Space of modular forms of level 288, weight 4, and Character 288.i

• Label: 288.4.i
• Level: $288$
• Weight: $4$
• Character orbit: 288.i
• Rep. character: $\chi_{288}(97,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $72$
• Newform subspaces: $5$
• Sturm bound: $192$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	288.i (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(192\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(288, [\chi])\).

	Total	New	Old
Modular forms	304	72	232
Cusp forms	272	72	200
Eisenstein series	32	0	32

Trace form

\( 72 q - 20 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(288, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
288.4.i.a	$288$	$4$	288.i	9.c	$3$	$8$	$4$	$16.993$	8.0.\(\cdots\).1	None		✓			288.4.i.a	$4$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$2^{2}\cdot 3^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{2}-\beta _{6}-\beta _{7})q^{3}+(1-\beta _{3}+\beta _{5}+\cdots)q^{5}+\cdots\)
288.4.i.b	$288$	$4$	288.i	9.c	$3$	$12$	$6$	$16.993$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		✓			288.4.i.b	$4$	$0$	\(0\)	\(0\)	\(-12\)	\(0\)	$2^{8}\cdot 3^{5}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{9}q^{3}+(-2+2\beta _{2}-\beta _{3})q^{5}+(\beta _{7}+\cdots)q^{7}+\cdots\)
288.4.i.c	$288$	$4$	288.i	9.c	$3$	$16$	$8$	$16.993$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		✓			288.4.i.c	$4$	$0$	\(0\)	\(0\)	\(-10\)	\(0\)	$2^{16}\cdot 3^{12}$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{10}q^{3}+(-1+\beta _{1}-\beta _{5}-\beta _{12}+\cdots)q^{5}+\cdots\)
288.4.i.d	$288$	$4$	288.i	9.c	$3$	$18$	$9$	$16.993$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None		✓			288.4.i.d	$2$	$0$	\(0\)	\(-5\)	\(10\)	\(-28\)	$2^{20}\cdot 3^{9}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1-\beta _{2}-\beta _{3}+\beta _{7})q^{3}+(-\beta _{2}+\cdots)q^{5}+\cdots\)
288.4.i.e	$288$	$4$	288.i	9.c	$3$	$18$	$9$	$16.993$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None		✓			288.4.i.d	$2$	$0$	\(0\)	\(5\)	\(10\)	\(28\)	$2^{20}\cdot 3^{9}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\beta _{2}+\beta _{3}-\beta _{7})q^{3}+(-\beta _{2}+\beta _{8}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(288, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(288, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(144, [\chi])\)\(^{\oplus 2}\)