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

• Label: 648.2.i
• Level: $648$
• Weight: $2$
• Character orbit: 648.i
• Rep. character: $\chi_{648}(217,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $24$
• Newform subspaces: $10$
• Sturm bound: $216$
• Trace bound: $7$

Defining parameters

Level:	$N$	$=$	$648 = 2^{3} \cdot 3^{4}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	648.i (of order $3$ and degree $2$)
Character conductor:	$\operatorname{cond}(\chi)$	$=$	$9$
Character field:			$\Q(\zeta_{3})$
Newform subspaces:			$10$
Sturm bound:			$216$
Trace bound:			$7$
Distinguishing $T_p$:			$5$, $7$

Dimensions

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

	Total	New	Old
Modular forms	264	24	240
Cusp forms	168	24	144
Eisenstein series	96	0	96

Trace form

$24 q - 12 q^{19} - 12 q^{25} + 30 q^{31} + 36 q^{43} + 60 q^{55} + 12 q^{61} + 6 q^{67} + 36 q^{73} - 30 q^{79} - 6 q^{85} - 60 q^{91} - 24 q^{97}+O(q^{100})$

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.i.a	$648$	$2$	648.i	9.c	$3$	$2$	$1$	$5.174$	$\Q(\sqrt{-3})$	None					216.2.a.a	$2$	$0$	$0$	$0$	$-4$	$3$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q-4\zeta_{6}q^{5}+(3-3\zeta_{6})q^{7}+(-4+4\zeta_{6})q^{11}+\cdots$
648.2.i.b	$648$	$2$	648.i	9.c	$3$	$2$	$1$	$5.174$	$\Q(\sqrt{-3})$	None					24.2.a.a	$2$	$0$	$0$	$0$	$-2$	$0$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q-2\zeta_{6}q^{5}+(4-4\zeta_{6})q^{11}+2\zeta_{6}q^{13}+\cdots$
648.2.i.c	$648$	$2$	648.i	9.c	$3$	$2$	$1$	$5.174$	$\Q(\sqrt{-3})$	None					216.2.a.b	$2$	$0$	$0$	$0$	$-1$	$-3$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q-\zeta_{6}q^{5}+(-3+3\zeta_{6})q^{7}+(5-5\zeta_{6})q^{11}+\cdots$
648.2.i.d	$648$	$2$	648.i	9.c	$3$	$2$	$1$	$5.174$	$\Q(\sqrt{-3})$	None		✓			648.2.a.b	$2$	$0$	$0$	$0$	$-1$	$0$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q-\zeta_{6}q^{5}+(-4+4\zeta_{6})q^{11}+5\zeta_{6}q^{13}+\cdots$
648.2.i.e	$648$	$2$	648.i	9.c	$3$	$2$	$1$	$5.174$	$\Q(\sqrt{-3})$	None					216.2.a.b	$2$	$0$	$0$	$0$	$1$	$-3$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+\zeta_{6}q^{5}+(-3+3\zeta_{6})q^{7}+(-5+5\zeta_{6})q^{11}+\cdots$
648.2.i.f	$648$	$2$	648.i	9.c	$3$	$2$	$1$	$5.174$	$\Q(\sqrt{-3})$	None		✓			648.2.a.b	$2$	$0$	$0$	$0$	$1$	$0$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+\zeta_{6}q^{5}+(4-4\zeta_{6})q^{11}+5\zeta_{6}q^{13}+\cdots$
648.2.i.g	$648$	$2$	648.i	9.c	$3$	$2$	$1$	$5.174$	$\Q(\sqrt{-3})$	None					24.2.a.a	$2$	$0$	$0$	$0$	$2$	$0$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+2\zeta_{6}q^{5}+(-4+4\zeta_{6})q^{11}+2\zeta_{6}q^{13}+\cdots$
648.2.i.h	$648$	$2$	648.i	9.c	$3$	$2$	$1$	$5.174$	$\Q(\sqrt{-3})$	None					216.2.a.a	$2$	$0$	$0$	$0$	$4$	$3$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+4\zeta_{6}q^{5}+(3-3\zeta_{6})q^{7}+(4-4\zeta_{6})q^{11}+\cdots$
648.2.i.i	$648$	$2$	648.i	9.c	$3$	$4$	$2$	$5.174$	$\Q(\zeta_{12})$	None		✓			648.2.a.e	$2$	$0$	$0$	$0$	$-4$	$0$	$3$	$\mathrm{SU}(2)[C_{3}]$	$q+(\beta_{2}-2\beta_1)q^{5}+(2\beta_{3}-2\beta_{2})q^{7}+\cdots$
648.2.i.j	$648$	$2$	648.i	9.c	$3$	$4$	$2$	$5.174$	$\Q(\zeta_{12})$	None		✓			648.2.a.e	$2$	$0$	$0$	$0$	$4$	$0$	$3$	$\mathrm{SU}(2)[C_{3}]$	$q+(\beta_{3}-\beta_{2}-2\beta_1+2)q^{5}-2\beta_{2} q^{7}+\cdots$

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

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