Space of modular forms of level 576, weight 7, and Character 576.b

• Label: 576.7.b
• Level: $576$
• Weight: $7$
• Character orbit: 576.b
• Rep. character: $\chi_{576}(415,\cdot)$
• Character field: $\Q$
• Dimension: $60$
• Newform subspaces: $8$
• Sturm bound: $672$
• Trace bound: $25$

Defining parameters

Level:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	576.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(672\)
Trace bound:			\(25\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(576, [\chi])\).

	Total	New	Old
Modular forms	600	60	540
Cusp forms	552	60	492
Eisenstein series	48	0	48

Trace form

\( 60 q + O(q^{10}) \)

Decomposition of \(S_{7}^{\mathrm{new}}(576, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
576.7.b.a	$576$	$7$	576.b	8.d	$2$	$4$	$4$	$132.511$	\(\Q(i, \sqrt{6})\)	\(\Q(\sqrt{-6}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{7}\cdot 3^{2}\cdot 7^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta _{2}q^{5}+235\beta _{1}q^{7}+\beta _{3}q^{11}-26711q^{25}+\cdots\)
576.7.b.b	$576$	$7$	576.b	8.d	$2$	$4$	$4$	$132.511$	\(\Q(\zeta_{12})\)	\(\Q(\sqrt{-3}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}\cdot 3^{4}\cdot 5^{2}\cdot 7^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+143\zeta_{12}q^{7}+\zeta_{12}^{2}q^{13}+\zeta_{12}^{3}q^{19}+\cdots\)
576.7.b.c	$576$	$7$	576.b	8.d	$2$	$4$	$4$	$132.511$	\(\Q(i, \sqrt{11})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+5\beta _{3}q^{5}+11\beta _{2}q^{7}+79\beta _{1}q^{11}+\cdots\)
576.7.b.d	$576$	$7$	576.b	8.d	$2$	$8$	$8$	$132.511$	8.0.\(\cdots\).4	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{24}\cdot 3^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{4}q^{5}+(-3\beta _{3}+\beta _{5})q^{7}+(-\beta _{1}+\cdots)q^{11}+\cdots\)
576.7.b.e	$576$	$7$	576.b	8.d	$2$	$8$	$8$	$132.511$	8.0.\(\cdots\).9	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{40}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{5}+(-\beta _{6}+\beta _{7})q^{7}+(7\beta _{1}+24\beta _{4}+\cdots)q^{11}+\cdots\)
576.7.b.f	$576$	$7$	576.b	8.d	$2$	$8$	$8$	$132.511$	8.0.\(\cdots\).10	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{24}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-10\beta _{1}+5\beta _{3})q^{5}+(-\beta _{5}+6\beta _{7})q^{7}+\cdots\)
576.7.b.g	$576$	$7$	576.b	8.d	$2$	$8$	$8$	$132.511$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{24}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-4\beta _{2}-\beta _{3})q^{5}+(-\beta _{4}+21\beta _{6}+\cdots)q^{7}+\cdots\)
576.7.b.h	$576$	$7$	576.b	8.d	$2$	$16$	$16$	$132.511$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{114}\cdot 3^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{9}q^{5}+(-\beta _{7}-\beta _{10})q^{7}-\beta _{2}q^{11}+\cdots\)

Decomposition of \(S_{7}^{\mathrm{old}}(576, [\chi])\) into lower level spaces

\( S_{7}^{\mathrm{old}}(576, [\chi]) \cong \) \(S_{7}^{\mathrm{new}}(8, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(32, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(144, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(288, [\chi])\)\(^{\oplus 2}\)