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

• Label: 648.2.l
• Level: $648$
• Weight: $2$
• Character orbit: 648.l
• Rep. character: $\chi_{648}(107,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $92$
• Newform subspaces: $7$
• Sturm bound: $216$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	648.l (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 72 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(216\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(7\), \(41\)

Dimensions

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

	Total	New	Old
Modular forms	240	100	140
Cusp forms	192	92	100
Eisenstein series	48	8	40

Trace form

92 * q + 2 * q^4 - 12 * q^10 + 2 * q^16 - 8 * q^19 - 2 * q^22 - 34 * q^25 + 12 * q^28 + 28 * q^34 + 42 * q^40 + 4 * q^43 - 72 * q^46 + 38 * q^49 + 12 * q^52 - 12 * q^58 + 20 * q^64 + 4 * q^67 - 6 * q^70 - 8 * q^73 + 40 * q^76 - 32 * q^82 + 46 * q^88 + 48 * q^91 - 72 * q^94 + 4 * q^97

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
648.2.l.a	$648$	$2$	648.l	72.l	$6$	$4$	$2$	$5.174$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None					72.2.f.a	$4$	$1$	\(0\)	\(0\)	\(0\)	\(-12\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{3}q^{2}-2q^{4}+(-\beta _{1}+2\beta _{3})q^{5}+\cdots\)
648.2.l.b	$648$	$2$	648.l	72.l	$6$	$4$	$2$	$5.174$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	\(\Q(\sqrt{-2}) \)					24.2.f.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+\beta _{1}q^{2}+2\beta _{2}q^{4}+2\beta _{3}q^{8}+2\beta _{1}q^{11}+\cdots\)
648.2.l.c	$648$	$2$	648.l	72.l	$6$	$4$	$2$	$5.174$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None					72.2.f.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(12\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{2}+2\beta _{2}q^{4}+(-\beta _{1}+2\beta _{3})q^{5}+\cdots\)
648.2.l.d	$648$	$2$	648.l	72.l	$6$	$8$	$4$	$5.174$	8.0.170772624.1	None					216.2.f.b	$4$	$0$	\(-3\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(1-\beta _{1}-\beta _{5}+\beta _{6})q^{2}+(-\beta _{2}-\beta _{7})q^{4}+\cdots\)
648.2.l.e	$648$	$2$	648.l	72.l	$6$	$8$	$4$	$5.174$	8.0.170772624.1	None					216.2.f.b	$4$	$0$	\(3\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{2}+\beta _{2}q^{4}+(\beta _{3}+\beta _{7})q^{5}+(2+\cdots)q^{7}+\cdots\)
648.2.l.f	$648$	$2$	648.l	72.l	$6$	$16$	$8$	$5.174$	16.0.\(\cdots\).1	None					216.2.f.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{6}+\beta _{12})q^{2}+(1-\beta _{3}-\beta _{8})q^{4}+\cdots\)
648.2.l.g	$648$	$2$	648.l	72.l	$6$	$48$	$24$	$5.174$		None		✓	✓		648.2.f.c	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{6}]$

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

\( S_{2}^{\mathrm{old}}(648, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(216, [\chi])\)\(^{\oplus 2}\)