Embedded newform 349.2.i.a.9.19

• Label: 349.2.i.a.9.19
• 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			9.19
Character	\(\chi\)	\(=\)	349.9
Dual form			349.2.i.a.194.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.07325 + 0.544407i) q^{2} +(-2.77045 + 1.04630i) q^{3} +(-0.325928 - 0.445212i) q^{4} +(3.94990 - 1.02025i) q^{5} +(-3.54299 - 0.385323i) q^{6} +(-2.35651 - 0.0851310i) q^{7} +(-0.496810 - 3.03041i) q^{8} +(4.32963 - 3.81430i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1568 q - 58 q^{2} - 58 q^{3} - 28 q^{4} - 62 q^{5} - 72 q^{6} - 60 q^{7} + 23 q^{8} - 30 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.07325	+	0.544407i	0.758899	+	0.384954i	0.788841	−	0.614597i	\(-0.210682\pi\)
							−0.0299415	+	0.999552i	\(0.509532\pi\)
\(3\)	−2.77045	+	1.04630i	−1.59952	+	0.604079i	−0.983057	−	0.183303i	\(-0.941321\pi\)
							−0.616466	+	0.787382i	\(0.711436\pi\)
\(5\)	3.94990	−	1.02025i	1.76645	−	0.456268i	0.780759	−	0.624833i	\(-0.214833\pi\)
							0.985691	+	0.168565i	\(0.0539133\pi\)
\(7\)	−2.35651	−	0.0851310i	−0.890675	−	0.0321765i	−0.413018	−	0.910723i	\(-0.635525\pi\)
							−0.477657	+	0.878546i	\(0.658514\pi\)
\(11\)	3.12230	+	1.44453i	0.941408	+	0.435542i	0.829706	−	0.558200i	\(-0.188508\pi\)
							0.111702	+	0.993742i	\(0.464370\pi\)
\(13\)	−0.0757993	−	4.19776i	−0.0210230	−	1.16425i	−0.823411	−	0.567446i	\(-0.807932\pi\)
							0.802388	−	0.596803i	\(-0.203563\pi\)
\(17\)	−0.0458946	+	0.279945i	−0.0111311	+	0.0678966i	−0.991808	−	0.127738i	\(-0.959228\pi\)
							0.980677	+	0.195635i	\(0.0626766\pi\)
\(19\)	8.13139	+	2.41662i	1.86547	+	0.554410i	0.998420	+	0.0561876i	\(0.0178945\pi\)
							0.867049	+	0.498222i	\(0.166014\pi\)
\(23\)	0.0419888	+	2.32534i	0.00875528	+	0.484866i	0.974378	+	0.224918i	\(0.0722113\pi\)
							−0.965622	+	0.259949i	\(0.916294\pi\)
\(29\)	−1.17111	+	0.763537i	−0.217470	+	0.141785i	−0.650267	−	0.759706i	\(-0.725343\pi\)
							0.432797	+	0.901491i	\(0.357527\pi\)
\(31\)	−4.33366	−	6.39168i	−0.778349	−	1.14798i	−0.985501	−	0.169670i	\(-0.945730\pi\)
							0.207152	−	0.978309i	\(-0.433581\pi\)
\(37\)	1.07063	−	0.644178i	0.176011	−	0.105902i	−0.424822	−	0.905277i	\(-0.639663\pi\)
							0.600833	+	0.799375i	\(0.294836\pi\)
\(41\)	1.30643	+	7.96885i	0.204030	+	1.24453i	0.867923	+	0.496698i	\(0.165454\pi\)
							−0.663894	+	0.747827i	\(0.731097\pi\)
\(43\)	−0.862775	−	9.53114i	−0.131572	−	1.45348i	−0.746754	−	0.665100i	\(-0.768389\pi\)
							0.615182	−	0.788385i	\(-0.289082\pi\)
\(47\)	−7.76049	−	5.89937i	−1.13198	−	0.860512i	−0.140415	−	0.990093i	\(-0.544844\pi\)
							−0.991569	+	0.129581i	\(0.958637\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	349.2.i.a.9.19	✓	1568
349.194	even	87	inner	349.2.i.a.194.19	yes	1568

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
349.2.i.a.9.19	✓	1568	1.1	even	1	trivial
349.2.i.a.194.19	yes	1568	349.194	even	87	inner