Embedded newform 147.3.l.a.8.16

• Label: 147.3.l.a.8.16
• Level: $147$
• Weight: $3$
• Character: 147.8
• Analytic conductor: $4.005$
• Analytic rank: $0$
• Dimension: $216$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	147.l (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.00545988610\)
Analytic rank:	\(0\)
Dimension:	\(216\)
Relative dimension:	\(36\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			8.16
Character	\(\chi\)	\(=\)	147.8
Dual form			147.3.l.a.92.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.497631 + 0.396848i) q^{2} +(1.93233 - 2.29480i) q^{3} +(-0.799935 + 3.50474i) q^{4} +(1.71979 + 3.57118i) q^{5} +(-0.0509003 + 1.90880i) q^{6} +(3.29295 + 6.17709i) q^{7} +(-2.09744 - 4.35537i) q^{8} +(-1.53222 - 8.86861i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q - 5 q^{3} + 62 q^{4} + 7 q^{6} - 14 q^{7} - 45 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).

\(n\)	\(50\)	\(52\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{6}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.497631	+	0.396848i	−0.248816	+	0.198424i	−0.739954	−	0.672658i	\(-0.765153\pi\)
							0.491138	+	0.871082i	\(0.336581\pi\)
\(3\)	1.93233	−	2.29480i	0.644109	−	0.764934i
							\(5\)	1.71979	+	3.57118i	0.343957	+	0.714235i	0.999150	−	0.0412270i	\(-0.0131267\pi\)
−0.655192	+	0.755462i	\(0.727412\pi\)
\(7\)	3.29295	+	6.17709i	0.470421	+	0.882442i
							\(11\)	−13.2410	+	10.5593i	−1.20372	+	0.959938i	−0.999818	−	0.0190520i	\(-0.993935\pi\)
−0.203906	+	0.978990i	\(0.565364\pi\)
\(13\)	7.24437	+	9.08415i	0.557259	+	0.698781i	0.978048	−	0.208377i	\(-0.0668183\pi\)
							−0.420789	+	0.907158i	\(0.638247\pi\)
\(17\)	22.4823	−	5.13143i	1.32249	−	0.301849i	0.497728	−	0.867333i	\(-0.334168\pi\)
							0.824759	+	0.565485i	\(0.191311\pi\)
\(19\)	5.71498			0.300788			0.150394	−	0.988626i	\(-0.451946\pi\)
							0.150394	+	0.988626i	\(0.451946\pi\)
\(23\)	41.0148	+	9.36137i	1.78325	+	0.407016i	0.981627	−	0.190809i	\(-0.0611112\pi\)
							0.801626	+	0.597825i	\(0.203968\pi\)
\(29\)	−30.6817	+	7.00289i	−1.05799	+	0.241479i	−0.715910	−	0.698192i	\(-0.753988\pi\)
							−0.342078	+	0.939671i	\(0.611131\pi\)
\(31\)	−30.2985			−0.977371			−0.488685	−	0.872460i	\(-0.662523\pi\)
							−0.488685	+	0.872460i	\(0.662523\pi\)
\(37\)	−12.9330	−	56.6631i	−0.349540	−	1.53144i	−0.778227	−	0.627983i	\(-0.783881\pi\)
							0.428687	−	0.903453i	\(-0.358976\pi\)
\(41\)	−4.88893	−	10.1520i	−0.119242	−	0.247609i	0.832801	−	0.553572i	\(-0.186736\pi\)
							−0.952043	+	0.305963i	\(0.901022\pi\)
\(43\)	−17.1138	−	8.24157i	−0.397995	−	0.191664i	0.224171	−	0.974550i	\(-0.428033\pi\)
							−0.622166	+	0.782886i	\(0.713747\pi\)
\(47\)	63.1772	−	50.3822i	1.34420	−	1.07196i	0.353563	−	0.935411i	\(-0.384970\pi\)
							0.990633	−	0.136550i	\(-0.0436014\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.3.l.a.8.16	✓	216
3.2	odd	2	inner	147.3.l.a.8.21	yes	216
49.43	even	7	inner	147.3.l.a.92.21	yes	216
147.92	odd	14	inner	147.3.l.a.92.16	yes	216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.3.l.a.8.16	✓	216	1.1	even	1	trivial
147.3.l.a.8.21	yes	216	3.2	odd	2	inner
147.3.l.a.92.16	yes	216	147.92	odd	14	inner
147.3.l.a.92.21	yes	216	49.43	even	7	inner