Space of modular forms of level 252, weight 9, and Character 252.z

• Label: 252.9.z
• Level: $252$
• Weight: $9$
• Character orbit: 252.z
• Rep. character: $\chi_{252}(73,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $54$
• Newform subspaces: $5$
• Sturm bound: $432$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	252.z (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(432\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(252, [\chi])\).

	Total	New	Old
Modular forms	792	54	738
Cusp forms	744	54	690
Eisenstein series	48	0	48

Trace form

\( 54 q - 837 q^{5} - 24 q^{7} + O(q^{10}) \)

Decomposition of \(S_{9}^{\mathrm{new}}(252, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
252.9.z.a	$252$	$9$	252.z	7.d	$6$	$2$	$1$	$102.659$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(4273\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(1504+1265\zeta_{6})q^{7}+(-5055+10110\zeta_{6})q^{13}+\cdots\)
252.9.z.b	$252$	$9$	252.z	7.d	$6$	$10$	$5$	$102.659$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(-1389\)	\(1217\)	$2^{10}\cdot 3^{8}\cdot 7^{3}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-92-93\beta _{1}-\beta _{5})q^{5}+(139-35\beta _{1}+\cdots)q^{7}+\cdots\)
252.9.z.c	$252$	$9$	252.z	7.d	$6$	$10$	$5$	$102.659$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(837\)	\(1526\)	$2^{12}\cdot 3^{9}\cdot 7^{6}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(112+56\beta _{1}-\beta _{3})q^{5}+(178+50\beta _{1}+\cdots)q^{7}+\cdots\)
252.9.z.d	$252$	$9$	252.z	7.d	$6$	$12$	$6$	$102.659$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(-285\)	\(198\)	$2^{20}\cdot 3^{10}\cdot 7^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-2^{4}-2^{4}\beta _{1}+\beta _{3})q^{5}+(114-14^{2}\beta _{1}+\cdots)q^{7}+\cdots\)
252.9.z.e	$252$	$9$	252.z	7.d	$6$	$20$	$10$	$102.659$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-7238\)	$2^{36}\cdot 3^{26}\cdot 7^{16}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\beta _{1}-\beta _{4})q^{5}+(-577-430\beta _{3}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{9}^{\mathrm{old}}(252, [\chi])\) into lower level spaces

\( S_{9}^{\mathrm{old}}(252, [\chi]) \cong \) \(S_{9}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 2}\)