Embedded newform 147.2.k.a.20.14

• Label: 147.2.k.a.20.14
• Level: $147$
• Weight: $2$
• Character: 147.20
• Analytic conductor: $1.174$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.17380090971\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(16\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			20.14
Character	\(\chi\)	\(=\)	147.20
Dual form			147.2.k.a.125.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.888371 - 1.84472i) q^{2} +(0.0997944 + 1.72917i) q^{3} +(-1.36681 - 1.71393i) q^{4} +(-0.654391 - 2.86707i) q^{5} +(3.27850 + 1.35205i) q^{6} +(2.64332 + 0.113383i) q^{7} +(-0.383662 + 0.0875683i) q^{8} +(-2.98008 + 0.345124i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 7 q^{3} + 2 q^{4} + 7 q^{6} - 14 q^{7} + 5 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(50\)	\(52\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{13}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.888371	−	1.84472i	0.628173	−	1.30442i	−0.307500	−	0.951548i	\(-0.599492\pi\)
							0.935673	−	0.352867i	\(-0.114793\pi\)
\(3\)	0.0997944	+	1.72917i	0.0576164	+	0.998339i
							\(5\)	−0.654391	−	2.86707i	−0.292653	−	1.28219i	−0.880818	−	0.473454i	\(-0.843007\pi\)
0.588166	−	0.808740i	\(-0.299850\pi\)
\(7\)	2.64332	+	0.113383i	0.999081	+	0.0428547i
							\(11\)	0.921564	−	1.91365i	0.277862	−	0.576986i	−0.714601	−	0.699532i	\(-0.753392\pi\)
0.992463	+	0.122546i	\(0.0391060\pi\)
\(13\)	−2.38541	+	4.95335i	−0.661593	+	1.37381i	0.252218	+	0.967670i	\(0.418840\pi\)
							−0.913811	+	0.406141i	\(0.866874\pi\)
\(17\)	−1.65026	+	2.06936i	−0.400246	+	0.501892i	−0.940587	−	0.339554i	\(-0.889724\pi\)
							0.540341	+	0.841446i	\(0.318295\pi\)
\(19\)			4.54073i			1.04171i	0.853644	+	0.520857i	\(0.174388\pi\)
							−0.853644	+	0.520857i	\(0.825612\pi\)
\(23\)	−6.09911	+	4.86388i	−1.27175	+	1.01419i	−0.273117	+	0.961981i	\(0.588055\pi\)
							−0.998636	+	0.0522083i	\(0.983374\pi\)
\(29\)	1.89494	+	1.51116i	0.351881	+	0.280616i	0.783438	−	0.621470i	\(-0.213464\pi\)
							−0.431557	+	0.902086i	\(0.642036\pi\)
\(31\)		−	1.91483i		−	0.343913i	−0.985105	−	0.171957i	\(-0.944991\pi\)
							0.985105	−	0.171957i	\(-0.0550089\pi\)
\(37\)	1.26707	−	1.58886i	0.208305	−	0.261207i	−0.666693	−	0.745332i	\(-0.732291\pi\)
							0.874998	+	0.484126i	\(0.160862\pi\)
\(41\)	0.544158	+	2.38411i	0.0849833	+	0.372336i	0.999480	−	0.0322582i	\(-0.0102699\pi\)
							−0.914496	+	0.404594i	\(0.867413\pi\)
\(43\)	1.85467	−	8.12583i	0.282834	−	1.23918i	−0.611307	−	0.791393i	\(-0.709356\pi\)
							0.894142	−	0.447784i	\(-0.147787\pi\)
\(47\)	3.00600	+	1.44761i	0.438470	+	0.211156i	0.640078	−	0.768310i	\(-0.278902\pi\)
							−0.201607	+	0.979466i	\(0.564617\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.2.k.a.20.14	yes	96
3.2	odd	2	inner	147.2.k.a.20.3	✓	96
49.27	odd	14	inner	147.2.k.a.125.3	yes	96
147.125	even	14	inner	147.2.k.a.125.14	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.2.k.a.20.3	✓	96	3.2	odd	2	inner
147.2.k.a.20.14	yes	96	1.1	even	1	trivial
147.2.k.a.125.3	yes	96	49.27	odd	14	inner
147.2.k.a.125.14	yes	96	147.125	even	14	inner