Embedded newform 349.2.k.a.3.19

• Label: 349.2.k.a.3.19
• Level: $349$
• Weight: $2$
• Character: 349.3
• Analytic conductor: $2.787$
• Analytic rank: $0$
• Dimension: $1624$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 349 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	349.k (of order \(174\), degree \(56\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.78677903054\)
Analytic rank:	\(0\)
Dimension:	\(1624\)
Relative dimension:	\(29\) over \(\Q(\zeta_{174})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{174}]$

Embedding invariants

Embedding label			3.19
Character	\(\chi\)	\(=\)	349.3
Dual form			349.2.k.a.233.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.815047 + 0.194897i) q^{2} +(-0.0632014 + 0.0115367i) q^{3} +(-1.15733 - 0.587061i) q^{4} +(-0.348458 - 2.74240i) q^{5} +(-0.0537606 - 0.00291482i) q^{6} +(-1.20872 - 0.0218260i) q^{7} +(-2.10629 - 1.78910i) q^{8} +(-2.80266 + 1.05846i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1624 q - 55 q^{2} - 58 q^{3} - 85 q^{4} - 60 q^{5} - 58 q^{6} - 58 q^{7} - 145 q^{8} - 29 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/349\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{13}{174}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.815047	+	0.194897i	0.576325	+	0.137813i	0.510287	−	0.860004i	\(-0.329539\pi\)
							0.0660376	+	0.997817i	\(0.478964\pi\)
\(3\)	−0.0632014	+	0.0115367i	−0.0364894	+	0.00666074i	−0.197786	−	0.980245i	\(-0.563375\pi\)
							0.161296	+	0.986906i	\(0.448432\pi\)
\(5\)	−0.348458	−	2.74240i	−0.155835	−	1.22644i	−0.857708	−	0.514137i	\(-0.828112\pi\)
							0.701873	−	0.712302i	\(-0.252348\pi\)
\(7\)	−1.20872	−	0.0218260i	−0.456853	−	0.00824944i	−0.210850	−	0.977518i	\(-0.567623\pi\)
							−0.246003	+	0.969269i	\(0.579117\pi\)
\(11\)	−0.554164	+	2.51759i	−0.167087	+	0.759083i	0.816310	+	0.577614i	\(0.196016\pi\)
							−0.983397	+	0.181469i	\(0.941915\pi\)
\(13\)	−2.44820	−	2.40440i	−0.679010	−	0.666859i	0.276772	−	0.960936i	\(-0.410735\pi\)
							−0.955782	+	0.294076i	\(0.904988\pi\)
\(17\)	−3.16962	−	3.73157i	−0.768746	−	0.905038i	0.228912	−	0.973447i	\(-0.426483\pi\)
							−0.997657	+	0.0684097i	\(0.978207\pi\)
\(19\)	4.76529	+	0.693130i	1.09323	+	0.159015i	0.666604	−	0.745412i	\(-0.267747\pi\)
							0.426628	+	0.904427i	\(0.359701\pi\)
\(23\)	1.84320	−	1.87678i	0.384333	−	0.391336i	−0.493939	−	0.869496i	\(-0.664444\pi\)
							0.878272	+	0.478161i	\(0.158696\pi\)
\(29\)	1.37104	−	0.407468i	0.254596	−	0.0756648i	−0.155060	−	0.987905i	\(-0.549557\pi\)
							0.409656	+	0.912240i	\(0.365649\pi\)
\(31\)	4.08546	−	7.70600i	0.733771	−	1.38404i	−0.182389	−	0.983226i	\(-0.558383\pi\)
							0.916160	−	0.400812i	\(-0.131272\pi\)
\(37\)	−0.617228	−	2.22306i	−0.101472	−	0.365468i	0.895328	−	0.445407i	\(-0.146941\pi\)
							−0.996800	+	0.0799393i	\(0.974527\pi\)
\(41\)	7.13193	−	8.39636i	1.11382	−	1.31129i	0.168297	−	0.985736i	\(-0.446173\pi\)
							0.945523	−	0.325554i	\(-0.105551\pi\)
\(43\)	−3.23093	−	2.95167i	−0.492713	−	0.450126i	0.389466	−	0.921041i	\(-0.372660\pi\)
							−0.882179	+	0.470915i	\(0.843924\pi\)
\(47\)	−1.42319	+	4.22387i	−0.207593	+	0.616114i	0.792388	+	0.610017i	\(0.208837\pi\)
							−0.999981	+	0.00609733i	\(0.998059\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	349.2.k.a.3.19	✓	1624
349.233	even	174	inner	349.2.k.a.233.19	yes	1624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
349.2.k.a.3.19	✓	1624	1.1	even	1	trivial
349.2.k.a.233.19	yes	1624	349.233	even	174	inner