Embedded newform 349.2.i.a.9.13

• Label: 349.2.i.a.9.13
• Level: $349$
• Weight: $2$
• Character: 349.9
• Analytic conductor: $2.787$
• Analytic rank: $0$
• Dimension: $1568$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 349 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	349.i (of order \(87\), degree \(56\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.78677903054\)
Analytic rank:	\(0\)
Dimension:	\(1568\)
Relative dimension:	\(28\) over \(\Q(\zeta_{87})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{87}]$

Embedding invariants

Embedding label			9.13
Character	\(\chi\)	\(=\)	349.9
Dual form			349.2.i.a.194.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.379345 - 0.192424i) q^{2} +(2.17990 - 0.823266i) q^{3} +(-1.07453 - 1.46779i) q^{4} +(2.33094 - 0.602075i) q^{5} +(-0.985352 - 0.107163i) q^{6} +(-2.42172 - 0.0874869i) q^{7} +(0.262811 + 1.60307i) q^{8} +(1.82315 - 1.60615i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1568 q - 58 q^{2} - 58 q^{3} - 28 q^{4} - 62 q^{5} - 72 q^{6} - 60 q^{7} + 23 q^{8} - 30 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{13}{87}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.379345	−	0.192424i	−0.268238	−	0.136064i	0.313117	−	0.949714i	\(-0.398627\pi\)
							−0.581355	+	0.813650i	\(0.697477\pi\)
\(3\)	2.17990	−	0.823266i	1.25857	−	0.475313i	0.367855	−	0.929883i	\(-0.380092\pi\)
							0.890711	+	0.454570i	\(0.150207\pi\)
\(5\)	2.33094	−	0.602075i	1.04243	−	0.269256i	0.310455	−	0.950588i	\(-0.399519\pi\)
							0.731975	+	0.681332i	\(0.238599\pi\)
\(7\)	−2.42172	−	0.0874869i	−0.915323	−	0.0330669i	−0.425567	−	0.904927i	\(-0.639925\pi\)
							−0.489756	+	0.871860i	\(0.662914\pi\)
\(11\)	5.64199	+	2.61026i	1.70112	+	0.787024i	0.997038	+	0.0769082i	\(0.0245048\pi\)
							0.704085	+	0.710115i	\(0.251357\pi\)
\(13\)	−0.0937233	−	5.19038i	−0.0259942	−	1.43955i	−0.706968	−	0.707246i	\(-0.749937\pi\)
							0.680974	−	0.732308i	\(-0.261557\pi\)
\(17\)	0.331017	−	2.01911i	0.0802833	−	0.489706i	−0.915826	−	0.401576i	\(-0.868463\pi\)
							0.996109	−	0.0881303i	\(-0.0280892\pi\)
\(19\)	−4.47525	−	1.33002i	−1.02669	−	0.305129i	−0.272761	−	0.962082i	\(-0.587937\pi\)
							−0.753931	+	0.656953i	\(0.771845\pi\)
\(23\)	0.0607694	+	3.36540i	0.0126713	+	0.701735i	0.942587	+	0.333961i	\(0.108386\pi\)
							−0.929915	+	0.367773i	\(0.880120\pi\)
\(29\)	−3.19863	+	2.08544i	−0.593972	+	0.387256i	−0.807662	−	0.589645i	\(-0.799268\pi\)
							0.213691	+	0.976901i	\(0.431451\pi\)
\(31\)	−2.78287	−	4.10442i	−0.499818	−	0.737176i	0.491179	−	0.871058i	\(-0.336566\pi\)
							−0.990997	+	0.133882i	\(0.957256\pi\)
\(37\)	6.23809	−	3.75333i	1.02554	−	0.617044i	0.0997185	−	0.995016i	\(-0.468206\pi\)
							0.925817	+	0.377971i	\(0.123378\pi\)
\(41\)	0.586711	+	3.57878i	0.0916289	+	0.558912i	0.992245	+	0.124296i	\(0.0396674\pi\)
							−0.900616	+	0.434615i	\(0.856884\pi\)
\(43\)	0.862720	+	9.53053i	0.131564	+	1.45339i	0.746799	+	0.665049i	\(0.231590\pi\)
							−0.615236	+	0.788343i	\(0.710939\pi\)
\(47\)	7.38327	+	5.61261i	1.07696	+	0.818684i	0.984210	−	0.177006i	\(-0.0566411\pi\)
							0.0927501	+	0.995689i	\(0.470434\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	349.2.i.a.9.13	✓	1568
349.194	even	87	inner	349.2.i.a.194.13	yes	1568

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
349.2.i.a.9.13	✓	1568	1.1	even	1	trivial
349.2.i.a.194.13	yes	1568	349.194	even	87	inner