Space of modular forms of level 162, weight 8, and trivial character

• Label: 162.8.a
• Level: $162$
• Weight: $8$
• Character orbit: 162.a
• Rep. character: $\chi_{162}(1,\cdot)$
• Character field: $\Q$
• Dimension: $28$
• Newform subspaces: $10$
• Sturm bound: $216$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	162.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(216\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(162))\).

	Total	New	Old
Modular forms	201	28	173
Cusp forms	177	28	149
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
\(+\)	\(+\)	\(+\)		\(52\)	\(7\)	\(45\)		\(46\)	\(7\)	\(39\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)		\(49\)	\(7\)	\(42\)		\(43\)	\(7\)	\(36\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)		\(50\)	\(6\)	\(44\)		\(44\)	\(6\)	\(38\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)		\(50\)	\(8\)	\(42\)		\(44\)	\(8\)	\(36\)		\(6\)	\(0\)	\(6\)
Plus space		\(+\)		\(102\)	\(15\)	\(87\)		\(90\)	\(15\)	\(75\)		\(12\)	\(0\)	\(12\)
Minus space		\(-\)		\(99\)	\(13\)	\(86\)		\(87\)	\(13\)	\(74\)		\(12\)	\(0\)	\(12\)

Trace form

28 * q + 1792 * q^4 + 332 * q^7 + 3984 * q^10 + 16286 * q^13 + 114688 * q^16 + 42590 * q^19 - 70512 * q^22 + 428602 * q^25 + 21248 * q^28 - 504832 * q^31 - 96768 * q^34 - 452590 * q^37 + 254976 * q^40 - 974458 * q^43 + 602016 * q^46 + 4578204 * q^49 + 1042304 * q^52 + 8204976 * q^55 + 206832 * q^58 - 6942430 * q^61 + 7340032 * q^64 + 11626394 * q^67 - 5569728 * q^70 + 11677052 * q^73 + 2725760 * q^76 + 1422440 * q^79 - 2862768 * q^82 - 25452666 * q^85 - 4512768 * q^88 - 18769424 * q^91 + 9219936 * q^94 - 15764302 * q^97

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(162))\) 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
162.8.a.a	$162$	$8$	162.a	1.a	$1$	$1$	$1$	$50.606$	\(\Q\)	None	✓	✓			162.8.a.a	$1$	$0$	\(-8\)	\(0\)	\(165\)	\(-508\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}+2^{6}q^{4}+165q^{5}-508q^{7}+\cdots\)
162.8.a.b	$162$	$8$	162.a	1.a	$1$	$1$	$1$	$50.606$	\(\Q\)	None	✓	✓			162.8.a.a	$1$	$1$	\(8\)	\(0\)	\(-165\)	\(-508\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}+2^{6}q^{4}-165q^{5}-508q^{7}+\cdots\)
162.8.a.c	$162$	$8$	162.a	1.a	$1$	$2$	$2$	$50.606$	\(\Q(\sqrt{1929}) \)	None	✓	✓			162.8.a.c	$1$	$0$	\(-16\)	\(0\)	\(114\)	\(280\)		$+$	$+$	$+$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q-8q^{2}+2^{6}q^{4}+(57-\beta )q^{5}+(140+\cdots)q^{7}+\cdots\)
162.8.a.d	$162$	$8$	162.a	1.a	$1$	$2$	$2$	$50.606$	\(\Q(\sqrt{1929}) \)	None	✓	✓			162.8.a.c	$1$	$1$	\(16\)	\(0\)	\(-114\)	\(280\)		$-$	$+$	$-$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+8q^{2}+2^{6}q^{4}+(-57-\beta )q^{5}+(140+\cdots)q^{7}+\cdots\)
162.8.a.e	$162$	$8$	162.a	1.a	$1$	$3$	$3$	$50.606$	3.3.69765.1	None	✓				18.8.c.a	$1$	$1$	\(-24\)	\(0\)	\(54\)	\(-210\)		$+$	$-$	$-$	$2\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q-8q^{2}+2^{6}q^{4}+(18+2\beta _{1}-3\beta _{2})q^{5}+\cdots\)
162.8.a.f	$162$	$8$	162.a	1.a	$1$	$3$	$3$	$50.606$	3.3.69765.1	None	✓		✓		18.8.c.a	$1$	$1$	\(24\)	\(0\)	\(-54\)	\(-210\)		$-$	$+$	$-$	$2\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+8q^{2}+2^{6}q^{4}+(-18-2\beta _{1}+3\beta _{2})q^{5}+\cdots\)
162.8.a.g	$162$	$8$	162.a	1.a	$1$	$4$	$4$	$50.606$	4.4.43103376.1	None	✓	✓	✓		162.8.a.g	$1$	$1$	\(-32\)	\(0\)	\(-528\)	\(560\)		$+$	$-$	$-$	$2^{4}\cdot 3^{8}$	$\mathrm{SU}(2)$	\(q-8q^{2}+2^{6}q^{4}+(-132+\beta _{1})q^{5}+\cdots\)
162.8.a.h	$162$	$8$	162.a	1.a	$1$	$4$	$4$	$50.606$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		18.8.c.b	$1$	$0$	\(-32\)	\(0\)	\(-54\)	\(44\)		$+$	$+$	$+$	$2^{2}\cdot 3^{8}$	$\mathrm{SU}(2)$	\(q-8q^{2}+2^{6}q^{4}+(-14-\beta _{1})q^{5}+(12+\cdots)q^{7}+\cdots\)
162.8.a.i	$162$	$8$	162.a	1.a	$1$	$4$	$4$	$50.606$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				18.8.c.b	$1$	$0$	\(32\)	\(0\)	\(54\)	\(44\)		$-$	$-$	$+$	$2^{2}\cdot 3^{8}$	$\mathrm{SU}(2)$	\(q+8q^{2}+2^{6}q^{4}+(14+\beta _{1})q^{5}+(12+\cdots)q^{7}+\cdots\)
162.8.a.j	$162$	$8$	162.a	1.a	$1$	$4$	$4$	$50.606$	4.4.43103376.1	None	✓	✓			162.8.a.g	$1$	$0$	\(32\)	\(0\)	\(528\)	\(560\)		$-$	$-$	$+$	$2^{4}\cdot 3^{8}$	$\mathrm{SU}(2)$	\(q+8q^{2}+2^{6}q^{4}+(132+\beta _{1})q^{5}+(140+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(162))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(162)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 5}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 2}\)