Space of modular forms of level 288, weight 2, and Character 288.i

• Label: 288.2.i
• Level: $288$
• Weight: $2$
• Character orbit: 288.i
• Rep. character: $\chi_{288}(97,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $24$
• Newform subspaces: $6$
• Sturm bound: $96$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	288.i (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(96\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(288, [\chi])\).

	Total	New	Old
Modular forms	112	24	88
Cusp forms	80	24	56
Eisenstein series	32	0	32

Trace form

\( 24 q + 4 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(288, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
288.2.i.a	$288$	$2$	288.i	9.c	$3$	$2$	$1$	$2.300$	\(\Q(\sqrt{-3}) \)	None		$2$	$1$	\(0\)	\(-3\)	\(-4\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1-\zeta_{6})q^{3}-4\zeta_{6}q^{5}+(-2+2\zeta_{6})q^{7}+\cdots\)
288.2.i.b	$288$	$2$	288.i	9.c	$3$	$2$	$1$	$2.300$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(3\)	\(-4\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\zeta_{6})q^{3}-4\zeta_{6}q^{5}+(2-2\zeta_{6})q^{7}+\cdots\)
288.2.i.c	$288$	$2$	288.i	9.c	$3$	$4$	$2$	$2.300$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		$2$	$0$	\(0\)	\(-4\)	\(2\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\beta _{3})q^{3}+\beta _{2}q^{5}+(1-\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
288.2.i.d	$288$	$2$	288.i	9.c	$3$	$4$	$2$	$2.300$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q-\zeta_{12}^{3}q^{3}+(1-\zeta_{12})q^{5}+\zeta_{12}^{2}q^{7}+\cdots\)
288.2.i.e	$288$	$2$	288.i	9.c	$3$	$4$	$2$	$2.300$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		$2$	$0$	\(0\)	\(4\)	\(2\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\beta _{3})q^{3}+(1-\beta _{2})q^{5}+(-\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
288.2.i.f	$288$	$2$	288.i	9.c	$3$	$8$	$4$	$2.300$	8.0.170772624.1	None		$4$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{1}+\beta _{5})q^{3}+(1-\beta _{4}+\beta _{7})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(288, [\chi])\) into lower level spaces

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