Space of modular forms of level 111, weight 4, and trivial character

• Label: 111.4.a
• Level: $111$
• Weight: $4$
• Character orbit: 111.a
• Rep. character: $\chi_{111}(1,\cdot)$
• Character field: $\Q$
• Dimension: $18$
• Newform subspaces: $6$
• Sturm bound: $50$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 111 = 3 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	111.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(50\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(111))\).

	Total	New	Old
Modular forms	40	18	22
Cusp forms	36	18	18
Eisenstein series	4	0	4

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)	\(37\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(6\)
\(+\)	\(-\)	$-$	\(3\)
\(-\)	\(+\)	$-$	\(2\)
\(-\)	\(-\)	$+$	\(7\)
Plus space		\(+\)	\(13\)
Minus space		\(-\)	\(5\)

Trace form

\( 18 q + 84 q^{4} + 16 q^{5} + 4 q^{7} + 84 q^{8} + 162 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(111))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	37
111.4.a.a	$111$	$4$	111.a	1.a	$1$	$1$	$1$	$6.549$	\(\Q\)	None	✓	$1$	$1$	\(-4\)	\(3\)	\(2\)	\(-28\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+3q^{3}+8q^{4}+2q^{5}-12q^{6}+\cdots\)
111.4.a.b	$111$	$4$	111.a	1.a	$1$	$1$	$1$	$6.549$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(-3\)	\(-4\)	\(-1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-3q^{3}-7q^{4}-4q^{5}+3q^{6}+\cdots\)
111.4.a.c	$111$	$4$	111.a	1.a	$1$	$1$	$1$	$6.549$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(3\)	\(-8\)	\(-13\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+3q^{3}-7q^{4}-8q^{5}+3q^{6}+\cdots\)
111.4.a.d	$111$	$4$	111.a	1.a	$1$	$3$	$3$	$6.549$	3.3.2700.1	None	✓	$1$	$1$	\(-3\)	\(-9\)	\(-6\)	\(15\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{2}-3q^{3}+(3-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
111.4.a.e	$111$	$4$	111.a	1.a	$1$	$5$	$5$	$6.549$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(4\)	\(-15\)	\(18\)	\(-12\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}-3q^{3}+(7+\beta _{2}+\beta _{4})q^{4}+\cdots\)
111.4.a.f	$111$	$4$	111.a	1.a	$1$	$7$	$7$	$6.549$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(3\)	\(21\)	\(14\)	\(43\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+3q^{3}+(6+\beta _{1}+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(111))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(111)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(37))\)\(^{\oplus 2}\)