Space of modular forms of level 162, weight 6, and trivial character

• Label: 162.6.a
• Level: $162$
• Weight: $6$
• Character orbit: 162.a
• Rep. character: $\chi_{162}(1,\cdot)$
• Character field: $\Q$
• Dimension: $20$
• Newform subspaces: $10$
• Sturm bound: $162$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	162.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(162\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(162))\).

	Total	New	Old
Modular forms	147	20	127
Cusp forms	123	20	103
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
\(+\)	\(+\)	\(+\)		\(35\)	\(5\)	\(30\)		\(29\)	\(5\)	\(24\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)		\(38\)	\(5\)	\(33\)		\(32\)	\(5\)	\(27\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)		\(37\)	\(6\)	\(31\)		\(31\)	\(6\)	\(25\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)		\(37\)	\(4\)	\(33\)		\(31\)	\(4\)	\(27\)		\(6\)	\(0\)	\(6\)
Plus space		\(+\)		\(72\)	\(9\)	\(63\)		\(60\)	\(9\)	\(51\)		\(12\)	\(0\)	\(12\)
Minus space		\(-\)		\(75\)	\(11\)	\(64\)		\(63\)	\(11\)	\(52\)		\(12\)	\(0\)	\(12\)

Trace form

20 * q + 320 * q^4 - 116 * q^7 - 24 * q^10 - 470 * q^13 + 5120 * q^16 - 770 * q^19 + 1896 * q^22 + 5414 * q^25 - 1856 * q^28 + 9952 * q^31 + 13392 * q^34 + 15358 * q^37 - 384 * q^40 + 56854 * q^43 - 18960 * q^46 + 4308 * q^49 - 7520 * q^52 - 64800 * q^55 + 36024 * q^58 + 90046 * q^61 + 81920 * q^64 - 8246 * q^67 + 126528 * q^70 - 154724 * q^73 - 12320 * q^76 - 139688 * q^79 + 13368 * q^82 - 196038 * q^85 + 30336 * q^88 + 19664 * q^91 + 206160 * q^94 - 29114 * q^97

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(162))\) 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
162.6.a.a	$162$	$6$	162.a	1.a	$1$	$1$	$1$	$25.982$	\(\Q\)	None	✓	✓			162.6.a.a	$1$	$1$	\(-4\)	\(0\)	\(-21\)	\(74\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+2^{4}q^{4}-21q^{5}+74q^{7}-2^{6}q^{8}+\cdots\)
162.6.a.b	$162$	$6$	162.a	1.a	$1$	$1$	$1$	$25.982$	\(\Q\)	None	✓	✓			162.6.a.a	$1$	$0$	\(4\)	\(0\)	\(21\)	\(74\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+2^{4}q^{4}+21q^{5}+74q^{7}+2^{6}q^{8}+\cdots\)
162.6.a.c	$162$	$6$	162.a	1.a	$1$	$2$	$2$	$25.982$	\(\Q(\sqrt{3}) \)	None	✓	✓			162.6.a.c	$1$	$0$	\(-8\)	\(0\)	\(12\)	\(-176\)		$+$	$-$	$-$	$3$	$\mathrm{SU}(2)$	\(q-4q^{2}+2^{4}q^{4}+(6+5\beta )q^{5}+(-88+\cdots)q^{7}+\cdots\)
162.6.a.d	$162$	$6$	162.a	1.a	$1$	$2$	$2$	$25.982$	\(\Q(\sqrt{921}) \)	None	✓	✓			162.6.a.d	$1$	$1$	\(-8\)	\(0\)	\(12\)	\(-14\)		$+$	$+$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q-4q^{2}+2^{4}q^{4}+(6+\beta )q^{5}+(-7-\beta )q^{7}+\cdots\)
162.6.a.e	$162$	$6$	162.a	1.a	$1$	$2$	$2$	$25.982$	\(\Q(\sqrt{6}) \)	None	✓				18.6.c.a	$1$	$1$	\(-8\)	\(0\)	\(54\)	\(-74\)		$+$	$+$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q-4q^{2}+2^{4}q^{4}+(3^{3}+2\beta )q^{5}+(-37+\cdots)q^{7}+\cdots\)
162.6.a.f	$162$	$6$	162.a	1.a	$1$	$2$	$2$	$25.982$	\(\Q(\sqrt{6}) \)	None	✓				18.6.c.a	$1$	$1$	\(8\)	\(0\)	\(-54\)	\(-74\)		$-$	$-$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+4q^{2}+2^{4}q^{4}+(-3^{3}+2\beta )q^{5}+(-37+\cdots)q^{7}+\cdots\)
162.6.a.g	$162$	$6$	162.a	1.a	$1$	$2$	$2$	$25.982$	\(\Q(\sqrt{3}) \)	None	✓	✓			162.6.a.c	$1$	$1$	\(8\)	\(0\)	\(-12\)	\(-176\)		$-$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q+4q^{2}+2^{4}q^{4}+(-6+5\beta )q^{5}+(-88+\cdots)q^{7}+\cdots\)
162.6.a.h	$162$	$6$	162.a	1.a	$1$	$2$	$2$	$25.982$	\(\Q(\sqrt{921}) \)	None	✓	✓			162.6.a.d	$1$	$0$	\(8\)	\(0\)	\(-12\)	\(-14\)		$-$	$+$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+4q^{2}+2^{4}q^{4}+(-6-\beta )q^{5}+(-7+\cdots)q^{7}+\cdots\)
162.6.a.i	$162$	$6$	162.a	1.a	$1$	$3$	$3$	$25.982$	3.3.125628.1	None	✓		✓		18.6.c.b	$1$	$0$	\(-12\)	\(0\)	\(-54\)	\(132\)		$+$	$-$	$-$	$2\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q-4q^{2}+2^{4}q^{4}+(-18+\beta _{1})q^{5}+(44+\cdots)q^{7}+\cdots\)
162.6.a.j	$162$	$6$	162.a	1.a	$1$	$3$	$3$	$25.982$	3.3.125628.1	None	✓		✓		18.6.c.b	$1$	$0$	\(12\)	\(0\)	\(54\)	\(132\)		$-$	$+$	$-$	$2\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+4q^{2}+2^{4}q^{4}+(18-\beta _{1})q^{5}+(44+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(162))\) into lower level spaces

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