Space of modular forms of level 222, weight 2, and trivial character

• Label: 222.2.a
• Level: $222$
• Weight: $2$
• Character orbit: 222.a
• Rep. character: $\chi_{222}(1,\cdot)$
• Character field: $\Q$
• Dimension: $5$
• Newform subspaces: $5$
• Sturm bound: $76$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 222 = 2 \cdot 3 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	222.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(76\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(222))\).

	Total	New	Old
Modular forms	42	5	37
Cusp forms	35	5	30
Eisenstein series	7	0	7

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

\(2\)	\(3\)	\(37\)	Fricke	Dim.
\(+\)	\(+\)	\(-\)	\(-\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)	\(1\)
Plus space			\(+\)	\(0\)
Minus space			\(-\)	\(5\)

Trace form

\( 5 q - q^{2} - q^{3} + 5 q^{4} + 2 q^{5} + q^{6} + 4 q^{7} - q^{8} + 5 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(222))\) 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	3	37
222.2.a.a	$222$	$2$	222.a	1.a	$1$	$1$	$1$	$1.773$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(-1\)	\(-4\)	\(3\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}-4q^{5}+q^{6}+3q^{7}+\cdots\)
222.2.a.b	$222$	$2$	222.a	1.a	$1$	$1$	$1$	$1.773$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(-1\)	\(2\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+2q^{5}+q^{6}-q^{8}+\cdots\)
222.2.a.c	$222$	$2$	222.a	1.a	$1$	$1$	$1$	$1.773$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(1\)	\(4\)	\(-1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}+4q^{5}-q^{6}-q^{7}+\cdots\)
222.2.a.d	$222$	$2$	222.a	1.a	$1$	$1$	$1$	$1.773$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(-1\)	\(0\)	\(3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-q^{6}+3q^{7}+q^{8}+\cdots\)
222.2.a.e	$222$	$2$	222.a	1.a	$1$	$1$	$1$	$1.773$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+q^{6}-q^{7}+q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(222))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(222)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(37))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(74))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(111))\)\(^{\oplus 2}\)