Space of modular forms of level 288, weight 9, and Character 288.g

• Label: 288.9.g
• Level: $288$
• Weight: $9$
• Character orbit: 288.g
• Rep. character: $\chi_{288}(127,\cdot)$
• Character field: $\Q$
• Dimension: $40$
• Newform subspaces: $6$
• Sturm bound: $432$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
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:			\(432\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(288, [\chi])\).

	Total	New	Old
Modular forms	400	40	360
Cusp forms	368	40	328
Eisenstein series	32	0	32

Trace form

40 * q + 336 * q^5 - 22832 * q^13 - 88368 * q^17 + 1956408 * q^25 + 1333968 * q^29 - 4894320 * q^37 + 4242768 * q^41 - 24791256 * q^49 - 21302832 * q^53 + 19165840 * q^61 - 3167712 * q^65 + 52357008 * q^73 + 91848192 * q^77 + 63365472 * q^85 + 23733840 * q^89 - 32739440 * q^97

Decomposition of \(S_{9}^{\mathrm{new}}(288, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
288.9.g.a	$288$	$9$	288.g	4.b	$2$	$4$	$4$	$117.325$	\(\Q(i, \sqrt{19})\)	None					32.9.c.b	$2$	$0$	\(0\)	\(0\)	\(-1064\)	\(0\)	$2^{18}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-266+\beta _{2})q^{5}+(-7\beta _{1}-14\beta _{3})q^{7}+\cdots\)
288.9.g.b	$288$	$9$	288.g	4.b	$2$	$4$	$4$	$117.325$	\(\Q(i, \sqrt{39})\)	None					32.9.c.a	$2$	$0$	\(0\)	\(0\)	\(728\)	\(0\)	$2^{18}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(182+\beta _{2})q^{5}+(17\beta _{1}-22\beta _{3})q^{7}+\cdots\)
288.9.g.c	$288$	$9$	288.g	4.b	$2$	$8$	$8$	$117.325$	8.0.3468738816.6	None					96.9.g.b	$2$	$0$	\(0\)	\(0\)	\(-560\)	\(0\)	$2^{43}\cdot 3^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-70-\beta _{4})q^{5}+(7\beta _{1}-7\beta _{3})q^{7}+\cdots\)
288.9.g.d	$288$	$9$	288.g	4.b	$2$	$8$	$8$	$117.325$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		✓			288.9.g.d	$2$	$0$	\(0\)	\(0\)	\(-448\)	\(0\)	$2^{48}\cdot 3^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-56-\beta _{3})q^{5}+\beta _{5}q^{7}+(3\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots\)
288.9.g.e	$288$	$9$	288.g	4.b	$2$	$8$	$8$	$117.325$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		✓			288.9.g.d	$2$	$0$	\(0\)	\(0\)	\(448\)	\(0\)	$2^{48}\cdot 3^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(56+\beta _{3})q^{5}+\beta _{5}q^{7}+(-3\beta _{1}+\beta _{2}+\cdots)q^{11}+\cdots\)
288.9.g.f	$288$	$9$	288.g	4.b	$2$	$8$	$8$	$117.325$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					96.9.g.a	$2$	$0$	\(0\)	\(0\)	\(1232\)	\(0\)	$2^{43}\cdot 3^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(154-\beta _{3})q^{5}+(4\beta _{1}+9\beta _{2}+\beta _{5}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{9}^{\mathrm{old}}(288, [\chi])\) into lower level spaces

\( S_{9}^{\mathrm{old}}(288, [\chi]) \simeq \) \(S_{9}^{\mathrm{new}}(4, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(32, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(144, [\chi])\)\(^{\oplus 2}\)