Space of modular forms of level 648, weight 2, and Character 648.f

• Label: 648.2.f
• Level: $648$
• Weight: $2$
• Character orbit: 648.f
• Rep. character: $\chi_{648}(323,\cdot)$
• Character field: $\Q$
• Dimension: $44$
• Newform subspaces: $3$
• Sturm bound: $216$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	648.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 24 \)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(216\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(648, [\chi])\).

	Total	New	Old
Modular forms	120	52	68
Cusp forms	96	44	52
Eisenstein series	24	8	16

Trace form

\( 44 q + 2 q^{4} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(648, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
648.2.f.a	$648$	$2$	648.f	24.f	$2$	$4$	$4$	$5.174$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	\(\Q(\sqrt{-2}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{1}q^{2}-2q^{4}-2\beta _{1}q^{8}+(-\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots\)
648.2.f.b	$648$	$2$	648.f	24.f	$2$	$16$	$16$	$5.174$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{2}+(1-\beta _{9})q^{4}-\beta _{4}q^{5}+(-\beta _{7}+\cdots)q^{7}+\cdots\)
648.2.f.c	$648$	$2$	648.f	24.f	$2$	$24$	$24$	$5.174$		None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{2}^{\mathrm{old}}(648, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(648, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 4}\)