Space of modular forms of level 220, weight 2, and trivial character

• Label: 220.2.a
• Level: $220$
• Weight: $2$
• Character orbit: 220.a
• Rep. character: $\chi_{220}(1,\cdot)$
• Character field: $\Q$
• Dimension: $2$
• Newform subspaces: $2$
• Sturm bound: $72$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 220 = 2^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	220.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(72\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(220))\).

	Total	New	Old
Modular forms	42	2	40
Cusp forms	31	2	29
Eisenstein series	11	0	11

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

\(2\)	\(5\)	\(11\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)		\(9\)	\(0\)	\(9\)		\(7\)	\(0\)	\(7\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(6\)	\(0\)	\(6\)		\(4\)	\(0\)	\(4\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(5\)	\(0\)	\(5\)		\(3\)	\(0\)	\(3\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(4\)	\(0\)	\(4\)		\(3\)	\(0\)	\(3\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)		\(6\)	\(0\)	\(6\)		\(5\)	\(0\)	\(5\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)		\(6\)	\(1\)	\(5\)		\(5\)	\(1\)	\(4\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)		\(4\)	\(1\)	\(3\)		\(3\)	\(1\)	\(2\)		\(1\)	\(0\)	\(1\)
Plus space			\(+\)		\(19\)	\(1\)	\(18\)		\(14\)	\(1\)	\(13\)		\(5\)	\(0\)	\(5\)
Minus space			\(-\)		\(23\)	\(1\)	\(22\)		\(17\)	\(1\)	\(16\)		\(6\)	\(0\)	\(6\)

Trace form

2 * q + 2 * q^5 - 4 * q^7 + 2 * q^9 - 4 * q^13 - 4 * q^17 - 8 * q^19 + 8 * q^21 + 2 * q^25 - 4 * q^29 + 8 * q^31 + 4 * q^33 - 4 * q^35 - 4 * q^37 + 8 * q^39 - 4 * q^41 + 12 * q^43 + 2 * q^45 + 16 * q^47 + 2 * q^49 - 8 * q^51 - 4 * q^53 - 16 * q^59 - 12 * q^61 - 4 * q^63 - 4 * q^65 - 8 * q^67 + 24 * q^69 - 8 * q^71 - 20 * q^73 + 4 * q^77 - 22 * q^81 + 12 * q^83 - 4 * q^85 + 16 * q^87 + 12 * q^89 + 16 * q^91 - 16 * q^93 - 8 * q^95 + 20 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(220))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	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	5	11
220.2.a.a	$220$	$2$	220.a	1.a	$1$	$1$	$1$	$1.757$	\(\Q\)	None	✓	✓	✓	✓	220.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(1\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{5}-4q^{7}+q^{9}-q^{11}-4q^{13}+\cdots\)
220.2.a.b	$220$	$2$	220.a	1.a	$1$	$1$	$1$	$1.757$	\(\Q\)	None	✓	✓	✓	✓	220.2.a.b	$1$	$0$	\(0\)	\(2\)	\(1\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{5}+q^{9}+q^{11}+2q^{15}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(220))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(220)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(110))\)\(^{\oplus 2}\)