Space of modular forms of level 162, weight 13, and Character 162.d

• Label: 162.13.d
• Level: $162$
• Weight: $13$
• Character orbit: 162.d
• Rep. character: $\chi_{162}(53,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $96$
• Newform subspaces: $8$
• Sturm bound: $351$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	162.d (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(351\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{13}(162, [\chi])\).

	Total	New	Old
Modular forms	672	96	576
Cusp forms	624	96	528
Eisenstein series	48	0	48

Trace form

\( 96 q + 98304 q^{4} + 343200 q^{7} + O(q^{10}) \)

Decomposition of \(S_{13}^{\mathrm{new}}(162, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
162.13.d.a	$162$	$13$	162.d	9.d	$6$	$4$	$2$	$148.067$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-67144\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-2^{5}\beta _{1}+2^{5}\beta _{3})q^{2}+(2^{11}-2^{11}\beta _{2}+\cdots)q^{4}+\cdots\)
162.13.d.b	$162$	$13$	162.d	9.d	$6$	$4$	$2$	$148.067$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(98744\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-2^{5}\beta _{1}+2^{5}\beta _{3})q^{2}+(2^{11}-2^{11}\beta _{2}+\cdots)q^{4}+\cdots\)
162.13.d.c	$162$	$13$	162.d	9.d	$6$	$8$	$4$	$148.067$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-271484\)	$2^{24}\cdot 3^{12}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{3}q^{2}-2^{11}\beta _{1}q^{4}+(\beta _{2}+139\beta _{3}+\cdots)q^{5}+\cdots\)
162.13.d.d	$162$	$13$	162.d	9.d	$6$	$8$	$4$	$148.067$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-153080\)	$2^{20}\cdot 3^{16}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\beta _{3}+\beta _{6})q^{2}+(2^{11}+2^{11}\beta _{1})q^{4}+\cdots\)
162.13.d.e	$162$	$13$	162.d	9.d	$6$	$8$	$4$	$148.067$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(68284\)	$2^{28}\cdot 3^{12}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{3}q^{2}+(2^{11}-2^{11}\beta _{1})q^{4}+(60\beta _{2}+\cdots)q^{5}+\cdots\)
162.13.d.f	$162$	$13$	162.d	9.d	$6$	$16$	$8$	$148.067$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(393320\)	$2^{40}\cdot 3^{72}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{8}q^{2}-2^{11}\beta _{1}q^{4}+(-10^{2}\beta _{8}+10^{2}\beta _{9}+\cdots)q^{5}+\cdots\)
162.13.d.g	$162$	$13$	162.d	9.d	$6$	$24$	$12$	$148.067$		None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(54336\)		$\mathrm{SU}(2)[C_{6}]$
162.13.d.h	$162$	$13$	162.d	9.d	$6$	$24$	$12$	$148.067$		None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(220224\)		$\mathrm{SU}(2)[C_{6}]$

Decomposition of \(S_{13}^{\mathrm{old}}(162, [\chi])\) into lower level spaces

\( S_{13}^{\mathrm{old}}(162, [\chi]) \cong \) \(S_{13}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{13}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{13}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{13}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{13}^{\mathrm{new}}(81, [\chi])\)\(^{\oplus 2}\)