Space of modular forms of level 576, weight 6, and Character 576.d

• Label: 576.6.d
• Level: $576$
• Weight: $6$
• Character orbit: 576.d
• Rep. character: $\chi_{576}(289,\cdot)$
• Character field: $\Q$
• Dimension: $50$
• Newform subspaces: $9$
• Sturm bound: $576$
• Trace bound: $25$

Defining parameters

Level:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	576.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(576\)
Trace bound:			\(25\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(576, [\chi])\).

	Total	New	Old
Modular forms	504	50	454
Cusp forms	456	50	406
Eisenstein series	48	0	48

Trace form

\( 50 q + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(576, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
576.6.d.a	$576$	$6$	576.d	8.b	$2$	$2$	$2$	$92.381$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-2}) \)			64.6.b.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+237iq^{11}-1914q^{17}-1441iq^{19}+\cdots\)
576.6.d.b	$576$	$6$	576.d	8.b	$2$	$4$	$4$	$92.381$	\(\Q(\zeta_{12})\)	\(\Q(\sqrt{-3}) \)		✓	576.6.d.b	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}$	$\mathrm{U}(1)[D_{2}]$	\(q+31\zeta_{12}q^{7}-29\zeta_{12}^{2}q^{13}+179\zeta_{12}^{3}q^{19}+\cdots\)
576.6.d.c	$576$	$6$	576.d	8.b	$2$	$4$	$4$	$92.381$	\(\Q(\zeta_{12})\)	None			192.6.d.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q-11\zeta_{12}^{2}q^{5}-\zeta_{12}q^{7}+60\zeta_{12}^{3}q^{11}+\cdots\)
576.6.d.d	$576$	$6$	576.d	8.b	$2$	$4$	$4$	$92.381$	\(\Q(i, \sqrt{51})\)	None			192.6.d.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{5}-5\beta _{2}q^{7}-60\beta _{1}q^{11}-4\beta _{3}q^{13}+\cdots\)
576.6.d.e	$576$	$6$	576.d	8.b	$2$	$4$	$4$	$92.381$	\(\Q(i, \sqrt{19})\)	None			192.6.d.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{5}+\beta _{2}q^{7}-24\beta _{1}q^{11}+8\beta _{3}q^{13}+\cdots\)
576.6.d.f	$576$	$6$	576.d	8.b	$2$	$8$	$8$	$92.381$	8.0.\(\cdots\).12	None			192.6.d.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{28}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{4}q^{5}+(-2\beta _{2}+3\beta _{5})q^{7}+(12\beta _{3}+\cdots)q^{11}+\cdots\)
576.6.d.g	$576$	$6$	576.d	8.b	$2$	$8$	$8$	$92.381$	8.0.\(\cdots\).2	None		✓	576.6.d.g	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{32}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{4}q^{5}-55\beta _{1}q^{7}+\beta _{5}q^{11}-57\beta _{2}q^{13}+\cdots\)
576.6.d.h	$576$	$6$	576.d	8.b	$2$	$8$	$8$	$92.381$	8.0.\(\cdots\).24	None		✓	576.6.d.h	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{30}\cdot 3^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{5}-\beta _{7}q^{7}+\beta _{3}q^{11}-5\beta _{6}q^{13}+\cdots\)
576.6.d.i	$576$	$6$	576.d	8.b	$2$	$8$	$8$	$92.381$	8.0.\(\cdots\).15	None			64.6.b.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{42}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{5}-\beta _{4}q^{7}+(\beta _{5}+9\beta _{6})q^{11}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(576, [\chi])\) into lower level spaces

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