Embedded newform 147.2.k.a.104.2

• Label: 147.2.k.a.104.2
• Level: $147$
• Weight: $2$
• Character: 147.104
• 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			104.2
Character	\(\chi\)	\(=\)	147.104
Dual form			147.2.k.a.41.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.78148 - 1.42068i) q^{2} +(0.0426424 + 1.73153i) q^{3} +(0.710285 + 3.11196i) q^{4} +(-1.09528 - 0.527460i) q^{5} +(2.38398 - 3.14525i) q^{6} +(-2.21851 - 1.44160i) q^{7} +(1.17846 - 2.44709i) q^{8} +(-2.99636 + 0.147673i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 7 q^{3} + 2 q^{4} + 7 q^{6} - 14 q^{7} + 5 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(50\)	\(52\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{9}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.78148	−	1.42068i	−1.25969	−	1.00457i	−0.999236	−	0.0390875i	\(-0.987555\pi\)
							−0.260458	−	0.965485i	\(-0.583874\pi\)
\(3\)	0.0426424	+	1.73153i	0.0246196	+	0.999697i
							\(5\)	−1.09528	−	0.527460i	−0.489825	−	0.235887i	0.172633	−	0.984986i	\(-0.444773\pi\)
−0.662458	+	0.749099i	\(0.730487\pi\)
\(7\)	−2.21851	−	1.44160i	−0.838517	−	0.544875i
							\(11\)	−3.63827	−	2.90142i	−1.09698	−	0.874812i	−0.104175	−	0.994559i	\(-0.533220\pi\)
−0.992804	+	0.119747i	\(0.961792\pi\)
\(13\)	−2.21274	−	1.76460i	−0.613705	−	0.489413i	0.266609	−	0.963805i	\(-0.414097\pi\)
							−0.880314	+	0.474392i	\(0.842668\pi\)
\(17\)	−0.896447	+	3.92759i	−0.217420	+	0.952581i	0.741955	+	0.670449i	\(0.233899\pi\)
							−0.959376	+	0.282132i	\(0.908959\pi\)
\(19\)			5.20859i			1.19493i	0.801894	+	0.597467i	\(0.203826\pi\)
							−0.801894	+	0.597467i	\(0.796174\pi\)
\(23\)	−3.01360	+	0.687835i	−0.628379	+	0.143423i	−0.524839	−	0.851202i	\(-0.675874\pi\)
							−0.103541	+	0.994625i	\(0.533017\pi\)
\(29\)	5.02894	+	1.14782i	0.933851	+	0.213145i	0.662266	−	0.749269i	\(-0.269595\pi\)
							0.271585	+	0.962414i	\(0.412452\pi\)
\(31\)		−	5.29517i		−	0.951041i	−0.879705	−	0.475521i	\(-0.842260\pi\)
							0.879705	−	0.475521i	\(-0.157740\pi\)
\(37\)	0.330947	−	1.44998i	0.0544074	−	0.238375i	−0.940411	−	0.340039i	\(-0.889560\pi\)
							0.994819	+	0.101665i	\(0.0324168\pi\)
\(41\)	5.32713	+	2.56541i	0.831958	+	0.400650i	0.800849	−	0.598866i	\(-0.204382\pi\)
							0.0311086	+	0.999516i	\(0.490096\pi\)
\(43\)	−9.88760	+	4.76162i	−1.50784	+	0.726140i	−0.991484	−	0.130232i	\(-0.958428\pi\)
							−0.516361	+	0.856371i	\(0.672714\pi\)
\(47\)	−0.464613	+	0.582607i	−0.0677708	+	0.0849819i	−0.814561	−	0.580077i	\(-0.803022\pi\)
							0.746791	+	0.665059i	\(0.231594\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.2.k.a.104.2	yes	96
3.2	odd	2	inner	147.2.k.a.104.15	yes	96
49.41	odd	14	inner	147.2.k.a.41.15	yes	96
147.41	even	14	inner	147.2.k.a.41.2	✓	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.2.k.a.41.2	✓	96	147.41	even	14	inner
147.2.k.a.41.15	yes	96	49.41	odd	14	inner
147.2.k.a.104.2	yes	96	1.1	even	1	trivial
147.2.k.a.104.15	yes	96	3.2	odd	2	inner