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

• Label: 252.9.d
• Level: $252$
• Weight: $9$
• Character orbit: 252.d
• Rep. character: $\chi_{252}(181,\cdot)$
• Character field: $\Q$
• Dimension: $26$
• Newform subspaces: $4$
• Sturm bound: $432$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	252.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(432\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(5\)

Dimensions

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

	Total	New	Old
Modular forms	396	26	370
Cusp forms	372	26	346
Eisenstein series	24	0	24

Trace form

\( 26 q + 1870 q^{7} + O(q^{10}) \)

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.d.a	$252$	$9$	252.d	7.b	$2$	$2$	$2$	$102.659$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-4034\)	$2^{5}$	$\mathrm{U}(1)[D_{2}]$	\(q+(-2017-47\zeta_{6})q^{7}-1606\zeta_{6}q^{13}+\cdots\)
252.9.d.b	$252$	$9$	252.d	7.b	$2$	$6$	$6$	$102.659$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(2166\)	$2^{12}\cdot 3^{5}\cdot 7^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{5}+(19^{2}-\beta _{3})q^{7}+(-4082+\cdots)q^{11}+\cdots\)
252.9.d.c	$252$	$9$	252.d	7.b	$2$	$8$	$8$	$102.659$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(6076\)	$2^{12}\cdot 3^{6}\cdot 7^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{5}+(756-7\beta _{2})q^{7}-\beta _{5}q^{11}+\cdots\)
252.9.d.d	$252$	$9$	252.d	7.b	$2$	$10$	$10$	$102.659$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-2338\)	$2^{19}\cdot 3^{15}\cdot 7^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}+\beta _{6})q^{5}+(-234-2\beta _{1}+\beta _{7}+\cdots)q^{7}+\cdots\)

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

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