Embedded newform 349.2.k.a.3.8

• Label: 349.2.k.a.3.8
• Level: $349$
• Weight: $2$
• Character: 349.3
• Analytic conductor: $2.787$
• Analytic rank: $0$
• Dimension: $1624$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 349 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	349.k (of order \(174\), degree \(56\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.78677903054\)
Analytic rank:	\(0\)
Dimension:	\(1624\)
Relative dimension:	\(29\) over \(\Q(\zeta_{174})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{174}]$

Embedding invariants

Embedding label			3.8
Character	\(\chi\)	\(=\)	349.3
Dual form			349.2.k.a.233.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.46055 - 0.349254i) q^{2} +(3.22574 - 0.588823i) q^{3} +(0.227590 + 0.115446i) q^{4} +(-0.420184 - 3.30689i) q^{5} +(-4.91701 - 0.266593i) q^{6} +(1.13449 + 0.0204856i) q^{7} +(1.99704 + 1.69630i) q^{8} +(7.25215 - 2.73886i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1624 q - 55 q^{2} - 58 q^{3} - 85 q^{4} - 60 q^{5} - 58 q^{6} - 58 q^{7} - 145 q^{8} - 29 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{13}{174}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.46055	−	0.349254i	−1.03277	−	0.246960i	−0.319304	−	0.947652i	\(-0.603449\pi\)
							−0.713463	+	0.700693i	\(0.752874\pi\)
\(3\)	3.22574	−	0.588823i	1.86238	−	0.339957i	0.873282	−	0.487215i	\(-0.161987\pi\)
							0.989098	+	0.147257i	\(0.0470445\pi\)
\(5\)	−0.420184	−	3.30689i	−0.187912	−	1.47889i	−0.755269	−	0.655415i	\(-0.772494\pi\)
							0.567357	−	0.823472i	\(-0.307966\pi\)
\(7\)	1.13449	+	0.0204856i	0.428797	+	0.00774282i	0.232033	−	0.972708i	\(-0.425462\pi\)
							0.196764	+	0.980451i	\(0.436957\pi\)
\(11\)	−0.722390	+	3.28185i	−0.217809	+	0.989515i	0.733119	+	0.680101i	\(0.238064\pi\)
							−0.950927	+	0.309414i	\(0.899867\pi\)
\(13\)	1.47878	+	1.45232i	0.410139	+	0.402800i	0.876196	−	0.481954i	\(-0.160073\pi\)
							−0.466057	+	0.884755i	\(0.654326\pi\)
\(17\)	−4.34143	−	5.11113i	−1.05295	−	1.23963i	−0.970007	−	0.243077i	\(-0.921843\pi\)
							−0.0829455	−	0.996554i	\(-0.526433\pi\)
\(19\)	1.92031	+	0.279317i	0.440549	+	0.0640796i	0.360603	−	0.932720i	\(-0.382571\pi\)
							0.0799464	+	0.996799i	\(0.474525\pi\)
\(23\)	−2.35429	+	2.39718i	−0.490903	+	0.499847i	−0.913707	−	0.406374i	\(-0.866793\pi\)
							0.422804	+	0.906221i	\(0.361046\pi\)
\(29\)	−5.14313	+	1.52852i	−0.955056	+	0.283838i	−0.724542	−	0.689231i	\(-0.757949\pi\)
							−0.230514	+	0.973069i	\(0.574041\pi\)
\(31\)	−2.49133	+	4.69915i	−0.447456	+	0.843992i	0.552489	+	0.833520i	\(0.313678\pi\)
							−0.999945	+	0.0104716i	\(0.996667\pi\)
\(37\)	−0.756014	−	2.72292i	−0.124288	−	0.447645i	0.875050	−	0.484033i	\(-0.160828\pi\)
							−0.999338	+	0.0363883i	\(0.988415\pi\)
\(41\)	7.65740	−	9.01499i	1.19589	−	1.40790i	0.305809	−	0.952093i	\(-0.401073\pi\)
							0.890076	−	0.455812i	\(-0.150651\pi\)
\(43\)	8.38390	+	7.65925i	1.27853	+	1.16803i	0.976687	+	0.214668i	\(0.0688671\pi\)
							0.301846	+	0.953357i	\(0.402397\pi\)
\(47\)	−1.97731	+	5.86846i	−0.288421	+	0.856003i	0.701284	+	0.712882i	\(0.252610\pi\)
							−0.989705	+	0.143121i	\(0.954286\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	349.2.k.a.3.8	✓	1624
349.233	even	174	inner	349.2.k.a.233.8	yes	1624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
349.2.k.a.3.8	✓	1624	1.1	even	1	trivial
349.2.k.a.233.8	yes	1624	349.233	even	174	inner