Space of modular forms of level 180, weight 2, and trivial character

• Label: 180.2.a
• Level: $180$
• Weight: $2$
• Character orbit: 180.a
• Rep. character: $\chi_{180}(1,\cdot)$
• Character field: $\Q$
• Dimension: $1$
• Newform subspaces: $1$
• Sturm bound: $72$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	180.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(72\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(180))\).

	Total	New	Old
Modular forms	48	1	47
Cusp forms	25	1	24
Eisenstein series	23	0	23

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

\(2\)	\(3\)	\(5\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(5\)	\(0\)	\(5\)		\(2\)	\(0\)	\(2\)		\(3\)	\(0\)	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)		\(7\)	\(0\)	\(7\)		\(3\)	\(0\)	\(3\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(+\)	\(-\)		\(7\)	\(0\)	\(7\)		\(3\)	\(0\)	\(3\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)		\(7\)	\(0\)	\(7\)		\(3\)	\(0\)	\(3\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(+\)	\(-\)		\(7\)	\(0\)	\(7\)		\(5\)	\(0\)	\(5\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)		\(5\)	\(0\)	\(5\)		\(3\)	\(0\)	\(3\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)		\(5\)	\(0\)	\(5\)		\(3\)	\(0\)	\(3\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)		\(5\)	\(1\)	\(4\)		\(3\)	\(1\)	\(2\)		\(2\)	\(0\)	\(2\)
Plus space			\(+\)		\(22\)	\(0\)	\(22\)		\(11\)	\(0\)	\(11\)		\(11\)	\(0\)	\(11\)
Minus space			\(-\)		\(26\)	\(1\)	\(25\)		\(14\)	\(1\)	\(13\)		\(12\)	\(0\)	\(12\)

Trace form

q + q^5 + 2 * q^7 + 2 * q^13 + 6 * q^17 - 4 * q^19 - 6 * q^23 + q^25 - 6 * q^29 - 4 * q^31 + 2 * q^35 + 2 * q^37 - 6 * q^41 - 10 * q^43 + 6 * q^47 - 3 * q^49 + 6 * q^53 - 12 * q^59 + 2 * q^61 + 2 * q^65 + 2 * q^67 + 12 * q^71 + 2 * q^73 + 8 * q^79 - 6 * q^83 + 6 * q^85 + 6 * q^89 + 4 * q^91 - 4 * q^95 + 2 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(180))\) 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	3	5
180.2.a.a	$180$	$2$	180.a	1.a	$1$	$1$	$1$	$1.437$	\(\Q\)	None	✓		✓	✓	20.2.a.a	$1$	$0$	\(0\)	\(0\)	\(1\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+2q^{7}+2q^{13}+6q^{17}-4q^{19}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(180))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(180)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 2}\)