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

• Label: 648.2.n
• Level: $648$
• Weight: $2$
• Character orbit: 648.n
• Rep. character: $\chi_{648}(109,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $92$
• Newform subspaces: $17$
• Sturm bound: $216$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	648.n (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 72 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 17 \)
Sturm bound:			\(216\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\), \(13\), \(17\)

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 + 4 * q^7 + 4 * q^10 + 2 * q^16 - 2 * q^22 + 42 * q^25 - 20 * q^28 + 4 * q^31 + 36 * q^34 + 22 * q^40 + 48 * q^46 - 30 * q^49 - 24 * q^52 + 32 * q^55 + 16 * q^58 - 100 * q^64 + 2 * q^70 - 8 * q^73 - 36 * q^76 + 4 * q^79 - 48 * q^82 - 74 * 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.n.a	$648$	$2$	648.n	72.n	$6$	$4$	$2$	$5.174$	\(\Q(\zeta_{12})\)	None		✓			648.2.d.e	$2$	$1$	\(-4\)	\(0\)	\(-6\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-1+\zeta_{12}^{3})q^{2}-2\zeta_{12}^{3}q^{4}+(-1+\cdots)q^{5}+\cdots\)
648.2.n.b	$648$	$2$	648.n	72.n	$6$	$4$	$2$	$5.174$	\(\Q(\zeta_{12})\)	None		✓			648.2.d.e	$2$	$0$	\(-2\)	\(0\)	\(-6\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+2\zeta_{12}q^{4}+\cdots\)
648.2.n.c	$648$	$2$	648.n	72.n	$6$	$4$	$2$	$5.174$	\(\Q(\zeta_{12})\)	None					24.2.d.a	$4$	$0$	\(-2\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+2\zeta_{12}q^{4}+\cdots\)
648.2.n.d	$648$	$2$	648.n	72.n	$6$	$4$	$2$	$5.174$	\(\Q(\sqrt{-3}, \sqrt{-7})\)	None		✓			648.2.d.b	$4$	$0$	\(-1\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{3}q^{2}+(1+\beta _{1}-\beta _{2})q^{4}+(1-2\beta _{1}+\cdots)q^{5}+\cdots\)
648.2.n.e	$648$	$2$	648.n	72.n	$6$	$4$	$2$	$5.174$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		✓			648.2.d.g	$2$	$1$	\(0\)	\(0\)	\(-6\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{3}q^{2}-2q^{4}+(-2-\beta _{1}+\beta _{2}+\beta _{3})q^{5}+\cdots\)
648.2.n.f	$648$	$2$	648.n	72.n	$6$	$4$	$2$	$5.174$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		✓			648.2.d.g	$2$	$0$	\(0\)	\(0\)	\(6\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{3}q^{2}-2q^{4}+(2-\beta _{1}-\beta _{2}+\beta _{3})q^{5}+\cdots\)
648.2.n.g	$648$	$2$	648.n	72.n	$6$	$4$	$2$	$5.174$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		✓			648.2.d.g	$2$	$0$	\(0\)	\(0\)	\(-6\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{2}+2\beta _{2}q^{4}+(-1-\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
648.2.n.h	$648$	$2$	648.n	72.n	$6$	$4$	$2$	$5.174$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	\(\Q(\sqrt{-6}) \)					72.2.d.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+\beta _{1}q^{2}+2\beta _{2}q^{4}+(-2\beta _{1}+2\beta _{3})q^{5}+\cdots\)
648.2.n.i	$648$	$2$	648.n	72.n	$6$	$4$	$2$	$5.174$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		✓			648.2.d.g	$2$	$0$	\(0\)	\(0\)	\(6\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{2}+2\beta _{2}q^{4}+(1-\beta _{1}+\beta _{2})q^{5}+\cdots\)
648.2.n.j	$648$	$2$	648.n	72.n	$6$	$4$	$2$	$5.174$	\(\Q(\sqrt{-3}, \sqrt{-7})\)	None		✓			648.2.d.b	$4$	$0$	\(1\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{3}q^{2}+(1+\beta _{1}-\beta _{2})q^{4}+(-1+2\beta _{1}+\cdots)q^{5}+\cdots\)
648.2.n.k	$648$	$2$	648.n	72.n	$6$	$4$	$2$	$5.174$	\(\Q(\zeta_{12})\)	None					24.2.d.a	$4$	$0$	\(2\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+2\zeta_{12}q^{4}+\cdots\)
648.2.n.l	$648$	$2$	648.n	72.n	$6$	$4$	$2$	$5.174$	\(\Q(\zeta_{12})\)	None		✓			648.2.d.e	$2$	$0$	\(2\)	\(0\)	\(6\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+2\zeta_{12}q^{4}+\cdots\)
648.2.n.m	$648$	$2$	648.n	72.n	$6$	$4$	$2$	$5.174$	\(\Q(\zeta_{12})\)	None		✓			648.2.d.e	$2$	$0$	\(4\)	\(0\)	\(6\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(1+\zeta_{12}^{3})q^{2}+2\zeta_{12}^{3}q^{4}+(1+2\zeta_{12}+\cdots)q^{5}+\cdots\)
648.2.n.n	$648$	$2$	648.n	72.n	$6$	$8$	$4$	$5.174$	\(\Q(i, \sqrt{3}, \sqrt{7})\)	None					216.2.d.b	$8$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{2}+(-1-\beta _{2}+\beta _{4}+\beta _{6})q^{4}+\cdots\)
648.2.n.o	$648$	$2$	648.n	72.n	$6$	$8$	$4$	$5.174$	\(\Q(i, \sqrt{3}, \sqrt{5})\)	None		✓			648.2.d.f	$8$	$0$	\(0\)	\(0\)	\(0\)	\(16\)	$3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{4}+\beta _{5})q^{2}+\beta _{6}q^{4}+(-\beta _{3}-\beta _{5}+\cdots)q^{5}+\cdots\)
648.2.n.p	$648$	$2$	648.n	72.n	$6$	$8$	$4$	$5.174$	\(\Q(\zeta_{24})\)	\(\Q(\sqrt{-6}) \)					216.2.d.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$2^{2}\cdot 3^{4}$	$\mathrm{U}(1)[D_{6}]$	\(q+\beta_{5} q^{2}+2\beta_{2} q^{4}+(\beta_{4}-\beta_{3}+\beta_1)q^{5}+\cdots\)
648.2.n.q	$648$	$2$	648.n	72.n	$6$	$16$	$8$	$5.174$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None			✓		216.2.d.c	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{5}q^{2}+(-\beta _{2}-\beta _{3})q^{4}+(-\beta _{4}+\beta _{5}+\cdots)q^{5}+\cdots\)

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

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