Space of modular forms of level 288, weight 5, and Character 288.e

• Label: 288.5.e
• Level: $288$
• Weight: $5$
• Character orbit: 288.e
• Rep. character: $\chi_{288}(161,\cdot)$
• Character field: $\Q$
• Dimension: $16$
• Newform subspaces: $5$
• Sturm bound: $240$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	288.e (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 3 \)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(240\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(288, [\chi])\).

	Total	New	Old
Modular forms	208	16	192
Cusp forms	176	16	160
Eisenstein series	32	0	32

Trace form

\( 16 q + 704 q^{13} - 3472 q^{25} + 3040 q^{37} + 8176 q^{49} - 14816 q^{61} + 3840 q^{73} - 7712 q^{85} + 17792 q^{97}+O(q^{100}) \)

Decomposition of \(S_{5}^{\mathrm{new}}(288, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
288.5.e.a	$288$	$5$	288.e	3.b	$2$	$2$	$2$	$29.771$	\(\Q(\sqrt{-2}) \)	\(\Q(\sqrt{-1}) \)		✓			288.5.e.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+17\beta q^{5}-240q^{13}+79\beta q^{17}+47q^{25}+\cdots\)
288.5.e.b	$288$	$5$	288.e	3.b	$2$	$2$	$2$	$29.771$	\(\Q(\sqrt{-2}) \)	\(\Q(\sqrt{-1}) \)		✓			288.5.e.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+31\beta q^{5}+240q^{13}+401\beta q^{17}+\cdots\)
288.5.e.c	$288$	$5$	288.e	3.b	$2$	$4$	$4$	$29.771$	\(\Q(\sqrt{-2}, \sqrt{7})\)	None		✓			288.5.e.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-96\)	$2^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-9\beta _{1}+\beta _{2})q^{5}+(-24+\beta _{3})q^{7}+\cdots\)
288.5.e.d	$288$	$5$	288.e	3.b	$2$	$4$	$4$	$29.771$	\(\Q(\sqrt{-2}, \sqrt{3})\)	None		✓			288.5.e.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{9}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{5}-\beta _{3}q^{7}+\beta _{2}q^{11}+48q^{13}+\cdots\)
288.5.e.e	$288$	$5$	288.e	3.b	$2$	$4$	$4$	$29.771$	\(\Q(\sqrt{-2}, \sqrt{7})\)	None		✓			288.5.e.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(96\)	$2^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-9\beta _{1}+\beta _{2})q^{5}+(24-\beta _{3})q^{7}+(40\beta _{1}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{5}^{\mathrm{old}}(288, [\chi])\) into lower level spaces

\( S_{5}^{\mathrm{old}}(288, [\chi]) \simeq \) \(S_{5}^{\mathrm{new}}(6, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(144, [\chi])\)\(^{\oplus 2}\)