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

\( 24 q - 6 q^{3} + 24 q^{7} - 108 q^{9} + O(q^{10}) \)

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

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

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