Space of modular forms of level 221, weight 2, and trivial character

• Label: 221.2.a
• Level: $221$
• Weight: $2$
• Character orbit: 221.a
• Rep. character: $\chi_{221}(1,\cdot)$
• Character field: $\Q$
• Dimension: $17$
• Newform subspaces: $7$
• Sturm bound: $42$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 221 = 13 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	221.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(42\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(221))\).

	Total	New	Old
Modular forms	22	17	5
Cusp forms	19	17	2
Eisenstein series	3	0	3

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

\(13\)	\(17\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(2\)
\(+\)	\(-\)	$-$	\(6\)
\(-\)	\(+\)	$-$	\(6\)
\(-\)	\(-\)	$+$	\(3\)
Plus space		\(+\)	\(5\)
Minus space		\(-\)	\(12\)

Trace form

\( 17 q - q^{2} + 19 q^{4} - 2 q^{5} - 4 q^{7} - 9 q^{8} + 17 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(221))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		13	17
221.2.a.a	$221$	$2$	221.a	1.a	$1$	$1$	$1$	$1.765$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(4\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+4q^{5}-2q^{7}+3q^{8}-3q^{9}+\cdots\)
221.2.a.b	$221$	$2$	221.a	1.a	$1$	$1$	$1$	$1.765$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(2\)	\(2\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+2q^{3}-q^{4}+2q^{5}+2q^{6}+2q^{7}+\cdots\)
221.2.a.c	$221$	$2$	221.a	1.a	$1$	$2$	$2$	$1.765$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(-1\)	\(-3\)	\(0\)	\(-1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-1-\beta )q^{3}+(-1+\beta )q^{4}+\cdots\)
221.2.a.d	$221$	$2$	221.a	1.a	$1$	$2$	$2$	$1.765$	\(\Q(\sqrt{21}) \)	None	✓	$1$	$0$	\(-1\)	\(1\)	\(-2\)	\(-5\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(1-\beta )q^{3}+(3+\beta )q^{4}-q^{5}+\cdots\)
221.2.a.e	$221$	$2$	221.a	1.a	$1$	$2$	$2$	$1.765$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(-2\)	\(4\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(1+\beta )q^{3}+3q^{4}+(-1-\beta )q^{5}+\cdots\)
221.2.a.f	$221$	$2$	221.a	1.a	$1$	$3$	$3$	$1.765$	3.3.229.1	None	✓	$1$	$1$	\(0\)	\(-3\)	\(-2\)	\(-9\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-1+\beta _{1})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
221.2.a.g	$221$	$2$	221.a	1.a	$1$	$6$	$6$	$1.765$	6.6.28134208.1	None	✓	$1$	$0$	\(1\)	\(1\)	\(-2\)	\(7\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{4}q^{2}+\beta _{1}q^{3}+(1-\beta _{3})q^{4}+\beta _{2}q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(221))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(221)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 2}\)