Space of modular forms of level 576, weight 8, and Character 576.d

• Label: 576.8.d
• Level: $576$
• Weight: $8$
• Character orbit: 576.d
• Rep. character: $\chi_{576}(289,\cdot)$
• Character field: $\Q$
• Dimension: $70$
• Newform subspaces: $10$
• Sturm bound: $768$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	576.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(768\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(576, [\chi])\).

	Total	New	Old
Modular forms	696	70	626
Cusp forms	648	70	578
Eisenstein series	48	0	48

Trace form

\( 70 q + O(q^{10}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(576, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
576.8.d.a	$576$	$8$	576.d	8.b	$2$	$2$	$2$	$179.934$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-2}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+4407iq^{11}+22182q^{17}+29861iq^{19}+\cdots\)
576.8.d.b	$576$	$8$	576.d	8.b	$2$	$4$	$4$	$179.934$	\(\Q(i, \sqrt{11})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-5\beta _{3}q^{5}+13\beta _{2}q^{7}-948\beta _{1}q^{11}+\cdots\)
576.8.d.c	$576$	$8$	576.d	8.b	$2$	$4$	$4$	$179.934$	\(\Q(i, \sqrt{291})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{5}+11\beta _{2}q^{7}+312\beta _{1}q^{11}+\cdots\)
576.8.d.d	$576$	$8$	576.d	8.b	$2$	$4$	$4$	$179.934$	\(\Q(\zeta_{12})\)	\(\Q(\sqrt{-3}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}$	$\mathrm{U}(1)[D_{2}]$	\(q-503\zeta_{12}q^{7}-883\zeta_{12}^{2}q^{13}-7181\zeta_{12}^{3}q^{19}+\cdots\)
576.8.d.e	$576$	$8$	576.d	8.b	$2$	$4$	$4$	$179.934$	\(\Q(i, \sqrt{435})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{5}+\beta _{2}q^{7}+2379\beta _{1}q^{11}+13\beta _{3}q^{13}+\cdots\)
576.8.d.f	$576$	$8$	576.d	8.b	$2$	$8$	$8$	$179.934$	8.0.\(\cdots\).10	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{42}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{5}-\beta _{7}q^{7}+(-5^{2}\beta _{5}-71\beta _{6}+\cdots)q^{11}+\cdots\)
576.8.d.g	$576$	$8$	576.d	8.b	$2$	$8$	$8$	$179.934$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{32}\cdot 3^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{5}-\beta _{5}q^{7}+\beta _{6}q^{11}-\beta _{2}q^{13}+\cdots\)
576.8.d.h	$576$	$8$	576.d	8.b	$2$	$8$	$8$	$179.934$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{28}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-8\beta _{1}+\beta _{3})q^{5}+(5\beta _{2}-\beta _{4})q^{7}+\cdots\)
576.8.d.i	$576$	$8$	576.d	8.b	$2$	$12$	$12$	$179.934$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{64}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{5}-\beta _{9}q^{7}+(316\beta _{3}-\beta _{10})q^{11}+\cdots\)
576.8.d.j	$576$	$8$	576.d	8.b	$2$	$16$	$16$	$179.934$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{110}\cdot 3^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{5}+(4\beta _{1}+\beta _{5})q^{7}-\beta _{9}q^{11}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(576, [\chi])\) into lower level spaces

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