Space of modular forms of level 288, weight 7, and Character 288.g

• Label: 288.7.g
• Level: $288$
• Weight: $7$
• Character orbit: 288.g
• Rep. character: $\chi_{288}(127,\cdot)$
• Character field: $\Q$
• Dimension: $30$
• Newform subspaces: $6$
• Sturm bound: $336$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
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:			\(336\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(288, [\chi])\).

	Total	New	Old
Modular forms	304	30	274
Cusp forms	272	30	242
Eisenstein series	32	0	32

Trace form

30 * q + 44 * q^5 + 5868 * q^13 + 14172 * q^17 + 133050 * q^25 + 1324 * q^29 - 41316 * q^37 - 120548 * q^41 - 729666 * q^49 - 327732 * q^53 - 437700 * q^61 - 47848 * q^65 - 25908 * q^73 - 268928 * q^77 + 1671720 * q^85 + 1408668 * q^89 - 710100 * q^97

Decomposition of \(S_{7}^{\mathrm{new}}(288, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
288.7.g.a	$288$	$7$	288.g	4.b	$2$	$2$	$2$	$66.256$	\(\Q(\sqrt{-1}) \)	None					32.7.c.a	$2$	$0$	\(0\)	\(0\)	\(-100\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-50 q^{5}+46\beta q^{7}-527\beta q^{11}+\cdots\)
288.7.g.b	$288$	$7$	288.g	4.b	$2$	$4$	$4$	$66.256$	\(\Q(i, \sqrt{6})\)	None					32.7.c.b	$2$	$0$	\(0\)	\(0\)	\(56\)	\(0\)	$2^{17}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(14+\beta _{3})q^{5}+(2\beta _{1}-5\beta _{2})q^{7}+(-17\beta _{1}+\cdots)q^{11}+\cdots\)
288.7.g.c	$288$	$7$	288.g	4.b	$2$	$4$	$4$	$66.256$	\(\Q(\zeta_{12})\)	None					96.7.g.a	$2$	$0$	\(0\)	\(0\)	\(200\)	\(0\)	$2^{8}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-5\beta_{2}+50)q^{5}+(-5\beta_{3}+73\beta_1)q^{7}+\cdots\)
288.7.g.d	$288$	$7$	288.g	4.b	$2$	$6$	$6$	$66.256$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None		✓			288.7.g.d	$2$	$0$	\(0\)	\(0\)	\(-312\)	\(0\)	$2^{25}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-52-\beta _{1})q^{5}+(\beta _{3}-\beta _{5})q^{7}+(\beta _{3}+\cdots)q^{11}+\cdots\)
288.7.g.e	$288$	$7$	288.g	4.b	$2$	$6$	$6$	$66.256$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None		✓			288.7.g.d	$2$	$0$	\(0\)	\(0\)	\(312\)	\(0\)	$2^{25}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(52+\beta _{1})q^{5}+(\beta _{3}-\beta _{5})q^{7}+(-\beta _{3}+\cdots)q^{11}+\cdots\)
288.7.g.f	$288$	$7$	288.g	4.b	$2$	$8$	$8$	$66.256$	8.0.\(\cdots\).5	None			✓		96.7.g.b	$2$	$0$	\(0\)	\(0\)	\(-112\)	\(0\)	$2^{41}\cdot 3^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-14+\beta _{1})q^{5}+(2\beta _{2}-\beta _{3}-\beta _{5}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{7}^{\mathrm{old}}(288, [\chi])\) into lower level spaces

\( S_{7}^{\mathrm{old}}(288, [\chi]) \simeq \) \(S_{7}^{\mathrm{new}}(4, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(32, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(144, [\chi])\)\(^{\oplus 2}\)