Embedded newform 289.2.f.a.154.7

• Label: 289.2.f.a.154.7
• Level: $289$
• Weight: $2$
• Character: 289.154
• Analytic conductor: $2.308$
• Analytic rank: $0$
• Dimension: $384$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 289 = 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	289.f (of order \(17\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.30767661842\)
Analytic rank:	\(0\)
Dimension:	\(384\)
Relative dimension:	\(24\) over \(\Q(\zeta_{17})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{17}]$

Embedding invariants

Embedding label			154.7
Character	\(\chi\)	\(=\)	289.154
Dual form			289.2.f.a.137.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.52526 - 0.285120i) q^{2} +(0.0891375 + 0.179012i) q^{3} +(0.380167 + 0.147278i) q^{4} +(2.68178 - 1.66049i) q^{5} +(-0.0849176 - 0.298454i) q^{6} +(-3.14329 + 4.16239i) q^{7} +(2.10066 + 1.30068i) q^{8} +(1.78380 - 2.36214i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q - 13 q^{2} - 13 q^{3} - 33 q^{4} - 43 q^{5} - 5 q^{6} - 13 q^{7} - 5 q^{8} - 25 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/289\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{6}{17}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.52526	−	0.285120i	−1.07852	−	0.201610i	−0.385629	−	0.922654i	\(-0.626016\pi\)
							−0.692889	+	0.721044i	\(0.743663\pi\)
\(3\)	0.0891375	+	0.179012i	0.0514636	+	0.103353i	0.919402	−	0.393319i	\(-0.128673\pi\)
							−0.867939	+	0.496671i	\(0.834555\pi\)
\(5\)	2.68178	−	1.66049i	1.19933	−	0.742593i	0.226476	−	0.974017i	\(-0.427280\pi\)
							0.972853	+	0.231424i	\(0.0743384\pi\)
\(7\)	−3.14329	+	4.16239i	−1.18805	+	1.57324i	−0.459628	+	0.888112i	\(0.652017\pi\)
							−0.728425	+	0.685125i	\(0.759747\pi\)
\(11\)	1.90002	+	0.736072i	0.572878	+	0.221934i	0.630215	−	0.776421i	\(-0.282967\pi\)
							−0.0573369	+	0.998355i	\(0.518261\pi\)
\(13\)	−0.700706	−	0.433859i	−0.194341	−	0.120331i	0.425812	−	0.904812i	\(-0.359988\pi\)
							−0.620153	+	0.784481i	\(0.712930\pi\)
\(17\)	−1.83557	+	3.69197i	−0.445192	+	0.895435i
							\(19\)	7.03406	−	1.31489i	1.61372	−	0.301657i	0.701912	−	0.712264i	\(-0.252330\pi\)
0.911812	+	0.410607i	\(0.134683\pi\)
\(23\)	3.19766	−	4.23438i	0.666758	−	0.882930i	−0.331407	−	0.943488i	\(-0.607523\pi\)
							0.998165	+	0.0605580i	\(0.0192880\pi\)
\(29\)	4.91577	+	1.90438i	0.912836	+	0.353634i	0.771428	−	0.636317i	\(-0.219543\pi\)
							0.141408	+	0.989951i	\(0.454837\pi\)
\(31\)	8.86011	+	5.48595i	1.59132	+	0.985306i	0.981199	+	0.193000i	\(0.0618217\pi\)
							0.610124	+	0.792306i	\(0.291119\pi\)
\(37\)	0.579671	−	6.25564i	0.0952973	−	1.02842i	−0.805038	−	0.593224i	\(-0.797855\pi\)
							0.900335	−	0.435198i	\(-0.143322\pi\)
\(41\)	−1.04544	−	2.09953i	−0.163270	−	0.327891i	0.798387	−	0.602145i	\(-0.205687\pi\)
							−0.961657	+	0.274253i	\(0.911569\pi\)
\(43\)	−2.45643	+	2.23933i	−0.374601	+	0.341494i	−0.838999	−	0.544132i	\(-0.816859\pi\)
							0.464398	+	0.885627i	\(0.346271\pi\)
\(47\)	−1.31131	−	1.73646i	−0.191275	−	0.253289i	0.692267	−	0.721642i	\(-0.256612\pi\)
							−0.883541	+	0.468353i	\(0.844848\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	289.2.f.a.154.7	yes	384
289.137	even	17	inner	289.2.f.a.137.7	✓	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
289.2.f.a.137.7	✓	384	289.137	even	17	inner
289.2.f.a.154.7	yes	384	1.1	even	1	trivial