Embedded newform 168.4.v.a.11.15

• Label: 168.4.v.a.11.15
• Level: $168$
• Weight: $4$
• Character: 168.11
• Analytic conductor: $9.912$
• Analytic rank: $0$
• Dimension: $184$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	168.v (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.91232088096\)
Analytic rank:	\(0\)
Dimension:	\(184\)
Relative dimension:	\(92\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			11.15
Character	\(\chi\)	\(=\)	168.11
Dual form			168.4.v.a.107.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.49348 + 1.33513i) q^{2} +(-4.07521 + 3.22376i) q^{3} +(4.43487 - 6.65822i) q^{4} +(-4.58233 - 7.93683i) q^{5} +(5.85732 - 13.4793i) q^{6} +(15.9950 + 9.33591i) q^{7} +(-2.16868 + 22.5233i) q^{8} +(6.21469 - 26.2750i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 2 q^{3} - 2 q^{4} - 20 q^{6} - 2 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.49348	+	1.33513i	−0.881578	+	0.472039i
							\(3\)	−4.07521	+	3.22376i	−0.784275	+	0.620414i
\(5\)	−4.58233	−	7.93683i	−0.409856	−	0.709892i								0.585017	−	0.811021i	\(-0.301088\pi\)
							−0.994873	+	0.101129i	\(0.967754\pi\)
\(7\)	15.9950	+	9.33591i	0.863650	+	0.504092i
							\(11\)	5.16860	+	2.98409i	0.141672	+	0.0817944i	0.569161	−	0.822226i	\(-0.307268\pi\)
−0.427489	+	0.904021i	\(0.640602\pi\)
\(13\)			64.6298i			1.37885i	0.724356	+	0.689427i	\(0.242137\pi\)
							−0.724356	+	0.689427i	\(0.757863\pi\)
\(17\)	−37.1834	−	21.4678i	−0.530488	−	0.306277i	0.210727	−	0.977545i	\(-0.432417\pi\)
							−0.741215	+	0.671268i	\(0.765750\pi\)
\(19\)	24.8818	+	43.0966i	0.300436	+	0.520371i	0.976235	−	0.216716i	\(-0.0695345\pi\)
							−0.675799	+	0.737086i	\(0.736201\pi\)
\(23\)	−16.3768	−	28.3654i	−0.148470	−	0.257157i	0.782192	−	0.623037i	\(-0.214101\pi\)
							−0.930662	+	0.365880i	\(0.880768\pi\)
\(29\)	−285.281			−1.82673			−0.913367	−	0.407137i	\(-0.866527\pi\)
							−0.913367	+	0.407137i	\(0.866527\pi\)
\(31\)	−179.243	−	103.486i	−1.03848	−	0.599568i	−0.119079	−	0.992885i	\(-0.537994\pi\)
							−0.919403	+	0.393317i	\(0.871328\pi\)
\(37\)	−173.117	+	99.9490i	−0.769195	+	0.444095i	−0.832587	−	0.553894i	\(-0.813141\pi\)
							0.0633923	+	0.997989i	\(0.479808\pi\)
\(41\)			264.700i			1.00827i	0.863624	+	0.504136i	\(0.168189\pi\)
							−0.863624	+	0.504136i	\(0.831811\pi\)
\(43\)	−339.097			−1.20260			−0.601300	−	0.799023i	\(-0.705350\pi\)
							−0.601300	+	0.799023i	\(0.705350\pi\)
\(47\)	198.945	+	344.583i	0.617428	+	1.06942i	0.989953	+	0.141395i	\(0.0451588\pi\)
							−0.372525	+	0.928022i	\(0.621508\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	168.4.v.a.11.15	✓	184
3.2	odd	2	inner	168.4.v.a.11.78	yes	184
7.2	even	3	inner	168.4.v.a.107.77	yes	184
8.3	odd	2	inner	168.4.v.a.11.16	yes	184
21.2	odd	6	inner	168.4.v.a.107.16	yes	184
24.11	even	2	inner	168.4.v.a.11.77	yes	184
56.51	odd	6	inner	168.4.v.a.107.78	yes	184
168.107	even	6	inner	168.4.v.a.107.15	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.4.v.a.11.15	✓	184	1.1	even	1	trivial
168.4.v.a.11.16	yes	184	8.3	odd	2	inner
168.4.v.a.11.77	yes	184	24.11	even	2	inner
168.4.v.a.11.78	yes	184	3.2	odd	2	inner
168.4.v.a.107.15	yes	184	168.107	even	6	inner
168.4.v.a.107.16	yes	184	21.2	odd	6	inner
168.4.v.a.107.77	yes	184	7.2	even	3	inner
168.4.v.a.107.78	yes	184	56.51	odd	6	inner