Embedded newform 147.3.l.a.8.8

• Label: 147.3.l.a.8.8
• 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			8.8
Character	\(\chi\)	\(=\)	147.8
Dual form			147.3.l.a.92.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.89369 + 1.51017i) q^{2} +(-2.37508 + 1.83275i) q^{3} +(0.415378 - 1.81989i) q^{4} +(-2.09584 - 4.35206i) q^{5} +(1.72992 - 7.05745i) q^{6} +(6.79676 - 1.67455i) q^{7} +(-2.24194 - 4.65543i) q^{8} +(2.28205 - 8.70587i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q - 5 q^{3} + 62 q^{4} + 7 q^{6} - 14 q^{7} - 45 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.89369	+	1.51017i	−0.946847	+	0.755085i	−0.969610	−	0.244656i	\(-0.921325\pi\)
							0.0227630	+	0.999741i	\(0.492754\pi\)
\(3\)	−2.37508	+	1.83275i	−0.791695	+	0.610917i
							\(5\)	−2.09584	−	4.35206i	−0.419168	−	0.870412i	−0.998470	−	0.0552987i	\(-0.982389\pi\)
0.579302	−	0.815113i	\(-0.303325\pi\)
\(7\)	6.79676	−	1.67455i	0.970965	−	0.239221i
							\(11\)	−5.74740	+	4.58340i	−0.522491	+	0.416673i	−0.848898	−	0.528557i	\(-0.822733\pi\)
0.326407	+	0.945229i	\(0.394162\pi\)
\(13\)	3.94978	+	4.95287i	0.303829	+	0.380990i	0.910184	−	0.414205i	\(-0.135940\pi\)
							−0.606355	+	0.795194i	\(0.707369\pi\)
\(17\)	21.0907	−	4.81380i	1.24063	−	0.283165i	0.448645	−	0.893710i	\(-0.351907\pi\)
							0.791982	+	0.610545i	\(0.209050\pi\)
\(19\)	−8.78908			−0.462583			−0.231292	−	0.972884i	\(-0.574295\pi\)
							−0.231292	+	0.972884i	\(0.574295\pi\)
\(23\)	−17.7831	−	4.05887i	−0.773178	−	0.176473i	−0.182310	−	0.983241i	\(-0.558357\pi\)
							−0.590868	+	0.806768i	\(0.701215\pi\)
\(29\)	51.2632	−	11.7005i	1.76770	−	0.403465i	0.789936	−	0.613190i	\(-0.210114\pi\)
							0.977761	+	0.209724i	\(0.0672567\pi\)
\(31\)	47.4032			1.52914			0.764568	−	0.644543i	\(-0.222952\pi\)
							0.764568	+	0.644543i	\(0.222952\pi\)
\(37\)	−15.8325	−	69.3667i	−0.427905	−	1.87478i	−0.481886	−	0.876234i	\(-0.660048\pi\)
							0.0539805	−	0.998542i	\(-0.482809\pi\)
\(41\)	11.0287	+	22.9013i	0.268992	+	0.558568i	0.991085	−	0.133228i	\(-0.0425341\pi\)
							−0.722093	+	0.691796i	\(0.756820\pi\)
\(43\)	−39.8844	−	19.2073i	−0.927545	−	0.446682i	−0.0917863	−	0.995779i	\(-0.529258\pi\)
							−0.835759	+	0.549096i	\(0.814972\pi\)
\(47\)	12.7547	−	10.1716i	0.271377	−	0.216416i	−0.478339	−	0.878175i	\(-0.658761\pi\)
							0.749717	+	0.661759i	\(0.230190\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.3.l.a.8.8	✓	216
3.2	odd	2	inner	147.3.l.a.8.29	yes	216
49.43	even	7	inner	147.3.l.a.92.29	yes	216
147.92	odd	14	inner	147.3.l.a.92.8	yes	216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.3.l.a.8.8	✓	216	1.1	even	1	trivial
147.3.l.a.8.29	yes	216	3.2	odd	2	inner
147.3.l.a.92.8	yes	216	147.92	odd	14	inner
147.3.l.a.92.29	yes	216	49.43	even	7	inner