Embedded newform 349.2.e.a.227.6

• Label: 349.2.e.a.227.6
• 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			227.6
Character	\(\chi\)	\(=\)	349.227
Dual form			349.2.e.a.123.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.71754 - 0.991624i) q^{2} +(-1.29117 - 2.23637i) q^{3} +(0.966636 + 1.67426i) q^{4} +(-0.796466 - 1.37952i) q^{5} +5.12141i q^{6} +(-1.81826 - 1.04977i) q^{7} +0.132337i q^{8} +(-1.83422 + 3.17696i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 58 q - 3 q^{2} + 27 q^{4} + 2 q^{5} - 29 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.71754	−	0.991624i	−1.21449	−	0.701184i	−0.250753	−	0.968051i	\(-0.580678\pi\)
							−0.963733	+	0.266867i	\(0.914012\pi\)
\(3\)	−1.29117	−	2.23637i	−0.745455	−	1.29117i	−0.949982	−	0.312305i	\(-0.898899\pi\)
							0.204527	−	0.978861i	\(-0.434435\pi\)
\(5\)	−0.796466	−	1.37952i	−0.356190	−	0.616940i	0.631130	−	0.775677i	\(-0.282591\pi\)
							−0.987321	+	0.158737i	\(0.949258\pi\)
\(7\)	−1.81826	−	1.04977i	−0.687238	−	0.396777i	0.115339	−	0.993326i	\(-0.463205\pi\)
							−0.802576	+	0.596549i	\(0.796538\pi\)
\(11\)			2.83760i			0.855569i	0.903881	+	0.427785i	\(0.140706\pi\)
							−0.903881	+	0.427785i	\(0.859294\pi\)
\(13\)	−3.23077	−	1.86529i	−0.896055	−	0.517337i	−0.0201366	−	0.999797i	\(-0.506410\pi\)
							−0.875918	+	0.482460i	\(0.839743\pi\)
\(17\)	−1.69939			−0.412162			−0.206081	−	0.978535i	\(-0.566071\pi\)
							−0.206081	+	0.978535i	\(0.566071\pi\)
\(19\)	0.686145	+	1.18844i	0.157412	+	0.272646i	0.933935	−	0.357443i	\(-0.116351\pi\)
							−0.776522	+	0.630090i	\(0.783018\pi\)
\(23\)	0.0635168	−	0.110014i	0.0132442	−	0.0229396i	−0.859327	−	0.511426i	\(-0.829117\pi\)
							0.872572	+	0.488486i	\(0.162451\pi\)
\(29\)	0.829833	+	1.43731i	0.154096	+	0.266902i	0.932730	−	0.360577i	\(-0.117420\pi\)
							−0.778633	+	0.627479i	\(0.784087\pi\)
\(31\)	1.80717			0.324577			0.162288	−	0.986743i	\(-0.448113\pi\)
							0.162288	+	0.986743i	\(0.448113\pi\)
\(37\)	4.87375			0.801239			0.400620	−	0.916244i	\(-0.368795\pi\)
							0.400620	+	0.916244i	\(0.368795\pi\)
\(41\)	−0.265830			−0.0415157			−0.0207578	−	0.999785i	\(-0.506608\pi\)
							−0.0207578	+	0.999785i	\(0.506608\pi\)
\(43\)	5.78504	−	3.34000i	0.882211	−	0.509345i	0.0108241	−	0.999941i	\(-0.496555\pi\)
							0.871387	+	0.490597i	\(0.163221\pi\)
\(47\)			7.15836i			1.04415i	0.852898	+	0.522077i	\(0.174843\pi\)
							−0.852898	+	0.522077i	\(0.825157\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	349.2.e.a.227.6	yes	58
349.123	even	6	inner	349.2.e.a.123.6	✓	58

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
349.2.e.a.123.6	✓	58	349.123	even	6	inner
349.2.e.a.227.6	yes	58	1.1	even	1	trivial