Embedded newform 349.2.k.a.3.13

• Label: 349.2.k.a.3.13
• 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			3.13
Character	\(\chi\)	\(=\)	349.3
Dual form			349.2.k.a.233.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.633957 - 0.151594i) q^{2} +(-2.19680 + 0.401003i) q^{3} +(-1.40473 - 0.712553i) q^{4} +(-0.512518 - 4.03357i) q^{5} +(1.45347 + 0.0788048i) q^{6} +(-5.00988 - 0.0904639i) q^{7} +(1.77612 + 1.50865i) q^{8} +(1.85862 - 0.701932i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1624 q - 55 q^{2} - 58 q^{3} - 85 q^{4} - 60 q^{5} - 58 q^{6} - 58 q^{7} - 145 q^{8} - 29 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.633957	−	0.151594i	−0.448275	−	0.107193i	0.00216977	−	0.999998i	\(-0.499309\pi\)
							−0.450445	+	0.892804i	\(0.648735\pi\)
\(3\)	−2.19680	+	0.401003i	−1.26833	+	0.231519i	−0.771443	−	0.636299i	\(-0.780465\pi\)
							−0.496883	+	0.867818i	\(0.665522\pi\)
\(5\)	−0.512518	−	4.03357i	−0.229205	−	1.80387i	−0.527674	−	0.849447i	\(-0.676936\pi\)
							0.298469	−	0.954419i	\(-0.403524\pi\)
\(7\)	−5.00988	−	0.0904639i	−1.89356	−	0.0341921i	−0.940975	−	0.338475i	\(-0.890089\pi\)
							−0.952582	+	0.304283i	\(0.901583\pi\)
\(11\)	0.624151	−	2.83555i	0.188189	−	0.854950i	−0.784002	−	0.620759i	\(-0.786825\pi\)
							0.972191	−	0.234191i	\(-0.0752442\pi\)
\(13\)	4.87003	+	4.78288i	1.35070	+	1.32653i	0.901294	+	0.433207i	\(0.142618\pi\)
							0.449408	+	0.893326i	\(0.351635\pi\)
\(17\)	−1.76471	−	2.07758i	−0.428006	−	0.503887i	0.505313	−	0.862936i	\(-0.331377\pi\)
							−0.933319	+	0.359049i	\(0.883101\pi\)
\(19\)	−4.46256	−	0.649097i	−1.02378	−	0.148913i	−0.388704	−	0.921363i	\(-0.627077\pi\)
							−0.635076	+	0.772450i	\(0.719031\pi\)
\(23\)	−0.447926	+	0.456087i	−0.0933990	+	0.0951008i	−0.758575	−	0.651586i	\(-0.774104\pi\)
							0.665176	+	0.746687i	\(0.268357\pi\)
\(29\)	4.34925	−	1.29258i	0.807635	−	0.240026i	0.145450	−	0.989366i	\(-0.453537\pi\)
							0.662186	+	0.749340i	\(0.269629\pi\)
\(31\)	−0.308731	+	0.582328i	−0.0554497	+	0.104589i	−0.909688	−	0.415293i	\(-0.863679\pi\)
							0.854238	+	0.519882i	\(0.174024\pi\)
\(37\)	1.05218	+	3.78960i	0.172977	+	0.623007i	0.998297	+	0.0583412i	\(0.0185811\pi\)
							−0.825320	+	0.564666i	\(0.809005\pi\)
\(41\)	0.440521	−	0.518622i	0.0687978	−	0.0809951i	−0.726686	−	0.686969i	\(-0.758941\pi\)
							0.795484	+	0.605974i	\(0.207217\pi\)
\(43\)	−1.27848	−	1.16798i	−0.194966	−	0.178115i	0.571092	−	0.820886i	\(-0.306520\pi\)
							−0.766058	+	0.642771i	\(0.777784\pi\)
\(47\)	−1.79756	+	5.33496i	−0.262201	+	0.778184i	0.732991	+	0.680238i	\(0.238124\pi\)
							−0.995192	+	0.0979458i	\(0.968773\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	349.2.k.a.3.13	✓	1624
349.233	even	174	inner	349.2.k.a.233.13	yes	1624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
349.2.k.a.3.13	✓	1624	1.1	even	1	trivial
349.2.k.a.233.13	yes	1624	349.233	even	174	inner