Embedded newform 349.2.i.a.9.11

• Label: 349.2.i.a.9.11
• 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			9.11
Character	\(\chi\)	\(=\)	349.9
Dual form			349.2.i.a.194.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.737293 - 0.373994i) q^{2} +(0.783668 - 0.295962i) q^{3} +(-0.777676 - 1.06229i) q^{4} +(-2.65376 + 0.685457i) q^{5} +(-0.688481 - 0.0748768i) q^{6} +(1.75900 + 0.0635456i) q^{7} +(0.443583 + 2.70573i) q^{8} +(-1.72451 + 1.51925i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1568 q - 58 q^{2} - 58 q^{3} - 28 q^{4} - 62 q^{5} - 72 q^{6} - 60 q^{7} + 23 q^{8} - 30 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.737293	−	0.373994i	−0.521345	−	0.264454i	0.171953	−	0.985105i	\(-0.444992\pi\)
							−0.693298	+	0.720651i	\(0.743843\pi\)
\(3\)	0.783668	−	0.295962i	0.452451	−	0.170874i	−0.116595	−	0.993180i	\(-0.537198\pi\)
							0.569046	+	0.822306i	\(0.307313\pi\)
\(5\)	−2.65376	+	0.685457i	−1.18680	+	0.306546i	−0.791014	−	0.611798i	\(-0.790446\pi\)
							−0.395783	+	0.918344i	\(0.629527\pi\)
\(7\)	1.75900	+	0.0635456i	0.664840	+	0.0240180i	0.366467	−	0.930431i	\(-0.380567\pi\)
							0.298373	+	0.954449i	\(0.403556\pi\)
\(11\)	−3.44475	−	1.59371i	−1.03863	−	0.480522i	−0.174960	−	0.984576i	\(-0.555980\pi\)
							−0.863673	+	0.504053i	\(0.831842\pi\)
\(13\)	−0.0713820	−	3.95313i	−0.0197978	−	1.09640i	−0.846051	−	0.533102i	\(-0.821026\pi\)
							0.826253	−	0.563299i	\(-0.190468\pi\)
\(17\)	−1.19339	+	7.27934i	−0.289439	+	1.76550i	0.296392	+	0.955066i	\(0.404217\pi\)
							−0.585831	+	0.810433i	\(0.699232\pi\)
\(19\)	−6.19179	−	1.84017i	−1.42049	−	0.422165i	−0.518933	−	0.854815i	\(-0.673670\pi\)
							−0.901562	+	0.432650i	\(0.857578\pi\)
\(23\)	0.124268	+	6.88194i	0.0259116	+	1.43498i	0.709288	+	0.704918i	\(0.249016\pi\)
							−0.683377	+	0.730066i	\(0.739489\pi\)
\(29\)	2.76180	−	1.80063i	0.512852	−	0.334368i	−0.263508	−	0.964657i	\(-0.584880\pi\)
							0.776361	+	0.630289i	\(0.217064\pi\)
\(31\)	−0.571679	−	0.843163i	−0.102676	−	0.151436i	0.772880	−	0.634552i	\(-0.218815\pi\)
							−0.875557	+	0.483116i	\(0.839505\pi\)
\(37\)	−3.86150	+	2.32339i	−0.634826	+	0.381962i	−0.796289	−	0.604917i	\(-0.793206\pi\)
							0.161463	+	0.986879i	\(0.448379\pi\)
\(41\)	−1.55521	−	9.48639i	−0.242884	−	1.48152i	−0.773437	−	0.633873i	\(-0.781464\pi\)
							0.530554	−	0.847651i	\(-0.321984\pi\)
\(43\)	0.203686	+	2.25013i	0.0310618	+	0.343142i	0.996569	+	0.0827657i	\(0.0263753\pi\)
							−0.965507	+	0.260376i	\(0.916153\pi\)
\(47\)	2.47988	+	1.88515i	0.361727	+	0.274978i	0.770213	−	0.637787i	\(-0.220150\pi\)
							−0.408486	+	0.912765i	\(0.633943\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	349.2.i.a.9.11	✓	1568
349.194	even	87	inner	349.2.i.a.194.11	yes	1568

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
349.2.i.a.9.11	✓	1568	1.1	even	1	trivial
349.2.i.a.194.11	yes	1568	349.194	even	87	inner