Embedded newform 168.2.v.a.11.1

• Label: 168.2.v.a.11.1
• Level: $168$
• Weight: $2$
• Character: 168.11
• Analytic conductor: $1.341$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	168.v (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.34148675396\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(28\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			11.1
Character	\(\chi\)	\(=\)	168.11
Dual form			168.2.v.a.107.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.40790 + 0.133456i) q^{2} +(-1.71959 - 0.207430i) q^{3} +(1.96438 - 0.375786i) q^{4} +(0.692094 + 1.19874i) q^{5} +(2.44869 + 0.0625523i) q^{6} +(-2.08863 - 1.62408i) q^{7} +(-2.71550 + 0.791228i) q^{8} +(2.91395 + 0.713387i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 2 q^{3} - 2 q^{4} - 8 q^{6} - 2 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/168\mathbb{Z}\right)^\times\).

\(n\)	\(73\)	\(85\)	\(113\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)	\(-1\)	\(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.40790	+	0.133456i	−0.995537	+	0.0943676i
							\(3\)	−1.71959	−	0.207430i	−0.992803	−	0.119760i
\(5\)	0.692094	+	1.19874i	0.309514	+	0.536094i								0.978256	−	0.207401i	\(-0.0665004\pi\)
							−0.668742	+	0.743494i	\(0.733167\pi\)
\(7\)	−2.08863	−	1.62408i	−0.789426	−	0.613845i
							\(11\)	3.82316	+	2.20731i	1.15273	+	0.665528i	0.949550	−	0.313614i	\(-0.101540\pi\)
0.203177	+	0.979142i	\(0.434873\pi\)
\(13\)			6.43609i			1.78505i	0.450999	+	0.892525i	\(0.351068\pi\)
							−0.450999	+	0.892525i	\(0.648932\pi\)
\(17\)	2.52866	+	1.45992i	0.613289	+	0.354083i	0.774252	−	0.632878i	\(-0.218126\pi\)
							−0.160962	+	0.986961i	\(0.551460\pi\)
\(19\)	1.58928	+	2.75271i	0.364606	+	0.631515i	0.988713	−	0.149823i	\(-0.0478705\pi\)
							−0.624107	+	0.781339i	\(0.714537\pi\)
\(23\)	1.84696	+	3.19902i	0.385117	+	0.667043i	0.991785	−	0.127912i	\(-0.0408277\pi\)
							−0.606668	+	0.794955i	\(0.707494\pi\)
\(29\)	−6.67884			−1.24023			−0.620115	−	0.784511i	\(-0.712914\pi\)
							−0.620115	+	0.784511i	\(0.712914\pi\)
\(31\)	−2.17580	−	1.25620i	−0.390786	−	0.225620i	0.291715	−	0.956505i	\(-0.405774\pi\)
							−0.682500	+	0.730885i	\(0.739107\pi\)
\(37\)	5.00149	−	2.88761i	0.822240	−	0.474720i	−0.0289485	−	0.999581i	\(-0.509216\pi\)
							0.851188	+	0.524861i	\(0.175883\pi\)
\(41\)		−	0.497427i		−	0.0776850i	−0.999245	−	0.0388425i	\(-0.987633\pi\)
							0.999245	−	0.0388425i	\(-0.0123671\pi\)
\(43\)	0.865898			0.132048			0.0660241	−	0.997818i	\(-0.478969\pi\)
							0.0660241	+	0.997818i	\(0.478969\pi\)
\(47\)	1.59001	+	2.75398i	0.231927	+	0.401710i	0.958375	−	0.285512i	\(-0.0921635\pi\)
							−0.726448	+	0.687221i	\(0.758830\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	168.2.v.a.11.1	✓	56
3.2	odd	2	inner	168.2.v.a.11.28	yes	56
4.3	odd	2		672.2.bd.a.431.28		56
7.2	even	3	inner	168.2.v.a.107.19	yes	56
8.3	odd	2	inner	168.2.v.a.11.10	yes	56
8.5	even	2		672.2.bd.a.431.27		56
12.11	even	2		672.2.bd.a.431.7		56
21.2	odd	6	inner	168.2.v.a.107.10	yes	56
24.5	odd	2		672.2.bd.a.431.8		56
24.11	even	2	inner	168.2.v.a.11.19	yes	56
28.23	odd	6		672.2.bd.a.527.8		56
56.37	even	6		672.2.bd.a.527.7		56
56.51	odd	6	inner	168.2.v.a.107.28	yes	56
84.23	even	6		672.2.bd.a.527.27		56
168.107	even	6	inner	168.2.v.a.107.1	yes	56
168.149	odd	6		672.2.bd.a.527.28		56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.v.a.11.1	✓	56	1.1	even	1	trivial
168.2.v.a.11.10	yes	56	8.3	odd	2	inner
168.2.v.a.11.19	yes	56	24.11	even	2	inner
168.2.v.a.11.28	yes	56	3.2	odd	2	inner
168.2.v.a.107.1	yes	56	168.107	even	6	inner
168.2.v.a.107.10	yes	56	21.2	odd	6	inner
168.2.v.a.107.19	yes	56	7.2	even	3	inner
168.2.v.a.107.28	yes	56	56.51	odd	6	inner
672.2.bd.a.431.7		56	12.11	even	2
672.2.bd.a.431.8		56	24.5	odd	2
672.2.bd.a.431.27		56	8.5	even	2
672.2.bd.a.431.28		56	4.3	odd	2
672.2.bd.a.527.7		56	56.37	even	6
672.2.bd.a.527.8		56	28.23	odd	6
672.2.bd.a.527.27		56	84.23	even	6
672.2.bd.a.527.28		56	168.149	odd	6