Space of modular forms of level 162, weight 7, and Character 162.d

• Label: 162.7.d
• Level: $162$
• Weight: $7$
• Character orbit: 162.d
• Rep. character: $\chi_{162}(53,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $48$
• Newform subspaces: $7$
• Sturm bound: $189$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	162.d (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(189\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(162, [\chi])\).

	Total	New	Old
Modular forms	348	48	300
Cusp forms	300	48	252
Eisenstein series	48	0	48

Trace form

\( 48 q + 768 q^{4} - 1200 q^{7} + O(q^{10}) \)

Decomposition of \(S_{7}^{\mathrm{new}}(162, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
162.7.d.a	$162$	$7$	162.d	9.d	$6$	$4$	$2$	$37.269$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-778\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(2\beta _{1}-2\beta _{3})q^{2}+(2^{5}-2^{5}\beta _{2})q^{4}+\cdots\)
162.7.d.b	$162$	$7$	162.d	9.d	$6$	$4$	$2$	$37.269$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$2^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{1}-\beta _{3})q^{2}+(2^{5}-2^{5}\beta _{2})q^{4}+30\beta _{1}q^{5}+\cdots\)
162.7.d.c	$162$	$7$	162.d	9.d	$6$	$4$	$2$	$37.269$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(410\)	$2^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\beta _{1}+\beta _{3})q^{2}+(2^{5}-2^{5}\beta _{2})q^{4}+\cdots\)
162.7.d.d	$162$	$7$	162.d	9.d	$6$	$4$	$2$	$37.269$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(968\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-4\beta _{1}+4\beta _{3})q^{2}+(2^{5}-2^{5}\beta _{2})q^{4}+\cdots\)
162.7.d.e	$162$	$7$	162.d	9.d	$6$	$8$	$4$	$37.269$	\(\Q(\zeta_{24})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-836\)	$2^{10}\cdot 3^{12}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\zeta_{24}^{5}q^{2}+2^{5}\zeta_{24}^{2}q^{4}+(5\zeta_{24}-5\zeta_{24}^{3}+\cdots)q^{5}+\cdots\)
162.7.d.f	$162$	$7$	162.d	9.d	$6$	$8$	$4$	$37.269$	\(\Q(\zeta_{24})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$3^{6}$	$\mathrm{SU}(2)[C_{6}]$	\(q+4\zeta_{24}^{3}q^{2}+(2^{5}-2^{5}\zeta_{24})q^{4}+(95\zeta_{24}^{5}+\cdots)q^{5}+\cdots\)
162.7.d.g	$162$	$7$	162.d	9.d	$6$	$16$	$8$	$37.269$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-964\)	$2^{4}\cdot 3^{36}$	$\mathrm{SU}(2)[C_{6}]$	\(q+4\beta _{8}q^{2}+(2^{5}-2^{5}\beta _{2})q^{4}+(-12\beta _{8}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{7}^{\mathrm{old}}(162, [\chi])\) into lower level spaces

\( S_{7}^{\mathrm{old}}(162, [\chi]) \cong \) \(S_{7}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(81, [\chi])\)\(^{\oplus 2}\)