Space of modular forms of level 180, weight 4, and trivial character

• Label: 180.4.a
• Level: $180$
• Weight: $4$
• Character orbit: 180.a
• Rep. character: $\chi_{180}(1,\cdot)$
• Character field: $\Q$
• Dimension: $5$
• Newform subspaces: $5$
• Sturm bound: $144$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	180.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(144\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(180))\).

	Total	New	Old
Modular forms	120	5	115
Cusp forms	96	5	91
Eisenstein series	24	0	24

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	Dim
\(-\)	\(+\)	\(+\)	$-$	\(1\)
\(-\)	\(+\)	\(-\)	$+$	\(1\)
\(-\)	\(-\)	\(+\)	$+$	\(2\)
\(-\)	\(-\)	\(-\)	$-$	\(1\)
Plus space			\(+\)	\(3\)
Minus space			\(-\)	\(2\)

Trace form

\( 5 q - 5 q^{5} - 8 q^{7} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(180))\) 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	5
180.4.a.a	$180$	$4$	180.a	1.a	$1$	$1$	$1$	$10.620$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-5\)	\(-16\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5q^{5}-2^{4}q^{7}+60q^{11}+86q^{13}+\cdots\)
180.4.a.b	$180$	$4$	180.a	1.a	$1$	$1$	$1$	$10.620$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-5\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{5}+2q^{7}-30q^{11}-4q^{13}-90q^{17}+\cdots\)
180.4.a.c	$180$	$4$	180.a	1.a	$1$	$1$	$1$	$10.620$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-5\)	\(32\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5q^{5}+2^{5}q^{7}-6^{2}q^{11}-10q^{13}+\cdots\)
180.4.a.d	$180$	$4$	180.a	1.a	$1$	$1$	$1$	$10.620$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(5\)	\(-28\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{5}-28q^{7}+24q^{11}-70q^{13}+\cdots\)
180.4.a.e	$180$	$4$	180.a	1.a	$1$	$1$	$1$	$10.620$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(5\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5q^{5}+2q^{7}+30q^{11}-4q^{13}+90q^{17}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(180))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(180)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 2}\)