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

• Label: 648.2.d
• Level: $648$
• Weight: $2$
• Character orbit: 648.d
• Rep. character: $\chi_{648}(325,\cdot)$
• Character field: $\Q$
• Dimension: $44$
• Newform subspaces: $11$
• Sturm bound: $216$
• Trace bound: $14$

Defining parameters

Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	648.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(216\)
Trace bound:			\(14\)
Distinguishing \(T_p\):			\(5\), \(17\)

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 + 4 * q^7 - 8 * q^10 - 10 * q^16 + 22 * q^22 - 24 * q^25 + 16 * q^28 + 4 * q^31 - 6 * q^34 - 20 * q^40 + 24 * q^46 + 24 * q^49 - 28 * q^55 + 16 * q^58 + 2 * q^64 - 4 * q^70 - 8 * q^73 - 30 * q^76 + 52 * q^79 + 54 * q^82 - 2 * q^88 - 48 * 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.d.a	$648$	$2$	648.d	8.b	$2$	$2$	$2$	$5.174$	\(\Q(\sqrt{-1}) \)	None					72.2.n.a	$2$	$0$	\(-2\)	\(0\)	\(0\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(i-1)q^{2}-2 i q^{4}-2 i q^{5}+4 q^{7}+\cdots\)
648.2.d.b	$648$	$2$	648.d	8.b	$2$	$2$	$2$	$5.174$	\(\Q(\sqrt{-7}) \)	None		✓			648.2.d.b	$2$	$1$	\(-1\)	\(0\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta q^{2}+(-2+\beta )q^{4}+(-1+2\beta )q^{5}+\cdots\)
648.2.d.c	$648$	$2$	648.d	8.b	$2$	$2$	$2$	$5.174$	\(\Q(\sqrt{-7}) \)	None		✓			648.2.d.b	$2$	$0$	\(1\)	\(0\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{2}+(-2+\beta )q^{4}+(1-2\beta )q^{5}+\cdots\)
648.2.d.d	$648$	$2$	648.d	8.b	$2$	$2$	$2$	$5.174$	\(\Q(\sqrt{-1}) \)	None					72.2.n.a	$2$	$0$	\(2\)	\(0\)	\(0\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(i+1)q^{2}+2 i q^{4}-2 i q^{5}+4 q^{7}+\cdots\)
648.2.d.e	$648$	$2$	648.d	8.b	$2$	$4$	$4$	$5.174$	\(\Q(\zeta_{12})\)	None		✓			648.2.d.e	$2$	$0$	\(-2\)	\(0\)	\(0\)	\(4\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta_{2}-\beta_1)q^{2}+(\beta_{3}+\beta_1)q^{4}+(-\beta_{3}+2\beta_{2}+\beta_1-1)q^{5}+\cdots\)
648.2.d.f	$648$	$2$	648.d	8.b	$2$	$4$	$4$	$5.174$	\(\Q(\sqrt{3}, \sqrt{-5})\)	None		✓			648.2.d.f	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-16\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+\beta _{2}q^{4}+(\beta _{1}+\beta _{3})q^{5}-4q^{7}+\cdots\)
648.2.d.g	$648$	$2$	648.d	8.b	$2$	$4$	$4$	$5.174$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		✓			648.2.d.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(1+\beta _{3})q^{4}+(\beta _{2}-\beta _{3})q^{5}+\cdots\)
648.2.d.h	$648$	$2$	648.d	8.b	$2$	$4$	$4$	$5.174$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		✓			648.2.d.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(1+\beta _{3})q^{4}+(\beta _{2}+\beta _{3})q^{5}+\cdots\)
648.2.d.i	$648$	$2$	648.d	8.b	$2$	$4$	$4$	$5.174$	\(\Q(\zeta_{12})\)	None		✓			648.2.d.e	$2$	$0$	\(2\)	\(0\)	\(0\)	\(4\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta_{2}+\beta_1)q^{2}+(\beta_{3}+\beta_1)q^{4}+\cdots\)
648.2.d.j	$648$	$2$	648.d	8.b	$2$	$8$	$8$	$5.174$	8.0.\(\cdots\).1	None					72.2.n.b	$2$	$0$	\(-1\)	\(0\)	\(0\)	\(-6\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{4}q^{2}-\beta _{5}q^{4}+(\beta _{1}+\beta _{4}-\beta _{7})q^{5}+\cdots\)
648.2.d.k	$648$	$2$	648.d	8.b	$2$	$8$	$8$	$5.174$	8.0.\(\cdots\).1	None					72.2.n.b	$2$	$0$	\(1\)	\(0\)	\(0\)	\(-6\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{4}q^{2}-\beta _{5}q^{4}+(-\beta _{1}-\beta _{4}+\beta _{7})q^{5}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(648, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(216, [\chi])\)\(^{\oplus 2}\)