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

• Label: 288.4.d
• Level: $288$
• Weight: $4$
• Character orbit: 288.d
• Rep. character: $\chi_{288}(145,\cdot)$
• Character field: $\Q$
• Dimension: $14$
• Newform subspaces: $4$
• Sturm bound: $192$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	288.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(192\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\)

Dimensions

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

	Total	New	Old
Modular forms	160	16	144
Cusp forms	128	14	114
Eisenstein series	32	2	30

Trace form

\( 14 q + 16 q^{7} + O(q^{10}) \)

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.d.a	$288$	$4$	288.d	8.b	$2$	$2$	$2$	$16.993$	\(\Q(\sqrt{-7}) \)	None					8.4.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(16\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-2\beta q^{5}+8q^{7}+3\beta q^{11}-10\beta q^{13}+\cdots\)
288.4.d.b	$288$	$4$	288.d	8.b	$2$	$2$	$2$	$16.993$	\(\Q(\sqrt{-2}) \)	\(\Q(\sqrt{-6}) \)					72.4.d.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(68\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+7\beta q^{5}+34q^{7}+2\beta q^{11}-267q^{25}+\cdots\)
288.4.d.c	$288$	$4$	288.d	8.b	$2$	$4$	$4$	$16.993$	\(\Q(\sqrt{-10}, \sqrt{22})\)	None					72.4.d.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-40\)	$2^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{5}-10q^{7}-6\beta _{1}q^{11}+\beta _{3}q^{13}+\cdots\)
288.4.d.d	$288$	$4$	288.d	8.b	$2$	$6$	$6$	$16.993$	6.0.8248384.1	None			✓		24.4.d.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-28\)	$2^{15}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{5}+(-5-\beta _{1})q^{7}+(-2\beta _{2}-2\beta _{3}+\cdots)q^{11}+\cdots\)

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

\( S_{4}^{\mathrm{old}}(288, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(8, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(32, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 2}\)