Embedded newform 147.3.l.a.8.20

• Label: 147.3.l.a.8.20
• 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			8.20
Character	\(\chi\)	\(=\)	147.8
Dual form			147.3.l.a.92.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.488839 - 0.389836i) q^{2} +(2.46056 + 1.71628i) q^{3} +(-0.803092 + 3.51858i) q^{4} +(1.15070 + 2.38946i) q^{5} +(1.87189 - 0.120232i) q^{6} +(-6.83519 - 1.51004i) q^{7} +(2.06423 + 4.28641i) q^{8} +(3.10876 + 8.44604i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q - 5 q^{3} + 62 q^{4} + 7 q^{6} - 14 q^{7} - 45 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.488839	−	0.389836i	0.244420	−	0.194918i	−0.493616	−	0.869680i	\(-0.664325\pi\)
							0.738035	+	0.674762i	\(0.235754\pi\)
\(3\)	2.46056	+	1.71628i	0.820188	+	0.572094i
							\(5\)	1.15070	+	2.38946i	0.230140	+	0.477891i	0.983776	−	0.179401i	\(-0.0574160\pi\)
−0.753636	+	0.657292i	\(0.771702\pi\)
\(7\)	−6.83519	−	1.51004i	−0.976455	−	0.215720i
							\(11\)	1.54332	−	1.23076i	0.140302	−	0.111887i	−0.550824	−	0.834621i	\(-0.685686\pi\)
0.691126	+	0.722734i	\(0.257115\pi\)
\(13\)	−2.42618	−	3.04233i	−0.186629	−	0.234025i	0.679711	−	0.733480i	\(-0.262105\pi\)
							−0.866340	+	0.499455i	\(0.833534\pi\)
\(17\)	13.8617	−	3.16384i	0.815392	−	0.186108i	0.205566	−	0.978643i	\(-0.434096\pi\)
							0.609826	+	0.792535i	\(0.291239\pi\)
\(19\)	−0.0919989			−0.00484205			−0.00242102	−	0.999997i	\(-0.500771\pi\)
							−0.00242102	+	0.999997i	\(0.500771\pi\)
\(23\)	39.2938	+	8.96856i	1.70843	+	0.389937i	0.961465	−	0.274929i	\(-0.0886543\pi\)
							0.746963	+	0.664866i	\(0.231511\pi\)
\(29\)	0.216461	−	0.0494058i	0.00746416	−	0.00170365i	−0.218787	−	0.975773i	\(-0.570210\pi\)
							0.226251	+	0.974069i	\(0.427353\pi\)
\(31\)	46.7877			1.50928			0.754640	−	0.656139i	\(-0.227811\pi\)
							0.754640	+	0.656139i	\(0.227811\pi\)
\(37\)	−5.17099	−	22.6556i	−0.139757	−	0.612313i	−0.995488	−	0.0948925i	\(-0.969749\pi\)
							0.855731	−	0.517421i	\(-0.173108\pi\)
\(41\)	−14.1753	−	29.4353i	−0.345739	−	0.717935i	0.653500	−	0.756926i	\(-0.273300\pi\)
							−0.999240	+	0.0389914i	\(0.987586\pi\)
\(43\)	−33.8569	−	16.3046i	−0.787370	−	0.379177i	−0.00341415	−	0.999994i	\(-0.501087\pi\)
							−0.783956	+	0.620817i	\(0.786801\pi\)
\(47\)	−40.6004	+	32.3777i	−0.863838	+	0.688888i	−0.951628	−	0.307251i	\(-0.900591\pi\)
							0.0877904	+	0.996139i	\(0.472019\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.3.l.a.8.20	yes	216
3.2	odd	2	inner	147.3.l.a.8.17	✓	216
49.43	even	7	inner	147.3.l.a.92.17	yes	216
147.92	odd	14	inner	147.3.l.a.92.20	yes	216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.3.l.a.8.17	✓	216	3.2	odd	2	inner
147.3.l.a.8.20	yes	216	1.1	even	1	trivial
147.3.l.a.92.17	yes	216	49.43	even	7	inner
147.3.l.a.92.20	yes	216	147.92	odd	14	inner