Space of modular forms of level 16, weight 36, and trivial character

• Label: 16.36.a
• Level: $16$
• Weight: $36$
• Character orbit: 16.a
• Rep. character: $\chi_{16}(1,\cdot)$
• Character field: $\Q$
• Dimension: $17$
• Newform subspaces: $6$
• Sturm bound: $72$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 36 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(72\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{36}(\Gamma_0(16))\).

	Total	New	Old
Modular forms	73	18	55
Cusp forms	67	17	50
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
\(+\)	\(9\)
\(-\)	\(8\)

Trace form

\( 17 q + 129140164 q^{3} + 740368852318 q^{5} - 124747377106072 q^{7} + 239150871423581125 q^{9} + O(q^{10}) \)

Decomposition of \(S_{36}^{\mathrm{new}}(\Gamma_0(16))\) 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
16.36.a.a	$16$	$36$	16.a	1.a	$1$	$1$	$1$	$124.152$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-159933852\)	\(-28\!\cdots\!90\)	\(78\!\cdots\!44\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-159933852q^{3}-2838742578690q^{5}+\cdots\)
16.36.a.b	$16$	$36$	16.a	1.a	$1$	$1$	$1$	$124.152$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-36494748\)	\(389070858750\)	\(12\!\cdots\!56\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-36494748q^{3}+389070858750q^{5}+\cdots\)
16.36.a.c	$16$	$36$	16.a	1.a	$1$	$3$	$3$	$124.152$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-50908884\)	\(280720890\)	\(55\!\cdots\!16\)		$-$	$-$	$2^{24}\cdot 3^{4}\cdot 5\cdot 7$	$\mathrm{SU}(2)$	\(q+(-16969628-\beta _{1})q^{3}+(93573630+\cdots)q^{5}+\cdots\)
16.36.a.d	$16$	$36$	16.a	1.a	$1$	$3$	$3$	$124.152$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(104875308\)	\(892652054010\)	\(-87\!\cdots\!56\)		$-$	$-$	$2^{24}\cdot 3^{3}\cdot 5\cdot 7$	$\mathrm{SU}(2)$	\(q+(34958436+\beta _{1})q^{3}+(297550684670+\cdots)q^{5}+\cdots\)
16.36.a.e	$16$	$36$	16.a	1.a	$1$	$4$	$4$	$124.152$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(229498128\)	\(12\!\cdots\!40\)	\(40\!\cdots\!64\)		$+$	$+$	$2^{42}\cdot 3^{6}\cdot 5^{2}\cdot 7$	$\mathrm{SU}(2)$	\(q+(57374532-\beta _{1})q^{3}+(306638518910+\cdots)q^{5}+\cdots\)
16.36.a.f	$16$	$36$	16.a	1.a	$1$	$5$	$5$	$124.152$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(42104212\)	\(10\!\cdots\!18\)	\(-56\!\cdots\!96\)		$+$	$+$	$2^{73}\cdot 3^{6}\cdot 5^{2}\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q+(8420842+\beta _{1})q^{3}+(214110743932+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{36}^{\mathrm{old}}(\Gamma_0(16))\) into lower level spaces

\( S_{36}^{\mathrm{old}}(\Gamma_0(16)) \cong \) \(S_{36}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 5}\)\(\oplus\)\(S_{36}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 4}\)\(\oplus\)\(S_{36}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 3}\)\(\oplus\)\(S_{36}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 2}\)