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

• Label: 288.7.e
• Level: $288$
• Weight: $7$
• Character orbit: 288.e
• Rep. character: $\chi_{288}(161,\cdot)$
• Character field: $\Q$
• Dimension: $24$
• Newform subspaces: $8$
• Sturm bound: $336$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	288.e (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 3 \)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(336\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

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

	Total	New	Old
Modular forms	304	24	280
Cusp forms	272	24	248
Eisenstein series	32	0	32

Trace form

\( 24 q + O(q^{10}) \)

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	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
288.7.e.a	$288$	$7$	288.e	3.b	$2$	$2$	$2$	$66.256$	\(\Q(\sqrt{-2}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-176\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+119\beta q^{5}-88q^{7}-872\beta q^{11}+472q^{13}+\cdots\)
288.7.e.b	$288$	$7$	288.e	3.b	$2$	$2$	$2$	$66.256$	\(\Q(\sqrt{-2}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+161\beta q^{5}-1656q^{13}+5383\beta q^{17}+\cdots\)
288.7.e.c	$288$	$7$	288.e	3.b	$2$	$2$	$2$	$66.256$	\(\Q(\sqrt{-2}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+73\beta q^{5}+1656q^{13}-4393\beta q^{17}+\cdots\)
288.7.e.d	$288$	$7$	288.e	3.b	$2$	$2$	$2$	$66.256$	\(\Q(\sqrt{-2}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(176\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+119\beta q^{5}+88q^{7}+872\beta q^{11}+472q^{13}+\cdots\)
288.7.e.e	$288$	$7$	288.e	3.b	$2$	$4$	$4$	$66.256$	\(\Q(\sqrt{-2}, \sqrt{-41})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-32\)	$2^{10}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+65\beta _{1}q^{5}+(-8-\beta _{2})q^{7}+(-520\beta _{1}+\cdots)q^{11}+\cdots\)
288.7.e.f	$288$	$7$	288.e	3.b	$2$	$4$	$4$	$66.256$	\(\Q(\sqrt{-2}, \sqrt{7})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{9}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+95\beta _{1}q^{5}+\beta _{3}q^{7}-\beta _{2}q^{11}-2040q^{13}+\cdots\)
288.7.e.g	$288$	$7$	288.e	3.b	$2$	$4$	$4$	$66.256$	\(\Q(\sqrt{-2}, \sqrt{3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{9}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-55\beta _{1}q^{5}-\beta _{3}q^{7}+7\beta _{2}q^{11}+504q^{13}+\cdots\)
288.7.e.h	$288$	$7$	288.e	3.b	$2$	$4$	$4$	$66.256$	\(\Q(\sqrt{-2}, \sqrt{-41})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(32\)	$2^{10}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+65\beta _{1}q^{5}+(8+\beta _{2})q^{7}+(520\beta _{1}+\beta _{3})q^{11}+\cdots\)

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

\( S_{7}^{\mathrm{old}}(288, [\chi]) \cong \) \(S_{7}^{\mathrm{new}}(3, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(6, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(144, [\chi])\)\(^{\oplus 2}\)