Space of modular forms of level 588, weight 3, and Character 588.c

• Label: 588.3.c
• Level: $588$
• Weight: $3$
• Character orbit: 588.c
• Rep. character: $\chi_{588}(197,\cdot)$
• Character field: $\Q$
• Dimension: $27$
• Newform subspaces: $10$
• Sturm bound: $336$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	588.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 3 \)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(336\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(5\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(588, [\chi])\).

	Total	New	Old
Modular forms	248	27	221
Cusp forms	200	27	173
Eisenstein series	48	0	48

Trace form

27 * q - 3 * q^3 - 3 * q^9 - 6 * q^13 + 2 * q^15 - 22 * q^19 - 61 * q^25 + 9 * q^27 + 6 * q^31 + 116 * q^33 + 68 * q^37 - 116 * q^39 - 184 * q^43 - 100 * q^45 - 62 * q^51 - 184 * q^55 + 114 * q^57 - 270 * q^61 + 172 * q^67 + 8 * q^69 + 46 * q^73 + 433 * q^75 + 440 * q^79 - 419 * q^81 - 116 * q^85 - 140 * q^87 - 46 * q^93 - 26 * q^97 + 62 * q^99

Decomposition of \(S_{3}^{\mathrm{new}}(588, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
588.3.c.a	$588$	$3$	588.c	3.b	$2$	$1$	$1$	$16.022$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				84.3.p.a	$2$	$0$	\(0\)	\(-3\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-3q^{3}+9q^{9}-q^{13}-37q^{19}+5^{2}q^{25}+\cdots\)
588.3.c.b	$588$	$3$	588.c	3.b	$2$	$1$	$1$	$16.022$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				84.3.p.a	$2$	$0$	\(0\)	\(3\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+3q^{3}+9q^{9}+q^{13}+37q^{19}+5^{2}q^{25}+\cdots\)
588.3.c.c	$588$	$3$	588.c	3.b	$2$	$1$	$1$	$16.022$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				12.3.c.a	$2$	$0$	\(0\)	\(3\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+3q^{3}+9q^{9}+22q^{13}-26q^{19}+\cdots\)
588.3.c.d	$588$	$3$	588.c	3.b	$2$	$2$	$2$	$16.022$	\(\Q(\sqrt{-5}) \)	None					84.3.p.b	$2$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-2+\beta )q^{3}+3\beta q^{5}+(-1-4\beta )q^{9}+\cdots\)
588.3.c.e	$588$	$3$	588.c	3.b	$2$	$2$	$2$	$16.022$	\(\Q(\sqrt{-35}) \)	None					84.3.p.c	$2$	$0$	\(0\)	\(-1\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1+\beta )q^{3}+(1-2\beta )q^{5}+(-8-\beta )q^{9}+\cdots\)
588.3.c.f	$588$	$3$	588.c	3.b	$2$	$2$	$2$	$16.022$	\(\Q(\sqrt{-35}) \)	None					84.3.p.c	$2$	$0$	\(0\)	\(1\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{3}+(1-2\beta )q^{5}+(-9+\beta )q^{9}+\cdots\)
588.3.c.g	$588$	$3$	588.c	3.b	$2$	$2$	$2$	$16.022$	\(\Q(\sqrt{-5}) \)	None					84.3.p.b	$2$	$0$	\(0\)	\(4\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(2+\beta )q^{3}+3\beta q^{5}+(-1+4\beta )q^{9}+\cdots\)
588.3.c.h	$588$	$3$	588.c	3.b	$2$	$4$	$4$	$16.022$	4.0.116032.1	None					84.3.c.a	$2$	$0$	\(0\)	\(-6\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-2+\beta _{1})q^{3}+(2-\beta _{1}+\beta _{2}+2\beta _{3})q^{5}+\cdots\)
588.3.c.i	$588$	$3$	588.c	3.b	$2$	$4$	$4$	$16.022$	\(\Q(\sqrt{-14}, \sqrt{22})\)	None		✓			588.3.c.i	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}+(-\beta _{1}-\beta _{2})q^{5}+(2-\beta _{3})q^{9}+\cdots\)
588.3.c.j	$588$	$3$	588.c	3.b	$2$	$8$	$8$	$16.022$	8.0.\(\cdots\).3	None		✓	✓		588.3.c.j	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{5}q^{3}-\beta _{3}q^{5}+(-\beta _{1}-\beta _{6})q^{9}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(588, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(588, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(147, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(294, [\chi])\)\(^{\oplus 2}\)