Space of modular forms of level 288, weight 3, and Character 288.b

• Label: 288.3.b
• Level: $288$
• Weight: $3$
• Character orbit: 288.b
• Rep. character: $\chi_{288}(271,\cdot)$
• Character field: $\Q$
• Dimension: $9$
• Newform subspaces: $3$
• Sturm bound: $144$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	288.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(144\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(288, [\chi])\).

	Total	New	Old
Modular forms	112	11	101
Cusp forms	80	9	71
Eisenstein series	32	2	30

Trace form

\( 9 q - 18 q^{11} + 6 q^{17} - 30 q^{19} - 15 q^{25} + 96 q^{35} + 6 q^{41} + 114 q^{43} - 39 q^{49} - 210 q^{59} - 96 q^{65} - 126 q^{67} + 18 q^{73} + 318 q^{83} + 54 q^{89} + 192 q^{91} + 162 q^{97}+O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(288, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
288.3.b.a	$288$	$3$	288.b	8.d	$2$	$1$	$1$	$7.847$	\(\Q\)	\(\Q(\sqrt{-2}) \)	✓				8.3.d.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+14q^{11}-2q^{17}+34q^{19}+5^{2}q^{25}+\cdots\)
288.3.b.b	$288$	$3$	288.b	8.d	$2$	$4$	$4$	$7.847$	\(\Q(\sqrt{-9 +4 \sqrt{3}})\)	None					24.3.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{5}+(-\beta _{1}-\beta _{2})q^{7}-8q^{11}+\cdots\)
288.3.b.c	$288$	$3$	288.b	8.d	$2$	$4$	$4$	$7.847$	\(\Q(\sqrt{-6}, \sqrt{10})\)	None					72.3.b.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{5}-\beta _{2}q^{7}-\beta _{3}q^{11}+2\beta _{2}q^{13}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(288, [\chi])\) into lower level spaces

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