Space of modular forms of level 162, weight 4, and trivial character

• Label: 162.4.a
• Level: $162$
• Weight: $4$
• Character orbit: 162.a
• Rep. character: $\chi_{162}(1,\cdot)$
• Character field: $\Q$
• Dimension: $12$
• Newform subspaces: $8$
• Sturm bound: $108$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	162.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(108\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(162))\).

	Total	New	Old
Modular forms	93	12	81
Cusp forms	69	12	57
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.

\(2\)	\(3\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(25\)	\(3\)	\(22\)		\(19\)	\(3\)	\(16\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)		\(22\)	\(3\)	\(19\)		\(16\)	\(3\)	\(13\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)		\(23\)	\(2\)	\(21\)		\(17\)	\(2\)	\(15\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)		\(23\)	\(4\)	\(19\)		\(17\)	\(4\)	\(13\)		\(6\)	\(0\)	\(6\)
Plus space		\(+\)		\(48\)	\(7\)	\(41\)		\(36\)	\(7\)	\(29\)		\(12\)	\(0\)	\(12\)
Minus space		\(-\)		\(45\)	\(5\)	\(40\)		\(33\)	\(5\)	\(28\)		\(12\)	\(0\)	\(12\)

Trace form

12 * q + 48 * q^4 + 24 * q^7 - 36 * q^10 - 30 * q^13 + 192 * q^16 - 30 * q^19 + 36 * q^22 + 822 * q^25 + 96 * q^28 + 132 * q^31 + 216 * q^34 - 678 * q^37 - 144 * q^40 - 1434 * q^43 + 504 * q^46 - 72 * q^49 - 120 * q^52 - 324 * q^55 + 1332 * q^58 - 1002 * q^61 + 768 * q^64 - 2838 * q^67 - 288 * q^70 + 2184 * q^73 - 120 * q^76 + 780 * q^79 - 1980 * q^82 + 3294 * q^85 + 144 * q^88 - 1140 * q^91 - 3096 * q^94 - 6078 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(162))\) 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
162.4.a.a	$162$	$4$	162.a	1.a	$1$	$1$	$1$	$9.558$	\(\Q\)	None	✓				18.4.c.a	$1$	$1$	\(-2\)	\(0\)	\(9\)	\(-31\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+9q^{5}-31q^{7}-8q^{8}+\cdots\)
162.4.a.b	$162$	$4$	162.a	1.a	$1$	$1$	$1$	$9.558$	\(\Q\)	None	✓	✓			162.4.a.b	$1$	$0$	\(-2\)	\(0\)	\(21\)	\(8\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+21q^{5}+8q^{7}-8q^{8}+\cdots\)
162.4.a.c	$162$	$4$	162.a	1.a	$1$	$1$	$1$	$9.558$	\(\Q\)	None	✓	✓			162.4.a.b	$1$	$1$	\(2\)	\(0\)	\(-21\)	\(8\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-21q^{5}+8q^{7}+8q^{8}+\cdots\)
162.4.a.d	$162$	$4$	162.a	1.a	$1$	$1$	$1$	$9.558$	\(\Q\)	None	✓				18.4.c.a	$1$	$1$	\(2\)	\(0\)	\(-9\)	\(-31\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-9q^{5}-31q^{7}+8q^{8}+\cdots\)
162.4.a.e	$162$	$4$	162.a	1.a	$1$	$2$	$2$	$9.558$	\(\Q(\sqrt{3}) \)	None	✓	✓	✓		162.4.a.e	$1$	$1$	\(-4\)	\(0\)	\(-12\)	\(16\)		$+$	$-$	$-$	$3$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(-6+\beta )q^{5}+(8-2\beta )q^{7}+\cdots\)
162.4.a.f	$162$	$4$	162.a	1.a	$1$	$2$	$2$	$9.558$	\(\Q(\sqrt{105}) \)	None	✓		✓		18.4.c.b	$1$	$0$	\(-4\)	\(0\)	\(-9\)	\(19\)		$+$	$+$	$+$	$3$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(-5-\beta )q^{5}+(9-\beta )q^{7}+\cdots\)
162.4.a.g	$162$	$4$	162.a	1.a	$1$	$2$	$2$	$9.558$	\(\Q(\sqrt{105}) \)	None	✓				18.4.c.b	$1$	$0$	\(4\)	\(0\)	\(9\)	\(19\)		$-$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(4-\beta )q^{5}+(10+\beta )q^{7}+\cdots\)
162.4.a.h	$162$	$4$	162.a	1.a	$1$	$2$	$2$	$9.558$	\(\Q(\sqrt{3}) \)	None	✓	✓			162.4.a.e	$1$	$0$	\(4\)	\(0\)	\(12\)	\(16\)		$-$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(6+\beta )q^{5}+(8+2\beta )q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(162))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(162)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 2}\)