Space of modular forms of level 162, weight 2, and trivial character

• Label: 162.2.a
• Level: $162$
• Weight: $2$
• Character orbit: 162.a
• Rep. character: $\chi_{162}(1,\cdot)$
• Character field: $\Q$
• Dimension: $4$
• Newform subspaces: $4$
• Sturm bound: $54$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	162.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(54\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(162))\).

	Total	New	Old
Modular forms	39	4	35
Cusp forms	16	4	12
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.

\(2\)	\(3\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(8\)	\(1\)	\(7\)		\(3\)	\(1\)	\(2\)		\(5\)	\(0\)	\(5\)
\(+\)	\(-\)	\(-\)		\(11\)	\(1\)	\(10\)		\(5\)	\(1\)	\(4\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)		\(10\)	\(2\)	\(8\)		\(4\)	\(2\)	\(2\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)		\(10\)	\(0\)	\(10\)		\(4\)	\(0\)	\(4\)		\(6\)	\(0\)	\(6\)
Plus space		\(+\)		\(18\)	\(1\)	\(17\)		\(7\)	\(1\)	\(6\)		\(11\)	\(0\)	\(11\)
Minus space		\(-\)		\(21\)	\(3\)	\(18\)		\(9\)	\(3\)	\(6\)		\(12\)	\(0\)	\(12\)

Trace form

4 * q + 4 * q^4 - 4 * q^7 + 6 * q^10 + 2 * q^13 + 4 * q^16 - 10 * q^19 - 6 * q^22 - 2 * q^25 - 4 * q^28 - 16 * q^31 - 10 * q^37 + 6 * q^40 + 14 * q^43 - 12 * q^46 + 12 * q^49 + 2 * q^52 - 6 * q^58 + 14 * q^61 + 4 * q^64 + 2 * q^67 - 24 * q^70 + 44 * q^73 - 10 * q^76 - 40 * q^79 + 6 * q^82 + 18 * q^85 - 6 * q^88 + 16 * q^91 + 12 * q^94 + 14 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(162))\) 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
162.2.a.a	$162$	$2$	162.a	1.a	$1$	$1$	$1$	$1.294$	\(\Q\)	None	✓	✓	✓	✓	162.2.a.a	$1$	$1$	\(-1\)	\(0\)	\(-3\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-3q^{5}-4q^{7}-q^{8}+3q^{10}+\cdots\)
162.2.a.b	$162$	$2$	162.a	1.a	$1$	$1$	$1$	$1.294$	\(\Q\)	None	✓		✓	✓	18.2.c.a	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+2q^{7}-q^{8}+3q^{11}+2q^{13}+\cdots\)
162.2.a.c	$162$	$2$	162.a	1.a	$1$	$1$	$1$	$1.294$	\(\Q\)	None	✓				18.2.c.a	$1$	$0$	\(1\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+2q^{7}+q^{8}-3q^{11}+2q^{13}+\cdots\)
162.2.a.d	$162$	$2$	162.a	1.a	$1$	$1$	$1$	$1.294$	\(\Q\)	None	✓	✓			162.2.a.a	$1$	$0$	\(1\)	\(0\)	\(3\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+3q^{5}-4q^{7}+q^{8}+3q^{10}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(162))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(162)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 2}\)