Space of modular forms of level 168, weight 4, and Character 168.q

• Label: 168.4.q
• Level: $168$
• Weight: $4$
• Character orbit: 168.q
• Rep. character: $\chi_{168}(25,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $24$
• Newform subspaces: $6$
• Sturm bound: $128$
• Trace bound: $5$

Defining parameters

Level:	$N$	$=$	$168 = 2^{3} \cdot 3 \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	168.q (of order $3$ and degree $2$)
Character conductor:	$\operatorname{cond}(\chi)$	$=$	$7$
Character field:			$\Q(\zeta_{3})$
Newform subspaces:			$6$
Sturm bound:			$128$
Trace bound:			$5$
Distinguishing $T_p$:			$5$

Dimensions

The following table gives the dimensions of various subspaces of $M_{4}(168, [\chi])$.

	Total	New	Old
Modular forms	208	24	184
Cusp forms	176	24	152
Eisenstein series	32	0	32

Trace form

24 * q - 6 * q^3 + 24 * q^7 - 108 * q^9 - 28 * q^11 - 140 * q^13 - 84 * q^15 - 4 * q^17 - 250 * q^19 - 60 * q^21 + 84 * q^23 - 138 * q^25 + 108 * q^27 - 280 * q^29 + 168 * q^31 + 114 * q^33 - 708 * q^35 - 318 * q^37 - 210 * q^39 - 888 * q^41 - 116 * q^43 + 268 * q^49 + 60 * q^51 + 1180 * q^53 + 3468 * q^55 + 612 * q^57 - 768 * q^59 + 1572 * q^61 + 54 * q^63 - 992 * q^65 - 1014 * q^67 - 1704 * q^69 - 1072 * q^71 + 1550 * q^73 - 1362 * q^75 - 1140 * q^77 + 1268 * q^79 - 972 * q^81 + 9032 * q^83 + 4360 * q^85 + 474 * q^87 + 2368 * q^89 - 450 * q^91 - 1146 * q^93 - 3392 * q^95 - 5860 * q^97 + 504 * q^99

Decomposition of $S_{4}^{\mathrm{new}}(168, [\chi])$ 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
168.4.q.a	$168$	$4$	168.q	7.c	$3$	$2$	$1$	$9.912$	$\Q(\sqrt{-3})$	None		✓			168.4.q.a	$2$	$0$	$0$	$3$	$-7$	$-35$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+3\zeta_{6}q^{3}+(-7+7\zeta_{6})q^{5}+(-14+\cdots)q^{7}+\cdots$
168.4.q.b	$168$	$4$	168.q	7.c	$3$	$2$	$1$	$9.912$	$\Q(\sqrt{-3})$	None		✓			168.4.q.b	$2$	$0$	$0$	$3$	$-2$	$35$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+3\zeta_{6}q^{3}+(-2+2\zeta_{6})q^{5}+(21-7\zeta_{6})q^{7}+\cdots$
168.4.q.c	$168$	$4$	168.q	7.c	$3$	$2$	$1$	$9.912$	$\Q(\sqrt{-3})$	None		✓			168.4.q.c	$2$	$0$	$0$	$3$	$11$	$7$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+3\zeta_{6}q^{3}+(11-11\zeta_{6})q^{5}+(14-21\zeta_{6})q^{7}+\cdots$
168.4.q.d	$168$	$4$	168.q	7.c	$3$	$4$	$2$	$9.912$	$\Q(\sqrt{-3}, \sqrt{505})$	None		✓			168.4.q.d	$2$	$0$	$0$	$6$	$-9$	$0$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(3-3\beta _{2})q^{3}+(\beta _{1}-5\beta _{2})q^{5}+(8-17\beta _{2}+\cdots)q^{7}+\cdots$
168.4.q.e	$168$	$4$	168.q	7.c	$3$	$6$	$3$	$9.912$	6.0.$\cdots$.1	None		✓			168.4.q.e	$2$	$0$	$0$	$-9$	$11$	$-1$	$2^{4}\cdot 3$	$\mathrm{SU}(2)[C_{3}]$	$q+(-3+3\beta _{3})q^{3}+(4\beta _{3}-\beta _{4}-\beta _{5})q^{5}+\cdots$
168.4.q.f	$168$	$4$	168.q	7.c	$3$	$8$	$4$	$9.912$	$\mathbb{Q}[x]/(x^{8} + \cdots)$	None		✓	✓		168.4.q.f	$2$	$0$	$0$	$-12$	$-4$	$18$	$2^{6}\cdot 7$	$\mathrm{SU}(2)[C_{3}]$	$q+3\beta _{1}q^{3}+(-1-\beta _{1}+\beta _{6})q^{5}+(2+\cdots)q^{7}+\cdots$

Decomposition of $S_{4}^{\mathrm{old}}(168, [\chi])$ into lower level spaces

$S_{4}^{\mathrm{old}}(168, [\chi]) \simeq$ $S_{4}^{\mathrm{new}}(7, [\chi])$$^{\oplus 8}$$\oplus$$S_{4}^{\mathrm{new}}(14, [\chi])$$^{\oplus 6}$$\oplus$$S_{4}^{\mathrm{new}}(21, [\chi])$$^{\oplus 4}$$\oplus$$S_{4}^{\mathrm{new}}(28, [\chi])$$^{\oplus 4}$$\oplus$$S_{4}^{\mathrm{new}}(42, [\chi])$$^{\oplus 3}$$\oplus$$S_{4}^{\mathrm{new}}(56, [\chi])$$^{\oplus 2}$$\oplus$$S_{4}^{\mathrm{new}}(84, [\chi])$$^{\oplus 2}$