Space of modular forms of level 648, weight 3, and Character 648.m

• Label: 648.3.m
• Level: $648$
• Weight: $3$
• Character orbit: 648.m
• Rep. character: $\chi_{648}(377,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $48$
• Newform subspaces: $7$
• Sturm bound: $324$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	648.m (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(324\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(648, [\chi])\).

	Total	New	Old
Modular forms	480	48	432
Cusp forms	384	48	336
Eisenstein series	96	0	96

Trace form

\( 48 q - 24 q^{19} + 120 q^{25} - 120 q^{31} + 180 q^{43} - 216 q^{49} + 120 q^{55} + 96 q^{61} + 84 q^{67} + 72 q^{73} + 300 q^{79} + 24 q^{85} + 960 q^{91} + 132 q^{97}+O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(648, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
648.3.m.a	$648$	$3$	648.m	9.d	$6$	$4$	$2$	$17.657$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None					72.3.e.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-24\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+5\beta _{1}q^{5}-12\beta _{2}q^{7}+(4\beta _{1}-4\beta _{3})q^{11}+\cdots\)
648.3.m.b	$648$	$3$	648.m	9.d	$6$	$4$	$2$	$17.657$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None					216.3.e.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-6\)	$2^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{5}-3\beta _{2}q^{7}+(-\beta _{1}+\beta _{3})q^{11}+\cdots\)
648.3.m.c	$648$	$3$	648.m	9.d	$6$	$4$	$2$	$17.657$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None					216.3.e.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(6\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{5}+3\beta _{2}q^{7}+(-7\beta _{1}+7\beta _{3})q^{11}+\cdots\)
648.3.m.d	$648$	$3$	648.m	9.d	$6$	$4$	$2$	$17.657$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None					24.3.e.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(12\)	$2^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{5}+6\beta _{2}q^{7}+(-\beta _{1}+\beta _{3})q^{11}+\cdots\)
648.3.m.e	$648$	$3$	648.m	9.d	$6$	$8$	$4$	$17.657$	\(\Q(\zeta_{24})\)	None		✓			648.3.e.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-12\)	$3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta_{6} q^{5}+(\beta_{3}-3\beta_{2})q^{7}+(-\beta_{7}+\beta_{6}+\cdots-\beta_1)q^{11}+\cdots\)
648.3.m.f	$648$	$3$	648.m	9.d	$6$	$8$	$4$	$17.657$	\(\Q(\zeta_{24})\)	None					216.3.e.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(12\)	$2^{6}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\beta_{4}-\beta_{3}+\beta_1)q^{5}+(-\beta_{7}+\beta_{6}-3\beta_{2}+3)q^{7}+\cdots\)
648.3.m.g	$648$	$3$	648.m	9.d	$6$	$16$	$8$	$17.657$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		✓	✓		648.3.e.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(12\)	$2^{4}\cdot 3^{12}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{11}+\beta _{12})q^{5}+(2+2\beta _{2}-\beta _{7})q^{7}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(648, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(648, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(81, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(162, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(216, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(324, [\chi])\)\(^{\oplus 2}\)