Space of modular forms of level 588, weight 6, and Character 588.i

• Label: 588.6.i
• Level: $588$
• Weight: $6$
• Character orbit: 588.i
• Rep. character: $\chi_{588}(361,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $66$
• Newform subspaces: $17$
• Sturm bound: $672$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	588.i (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 17 \)
Sturm bound:			\(672\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(588, [\chi])\).

	Total	New	Old
Modular forms	1168	66	1102
Cusp forms	1072	66	1006
Eisenstein series	96	0	96

Trace form

\( 66 q - 9 q^{3} - 22 q^{5} - 2673 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(588, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
588.6.i.a	$588$	$6$	588.i	7.c	$3$	$2$	$1$	$94.306$	\(\Q(\sqrt{-3}) \)	None		✓			588.6.a.a	$2$	$0$	\(0\)	\(-9\)	\(-68\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-9+9\zeta_{6})q^{3}-68\zeta_{6}q^{5}-3^{4}\zeta_{6}q^{9}+\cdots\)
588.6.i.b	$588$	$6$	588.i	7.c	$3$	$2$	$1$	$94.306$	\(\Q(\sqrt{-3}) \)	None					84.6.a.a	$2$	$0$	\(0\)	\(-9\)	\(6\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-9+9\zeta_{6})q^{3}+6\zeta_{6}q^{5}-3^{4}\zeta_{6}q^{9}+\cdots\)
588.6.i.c	$588$	$6$	588.i	7.c	$3$	$2$	$1$	$94.306$	\(\Q(\sqrt{-3}) \)	None					84.6.a.b	$2$	$0$	\(0\)	\(-9\)	\(34\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-9+9\zeta_{6})q^{3}+34\zeta_{6}q^{5}-3^{4}\zeta_{6}q^{9}+\cdots\)
588.6.i.d	$588$	$6$	588.i	7.c	$3$	$2$	$1$	$94.306$	\(\Q(\sqrt{-3}) \)	None					84.6.i.a	$2$	$0$	\(0\)	\(9\)	\(-69\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(9-9\zeta_{6})q^{3}-69\zeta_{6}q^{5}-3^{4}\zeta_{6}q^{9}+\cdots\)
588.6.i.e	$588$	$6$	588.i	7.c	$3$	$2$	$1$	$94.306$	\(\Q(\sqrt{-3}) \)	None					84.6.a.b	$2$	$0$	\(0\)	\(9\)	\(-34\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(9-9\zeta_{6})q^{3}-34\zeta_{6}q^{5}-3^{4}\zeta_{6}q^{9}+\cdots\)
588.6.i.f	$588$	$6$	588.i	7.c	$3$	$2$	$1$	$94.306$	\(\Q(\sqrt{-3}) \)	None					84.6.a.a	$2$	$0$	\(0\)	\(9\)	\(-6\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(9-9\zeta_{6})q^{3}-6\zeta_{6}q^{5}-3^{4}\zeta_{6}q^{9}+\cdots\)
588.6.i.g	$588$	$6$	588.i	7.c	$3$	$2$	$1$	$94.306$	\(\Q(\sqrt{-3}) \)	None		✓			588.6.a.a	$2$	$0$	\(0\)	\(9\)	\(68\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(9-9\zeta_{6})q^{3}+68\zeta_{6}q^{5}-3^{4}\zeta_{6}q^{9}+\cdots\)
588.6.i.h	$588$	$6$	588.i	7.c	$3$	$4$	$2$	$94.306$	\(\Q(\sqrt{-3}, \sqrt{505})\)	None					84.6.a.d	$2$	$0$	\(0\)	\(-18\)	\(-78\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-9+9\beta _{1})q^{3}+(-39\beta _{1}-\beta _{2})q^{5}+\cdots\)
588.6.i.i	$588$	$6$	588.i	7.c	$3$	$4$	$2$	$94.306$	\(\Q(\sqrt{-3}, \sqrt{5569})\)	None					84.6.a.c	$2$	$0$	\(0\)	\(-18\)	\(-6\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q-9\beta _{1}q^{3}+(-3+3\beta _{1}+\beta _{2}+\beta _{3})q^{5}+\cdots\)
588.6.i.j	$588$	$6$	588.i	7.c	$3$	$4$	$2$	$94.306$	\(\Q(\sqrt{-3}, \sqrt{2641})\)	None		✓			588.6.a.i	$2$	$0$	\(0\)	\(-18\)	\(90\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-9+9\beta _{1})q^{3}+(45\beta _{1}+\beta _{2})q^{5}+\cdots\)
588.6.i.k	$588$	$6$	588.i	7.c	$3$	$4$	$2$	$94.306$	\(\Q(\sqrt{-3}, \sqrt{2641})\)	None		✓			588.6.a.i	$2$	$0$	\(0\)	\(18\)	\(-90\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(9-9\beta _{1})q^{3}+(-45\beta _{1}-\beta _{2})q^{5}+\cdots\)
588.6.i.l	$588$	$6$	588.i	7.c	$3$	$4$	$2$	$94.306$	\(\Q(\sqrt{-3}, \sqrt{5569})\)	None					84.6.a.c	$2$	$0$	\(0\)	\(18\)	\(6\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(9-9\beta _{1})q^{3}+(3\beta _{1}+\beta _{2})q^{5}-3^{4}\beta _{1}q^{9}+\cdots\)
588.6.i.m	$588$	$6$	588.i	7.c	$3$	$4$	$2$	$94.306$	\(\Q(\sqrt{-3}, \sqrt{7081})\)	None					84.6.i.b	$2$	$0$	\(0\)	\(18\)	\(47\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(9-9\beta _{2})q^{3}+(-\beta _{1}+24\beta _{2})q^{5}+\cdots\)
588.6.i.n	$588$	$6$	588.i	7.c	$3$	$4$	$2$	$94.306$	\(\Q(\sqrt{-3}, \sqrt{505})\)	None					84.6.a.d	$2$	$0$	\(0\)	\(18\)	\(78\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(9-9\beta _{1})q^{3}+(39\beta _{1}+\beta _{2})q^{5}-3^{4}\beta _{1}q^{9}+\cdots\)
588.6.i.o	$588$	$6$	588.i	7.c	$3$	$8$	$4$	$94.306$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					84.6.i.c	$2$	$0$	\(0\)	\(-36\)	\(0\)	\(0\)	$2^{6}\cdot 3^{4}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-9+9\beta _{1})q^{3}+(-\beta _{3}+\beta _{4})q^{5}+\cdots\)
588.6.i.p	$588$	$6$	588.i	7.c	$3$	$8$	$4$	$94.306$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		✓			588.6.a.m	$2$	$0$	\(0\)	\(-36\)	\(0\)	\(0\)	$2^{8}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-9+9\beta _{1})q^{3}+(-\beta _{2}-2\beta _{3}-2\beta _{4}+\cdots)q^{5}+\cdots\)
588.6.i.q	$588$	$6$	588.i	7.c	$3$	$8$	$4$	$94.306$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		✓			588.6.a.m	$2$	$0$	\(0\)	\(36\)	\(0\)	\(0\)	$2^{8}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(9-9\beta _{1})q^{3}+(\beta _{2}+2\beta _{3}+2\beta _{4}+\beta _{5}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(588, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(588, [\chi]) \simeq \) \(S_{6}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(98, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(147, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(196, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(294, [\chi])\)\(^{\oplus 2}\)