Embedded newform 349.2.e.a.227.13

• Label: 349.2.e.a.227.13
• Level: $349$
• Weight: $2$
• Character: 349.227
• 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			227.13
Character	\(\chi\)	\(=\)	349.227
Dual form			349.2.e.a.123.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.529073 - 0.305460i) q^{2} +(0.484492 + 0.839165i) q^{3} +(-0.813388 - 1.40883i) q^{4} +(-0.0329412 - 0.0570558i) q^{5} -0.591972i q^{6} +(-3.18284 - 1.83761i) q^{7} +2.21567i q^{8} +(1.03054 - 1.78494i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 58 q - 3 q^{2} + 27 q^{4} + 2 q^{5} - 29 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.529073	−	0.305460i	−0.374111	−	0.215993i	0.301142	−	0.953579i	\(-0.402632\pi\)
							−0.675253	+	0.737586i	\(0.735966\pi\)
\(3\)	0.484492	+	0.839165i	0.279722	+	0.484492i	0.971315	−	0.237795i	\(-0.0764245\pi\)
							−0.691594	+	0.722287i	\(0.743091\pi\)
\(5\)	−0.0329412	−	0.0570558i	−0.0147317	−	0.0255161i	0.858566	−	0.512704i	\(-0.171356\pi\)
							−0.873297	+	0.487188i	\(0.838023\pi\)
\(7\)	−3.18284	−	1.83761i	−1.20300	−	0.694553i	−0.241780	−	0.970331i	\(-0.577731\pi\)
							−0.961221	+	0.275778i	\(0.911064\pi\)
\(11\)			2.60755i			0.786207i	0.919494	+	0.393103i	\(0.128599\pi\)
							−0.919494	+	0.393103i	\(0.871401\pi\)
\(13\)	−3.80680	−	2.19786i	−1.05582	−	0.609575i	−0.131544	−	0.991310i	\(-0.541993\pi\)
							−0.924272	+	0.381735i	\(0.875327\pi\)
\(17\)	−5.89706			−1.43025			−0.715124	−	0.698998i	\(-0.753630\pi\)
							−0.715124	+	0.698998i	\(0.753630\pi\)
\(19\)	−1.73717	−	3.00886i	−0.398533	−	0.690280i	0.595012	−	0.803717i	\(-0.297147\pi\)
							−0.993545	+	0.113437i	\(0.963814\pi\)
\(23\)	−0.241590	+	0.418446i	−0.0503750	+	0.0872521i	−0.890113	−	0.455739i	\(-0.849375\pi\)
							0.839738	+	0.542991i	\(0.182708\pi\)
\(29\)	−1.77131	−	3.06800i	−0.328924	−	0.569713i	0.653375	−	0.757035i	\(-0.273353\pi\)
							−0.982299	+	0.187322i	\(0.940019\pi\)
\(31\)	2.61433			0.469548			0.234774	−	0.972050i	\(-0.424565\pi\)
							0.234774	+	0.972050i	\(0.424565\pi\)
\(37\)	−10.1686			−1.67170			−0.835850	−	0.548958i	\(-0.815025\pi\)
							−0.835850	+	0.548958i	\(0.815025\pi\)
\(41\)	0.703340			0.109843			0.0549216	−	0.998491i	\(-0.482509\pi\)
							0.0549216	+	0.998491i	\(0.482509\pi\)
\(43\)	3.36545	−	1.94304i	0.513226	−	0.296311i	−0.220933	−	0.975289i	\(-0.570910\pi\)
							0.734159	+	0.678978i	\(0.237577\pi\)
\(47\)		−	6.07101i		−	0.885548i	−0.896633	−	0.442774i	\(-0.853994\pi\)
							0.896633	−	0.442774i	\(-0.146006\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	349.2.e.a.227.13	yes	58
349.123	even	6	inner	349.2.e.a.123.13	✓	58

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
349.2.e.a.123.13	✓	58	349.123	even	6	inner
349.2.e.a.227.13	yes	58	1.1	even	1	trivial