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

• Label: 288.5.g
• Level: $288$
• Weight: $5$
• Character orbit: 288.g
• Rep. character: $\chi_{288}(127,\cdot)$
• Character field: $\Q$
• Dimension: $20$
• Newform subspaces: $6$
• Sturm bound: $240$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	288.g (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 4 \)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(240\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

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

	Total	New	Old
Modular forms	208	20	188
Cusp forms	176	20	156
Eisenstein series	32	0	32

Trace form

20 * q - 24 * q^5 - 472 * q^13 - 408 * q^17 + 2268 * q^25 - 216 * q^29 + 1800 * q^37 - 984 * q^41 - 8172 * q^49 + 5928 * q^53 - 4600 * q^61 + 4368 * q^65 + 5448 * q^73 - 8448 * q^77 - 9168 * q^85 - 9816 * q^89 + 27080 * q^97

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.g.a	$288$	$5$	288.g	4.b	$2$	$2$	$2$	$29.771$	\(\Q(\sqrt{-1}) \)	None					32.5.c.b	$2$	$0$	\(0\)	\(0\)	\(-52\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-26 q^{5}+22\beta q^{7}+25\beta q^{11}-38 q^{13}+\cdots\)
288.5.g.b	$288$	$5$	288.g	4.b	$2$	$2$	$2$	$29.771$	\(\Q(\sqrt{-1}) \)	None					32.5.c.a	$2$	$0$	\(0\)	\(0\)	\(76\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+38 q^{5}-2\beta q^{7}+53\beta q^{11}+154 q^{13}+\cdots\)
288.5.g.c	$288$	$5$	288.g	4.b	$2$	$4$	$4$	$29.771$	\(\Q(\zeta_{12})\)	None					96.5.g.b	$2$	$0$	\(0\)	\(0\)	\(-88\)	\(0\)	$2^{8}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta_{2}-22)q^{5}+(\beta_{3}-\beta_1)q^{7}+\cdots\)
288.5.g.d	$288$	$5$	288.g	4.b	$2$	$4$	$4$	$29.771$	\(\Q(i, \sqrt{5})\)	None		✓			288.5.g.d	$2$	$0$	\(0\)	\(0\)	\(-32\)	\(0\)	$2^{10}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-8+\beta _{3})q^{5}+(-3\beta _{1}-\beta _{2})q^{7}+\cdots\)
288.5.g.e	$288$	$5$	288.g	4.b	$2$	$4$	$4$	$29.771$	\(\Q(i, \sqrt{5})\)	None		✓			288.5.g.d	$2$	$0$	\(0\)	\(0\)	\(32\)	\(0\)	$2^{10}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(8-\beta _{3})q^{5}+(3\beta _{1}+\beta _{2})q^{7}+(9\beta _{1}+\cdots)q^{11}+\cdots\)
288.5.g.f	$288$	$5$	288.g	4.b	$2$	$4$	$4$	$29.771$	\(\Q(\zeta_{12})\)	None					96.5.g.a	$2$	$0$	\(0\)	\(0\)	\(40\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta_{2}+10)q^{5}+(5\beta_{3}+11\beta_1)q^{7}+\cdots\)

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

\( S_{5}^{\mathrm{old}}(288, [\chi]) \simeq \) \(S_{5}^{\mathrm{new}}(4, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(32, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(144, [\chi])\)\(^{\oplus 2}\)