Embedded newform 349.2.e.a.123.17

• Label: 349.2.e.a.123.17
• Level: $349$
• Weight: $2$
• Character: 349.123
• Analytic conductor: $2.787$
• Analytic rank: $0$
• Dimension: $58$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 349 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	349.e (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.78677903054\)
Analytic rank:	\(0\)
Dimension:	\(58\)
Relative dimension:	\(29\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			123.17
Character	\(\chi\)	\(=\)	349.123
Dual form			349.2.e.a.227.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.499056 - 0.288130i) q^{2} +(-0.844268 + 1.46231i) q^{3} +(-0.833962 + 1.44447i) q^{4} +(0.707762 - 1.22588i) q^{5} +0.973035i q^{6} +(1.68926 - 0.975293i) q^{7} +2.11368i q^{8} +(0.0744233 + 0.128905i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 58 q - 3 q^{2} + 27 q^{4} + 2 q^{5} - 29 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.499056	−	0.288130i	0.352886	−	0.203739i	−0.313070	−	0.949730i	\(-0.601357\pi\)
							0.665955	+	0.745991i	\(0.268024\pi\)
\(3\)	−0.844268	+	1.46231i	−0.487438	+	0.844268i	−0.999896	−	0.0144447i	\(-0.995402\pi\)
							0.512457	+	0.858713i	\(0.328735\pi\)
\(5\)	0.707762	−	1.22588i	0.316521	−	0.548230i	−0.663239	−	0.748408i	\(-0.730819\pi\)
							0.979760	+	0.200178i	\(0.0641520\pi\)
\(7\)	1.68926	−	0.975293i	0.638479	−	0.368626i	−0.145549	−	0.989351i	\(-0.546495\pi\)
							0.784028	+	0.620725i	\(0.213162\pi\)
\(11\)			4.02851i			1.21464i	0.794457	+	0.607320i	\(0.207756\pi\)
							−0.794457	+	0.607320i	\(0.792244\pi\)
\(13\)	−2.31456	+	1.33631i	−0.641942	+	0.370626i	−0.785362	−	0.619036i	\(-0.787523\pi\)
							0.143420	+	0.989662i	\(0.454190\pi\)
\(17\)	−4.81926			−1.16884			−0.584421	−	0.811451i	\(-0.698678\pi\)
							−0.584421	+	0.811451i	\(0.698678\pi\)
\(19\)	−0.128339	+	0.222289i	−0.0294430	+	0.0509967i	−0.880371	−	0.474285i	\(-0.842707\pi\)
							0.850928	+	0.525282i	\(0.176040\pi\)
\(23\)	4.23068	+	7.32776i	0.882158	+	1.52794i	0.848937	+	0.528495i	\(0.177243\pi\)
							0.0332215	+	0.999448i	\(0.489423\pi\)
\(29\)	4.16127	−	7.20752i	0.772728	−	1.33840i	−0.163335	−	0.986571i	\(-0.552225\pi\)
							0.936063	−	0.351833i	\(-0.114442\pi\)
\(31\)	−2.97089			−0.533588			−0.266794	−	0.963754i	\(-0.585964\pi\)
							−0.266794	+	0.963754i	\(0.585964\pi\)
\(37\)	3.32524			0.546666			0.273333	−	0.961919i	\(-0.411874\pi\)
							0.273333	+	0.961919i	\(0.411874\pi\)
\(41\)	8.59566			1.34242			0.671208	−	0.741269i	\(-0.265776\pi\)
							0.671208	+	0.741269i	\(0.265776\pi\)
\(43\)	5.22416	+	3.01617i	0.796677	+	0.459962i	0.842308	−	0.538997i	\(-0.181196\pi\)
							−0.0456307	+	0.998958i	\(0.514530\pi\)
\(47\)		−	6.48701i		−	0.946228i	−0.881001	−	0.473114i	\(-0.843130\pi\)
							0.881001	−	0.473114i	\(-0.156870\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	349.2.e.a.123.17	✓	58
349.227	even	6	inner	349.2.e.a.227.17	yes	58

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
349.2.e.a.123.17	✓	58	1.1	even	1	trivial
349.2.e.a.227.17	yes	58	349.227	even	6	inner