Space of modular forms of level 16, weight 40, and trivial character

• Label: 16.40.a
• Level: $16$
• Weight: $40$
• Character orbit: 16.a
• Rep. character: $\chi_{16}(1,\cdot)$
• Character field: $\Q$
• Dimension: $19$
• Newform subspaces: $6$
• Sturm bound: $80$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 40 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(80\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{40}(\Gamma_0(16))\).

	Total	New	Old
Modular forms	81	20	61
Cusp forms	75	19	56
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
\(+\)	\(10\)
\(-\)	\(9\)

Trace form

\( 19 q + 1162261468 q^{3} - 15349343544342 q^{5} - 26203001189979624 q^{7} + 22100283135419643871 q^{9} + O(q^{10}) \)

Decomposition of \(S_{40}^{\mathrm{new}}(\Gamma_0(16))\) 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
16.40.a.a	$16$	$40$	16.a	1.a	$1$	$1$	$1$	$154.143$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(735458292\)	\(-16\!\cdots\!50\)	\(-16\!\cdots\!64\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+735458292q^{3}-16226178983250q^{5}+\cdots\)
16.40.a.b	$16$	$40$	16.a	1.a	$1$	$2$	$2$	$154.143$	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-287418264\)	\(53\!\cdots\!20\)	\(-74\!\cdots\!12\)		$-$	$-$	$2^{7}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(-143709132-\beta )q^{3}+(26811369083310+\cdots)q^{5}+\cdots\)
16.40.a.c	$16$	$40$	16.a	1.a	$1$	$3$	$3$	$154.143$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-1109442852\)	\(17\!\cdots\!90\)	\(17\!\cdots\!44\)		$-$	$-$	$2^{22}\cdot 3^{5}\cdot 5\cdot 13$	$\mathrm{SU}(2)$	\(q+(-369814284-\beta _{2})q^{3}+(5808972166830+\cdots)q^{5}+\cdots\)
16.40.a.d	$16$	$40$	16.a	1.a	$1$	$3$	$3$	$154.143$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(1269987036\)	\(-81\!\cdots\!90\)	\(-26\!\cdots\!24\)		$-$	$-$	$2^{22}\cdot 3^{3}\cdot 5\cdot 7\cdot 13$	$\mathrm{SU}(2)$	\(q+(423329012-\beta _{1})q^{3}+(-27008612411730+\cdots)q^{5}+\cdots\)
16.40.a.e	$16$	$40$	16.a	1.a	$1$	$5$	$5$	$154.143$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-2193821276\)	\(-10\!\cdots\!30\)	\(27\!\cdots\!08\)		$+$	$+$	$2^{72}\cdot 3^{7}\cdot 5^{3}\cdot 13^{2}$	$\mathrm{SU}(2)$	\(q+(-438764255-\beta _{1})q^{3}+(-2089929307363+\cdots)q^{5}+\cdots\)
16.40.a.f	$16$	$40$	16.a	1.a	$1$	$5$	$5$	$154.143$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(2747498532\)	\(21\!\cdots\!18\)	\(29\!\cdots\!24\)		$+$	$+$	$2^{73}\cdot 3^{9}\cdot 5^{2}\cdot 7\cdot 13\cdot 19$	$\mathrm{SU}(2)$	\(q+(549499706+\beta _{1})q^{3}+(4260532908373+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{40}^{\mathrm{old}}(\Gamma_0(16))\) into lower level spaces

\( S_{40}^{\mathrm{old}}(\Gamma_0(16)) \cong \) \(S_{40}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 5}\)\(\oplus\)\(S_{40}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 4}\)\(\oplus\)\(S_{40}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 3}\)\(\oplus\)\(S_{40}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 2}\)