Space of modular forms of level 288, weight 6, and Character 288.d

• Label: 288.6.d
• Level: $288$
• Weight: $6$
• Character orbit: 288.d
• Rep. character: $\chi_{288}(145,\cdot)$
• Character field: $\Q$
• Dimension: $24$
• Newform subspaces: $4$
• Sturm bound: $288$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	288.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(288\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(288, [\chi])\).

	Total	New	Old
Modular forms	256	26	230
Cusp forms	224	24	200
Eisenstein series	32	2	30

Trace form

24 * q - 96 * q^7 + 204 * q^17 + 4008 * q^23 - 10944 * q^25 - 1392 * q^31 + 7044 * q^41 + 10536 * q^47 + 28584 * q^49 + 24576 * q^55 - 32016 * q^65 - 65544 * q^71 - 65160 * q^73 - 7104 * q^79 - 90948 * q^89 - 16560 * q^95 + 179976 * q^97

Decomposition of \(S_{6}^{\mathrm{new}}(288, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
288.6.d.a	$288$	$6$	288.d	8.b	$2$	$2$	$2$	$46.191$	\(\Q(\sqrt{-2}) \)	\(\Q(\sqrt{-6}) \)					72.6.d.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-484\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+11\beta q^{5}-242q^{7}+262\beta q^{11}+2157q^{25}+\cdots\)
288.6.d.b	$288$	$6$	288.d	8.b	$2$	$4$	$4$	$46.191$	\(\Q(\sqrt{-54 +2 \sqrt{73}})\)	None					8.6.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-96\)	$2^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{5}+(-24-\beta _{3})q^{7}+(5\beta _{1}+8\beta _{2}+\cdots)q^{11}+\cdots\)
288.6.d.c	$288$	$6$	288.d	8.b	$2$	$8$	$8$	$46.191$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					72.6.d.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(288\)	$2^{30}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{5}+(6^{2}-\beta _{3})q^{7}+(-5\beta _{1}+2\beta _{2}+\cdots)q^{11}+\cdots\)
288.6.d.d	$288$	$6$	288.d	8.b	$2$	$10$	$10$	$46.191$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None			✓		24.6.d.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(196\)	$2^{44}\cdot 3^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{5}+(20-\beta _{5})q^{7}+(\beta _{1}-3\beta _{3}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(288, [\chi])\) into lower level spaces

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