Embedded newform 147.3.l.a.8.10

• Label: 147.3.l.a.8.10
• 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			8.10
Character	\(\chi\)	\(=\)	147.8
Dual form			147.3.l.a.92.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.70385 + 1.35878i) q^{2} +(-2.98293 - 0.319571i) q^{3} +(0.166754 - 0.730599i) q^{4} +(-1.36590 - 2.83633i) q^{5} +(5.51670 - 3.50864i) q^{6} +(-6.10925 + 3.41717i) q^{7} +(-3.07367 - 6.38254i) q^{8} +(8.79575 + 1.90652i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q - 5 q^{3} + 62 q^{4} + 7 q^{6} - 14 q^{7} - 45 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.70385	+	1.35878i	−0.851926	+	0.679388i	−0.948789	−	0.315909i	\(-0.897690\pi\)
							0.0968632	+	0.995298i	\(0.469119\pi\)
\(3\)	−2.98293	−	0.319571i	−0.994310	−	0.106524i
							\(5\)	−1.36590	−	2.83633i	−0.273181	−	0.567265i	0.718569	−	0.695456i	\(-0.244797\pi\)
−0.991750	+	0.128190i	\(0.959083\pi\)
\(7\)	−6.10925	+	3.41717i	−0.872750	+	0.488168i
							\(11\)	8.50965	−	6.78622i	0.773605	−	0.616929i	−0.155036	−	0.987909i	\(-0.549549\pi\)
0.928641	+	0.370980i	\(0.120978\pi\)
\(13\)	3.41419	+	4.28126i	0.262630	+	0.329328i	0.895610	−	0.444841i	\(-0.146740\pi\)
							−0.632979	+	0.774169i	\(0.718168\pi\)
\(17\)	−18.4891	+	4.22002i	−1.08760	+	0.248237i	−0.728487	−	0.685059i	\(-0.759776\pi\)
							−0.359108	+	0.933296i	\(0.616919\pi\)
\(19\)	37.2890			1.96258			0.981290	−	0.192538i	\(-0.0616718\pi\)
							0.981290	+	0.192538i	\(0.0616718\pi\)
\(23\)	30.7721	+	7.02352i	1.33792	+	0.305370i	0.830830	−	0.556526i	\(-0.187866\pi\)
							0.507085	+	0.861896i	\(0.330723\pi\)
\(29\)	7.40553	−	1.69026i	0.255363	−	0.0582849i	−0.0929228	−	0.995673i	\(-0.529621\pi\)
							0.348286	+	0.937388i	\(0.386764\pi\)
\(31\)	−27.8638			−0.898833			−0.449417	−	0.893322i	\(-0.648368\pi\)
							−0.449417	+	0.893322i	\(0.648368\pi\)
\(37\)	−12.2994	−	53.8871i	−0.332416	−	1.45641i	−0.814438	−	0.580250i	\(-0.802955\pi\)
							0.482022	−	0.876159i	\(-0.339902\pi\)
\(41\)	−31.1458	−	64.6750i	−0.759655	−	1.57744i	−0.815334	−	0.578991i	\(-0.803447\pi\)
							0.0556791	−	0.998449i	\(-0.482268\pi\)
\(43\)	49.4113	+	23.7952i	1.14910	+	0.553377i	0.908763	−	0.417313i	\(-0.137028\pi\)
							0.240336	+	0.970690i	\(0.422742\pi\)
\(47\)	27.2969	−	21.7686i	0.580786	−	0.463161i	−0.288494	−	0.957482i	\(-0.593154\pi\)
							0.869280	+	0.494320i	\(0.164583\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.3.l.a.8.10	✓	216
3.2	odd	2	inner	147.3.l.a.8.27	yes	216
49.43	even	7	inner	147.3.l.a.92.27	yes	216
147.92	odd	14	inner	147.3.l.a.92.10	yes	216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.3.l.a.8.10	✓	216	1.1	even	1	trivial
147.3.l.a.8.27	yes	216	3.2	odd	2	inner
147.3.l.a.92.10	yes	216	147.92	odd	14	inner
147.3.l.a.92.27	yes	216	49.43	even	7	inner