Space of modular forms of level 216, weight 4, and Character 216.d

• Label: 216.4.d
• Level: $216$
• Weight: $4$
• Character orbit: 216.d
• Rep. character: $\chi_{216}(109,\cdot)$
• Character field: $\Q$
• Dimension: $48$
• Newform subspaces: $4$
• Sturm bound: $144$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 216 = 2^{3} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	216.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(144\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(216, [\chi])\).

	Total	New	Old
Modular forms	114	48	66
Cusp forms	102	48	54
Eisenstein series	12	0	12

Trace form

48 * q - 6 * q^4 - 66 * q^10 - 270 * q^16 - 30 * q^22 - 1200 * q^25 - 534 * q^28 + 264 * q^31 + 516 * q^34 + 882 * q^40 - 756 * q^46 + 1992 * q^49 - 696 * q^52 - 144 * q^55 + 516 * q^58 + 6 * q^64 - 2298 * q^70 - 216 * q^73 - 1968 * q^76 - 456 * q^79 + 1140 * q^82 + 1110 * q^88 + 3552 * q^94 + 1536 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(216, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
216.4.d.a	$216$	$4$	216.d	8.b	$2$	$4$	$4$	$12.744$	\(\Q(\zeta_{8})\)	\(\Q(\sqrt{-6}) \)		✓			216.4.d.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(68\)	$2\cdot 3^{4}$	$\mathrm{U}(1)[D_{2}]$	\(q-2\beta_{2} q^{2}-8 q^{4}+(-7\beta_{2}+\beta_1)q^{5}+\cdots\)
216.4.d.b	$216$	$4$	216.d	8.b	$2$	$4$	$4$	$12.744$	\(\Q(i, \sqrt{7})\)	None		✓			216.4.d.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(100\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}+\beta _{2})q^{2}+(6+2\beta _{3})q^{4}-5\beta _{1}q^{5}+\cdots\)
216.4.d.c	$216$	$4$	216.d	8.b	$2$	$16$	$16$	$12.744$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		✓			216.4.d.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-168\)	$2^{14}\cdot 3^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+\beta _{13}q^{5}+(-11+\cdots)q^{7}+\cdots\)
216.4.d.d	$216$	$4$	216.d	8.b	$2$	$24$	$24$	$12.744$		None		✓	✓		216.4.d.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{4}^{\mathrm{old}}(216, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(216, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(8, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 2}\)