Embedded newform 349.2.k.a.3.2

• Label: 349.2.k.a.3.2
• 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			3.2
Character	\(\chi\)	\(=\)	349.3
Dual form			349.2.k.a.233.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.68985 - 0.643207i) q^{2} +(-1.01045 + 0.184447i) q^{3} +(5.03791 + 2.55550i) q^{4} +(0.223073 + 1.75561i) q^{5} +(2.83660 + 0.153796i) q^{6} +(-3.93989 - 0.0711429i) q^{7} +(-7.69170 - 6.53339i) q^{8} +(-1.81953 + 0.687167i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1624 q - 55 q^{2} - 58 q^{3} - 85 q^{4} - 60 q^{5} - 58 q^{6} - 58 q^{7} - 145 q^{8} - 29 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.68985	−	0.643207i	−1.90201	−	0.454816i	−0.999719	−	0.0236859i	\(-0.992460\pi\)
							−0.902290	−	0.431130i	\(-0.858115\pi\)
\(3\)	−1.01045	+	0.184447i	−0.583386	+	0.106491i	−0.463189	−	0.886259i	\(-0.653295\pi\)
							−0.120196	+	0.992750i	\(0.538352\pi\)
\(5\)	0.223073	+	1.75561i	0.0997614	+	0.785132i	0.960926	+	0.276806i	\(0.0892759\pi\)
							−0.861164	+	0.508326i	\(0.830264\pi\)
\(7\)	−3.93989	−	0.0711429i	−1.48914	−	0.0268895i	−0.732519	−	0.680746i	\(-0.761656\pi\)
							−0.756618	+	0.653857i	\(0.773150\pi\)
\(11\)	0.176390	−	0.801346i	0.0531835	−	0.241615i	−0.942528	−	0.334127i	\(-0.891559\pi\)
							0.995712	+	0.0925120i	\(0.0294896\pi\)
\(13\)	1.96724	+	1.93204i	0.545614	+	0.535850i	0.920255	−	0.391319i	\(-0.127981\pi\)
							−0.374641	+	0.927170i	\(0.622234\pi\)
\(17\)	−3.21081	−	3.78005i	−0.778735	−	0.916798i	0.219534	−	0.975605i	\(-0.429546\pi\)
							−0.998269	+	0.0588071i	\(0.981270\pi\)
\(19\)	3.38928	+	0.492984i	0.777554	+	0.113098i	0.521143	−	0.853470i	\(-0.325506\pi\)
							0.256411	+	0.966568i	\(0.417460\pi\)
\(23\)	3.32672	−	3.38733i	0.693669	−	0.706308i	−0.273092	−	0.961988i	\(-0.588046\pi\)
							0.966762	+	0.255680i	\(0.0822993\pi\)
\(29\)	−9.61743	+	2.85826i	−1.78591	+	0.530766i	−0.996537	−	0.0831457i	\(-0.973503\pi\)
							−0.789375	+	0.613911i	\(0.789595\pi\)
\(31\)	2.80785	−	5.29616i	0.504304	−	0.951218i	−0.492421	−	0.870357i	\(-0.663888\pi\)
							0.996725	−	0.0808613i	\(-0.0257671\pi\)
\(37\)	−0.375640	−	1.35293i	−0.0617548	−	0.222421i	0.926233	−	0.376951i	\(-0.123027\pi\)
							−0.987988	+	0.154530i	\(0.950614\pi\)
\(41\)	4.31197	−	5.07644i	0.673416	−	0.792807i	−0.314257	−	0.949338i	\(-0.601755\pi\)
							0.987673	+	0.156531i	\(0.0500311\pi\)
\(43\)	0.140614	+	0.128460i	0.0214434	+	0.0195900i	0.685131	−	0.728420i	\(-0.259745\pi\)
							−0.663687	+	0.748010i	\(0.731009\pi\)
\(47\)	3.74922	−	11.1273i	0.546880	−	1.62308i	−0.215895	−	0.976417i	\(-0.569267\pi\)
							0.762775	−	0.646664i	\(-0.223836\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	349.2.k.a.3.2	✓	1624
349.233	even	174	inner	349.2.k.a.233.2	yes	1624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
349.2.k.a.3.2	✓	1624	1.1	even	1	trivial
349.2.k.a.233.2	yes	1624	349.233	even	174	inner