{\mathrm{old}}(\Gamma_0(16))\) into lower level spaces

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