Embedded newform 959.2.o.b.50.16

• Label: 959.2.o.b.50.16
• Level: $959$
• Weight: $2$
• Character: 959.50
• Analytic conductor: $7.658$
• Analytic rank: $0$
• Dimension: $576$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 959 = 7 \cdot 137 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	959.o (of order \(17\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.65765355384\)
Analytic rank:	\(0\)
Dimension:	\(576\)
Relative dimension:	\(36\) over \(\Q(\zeta_{17})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{17}]$

Embedding invariants

Embedding label			50.16
Character	\(\chi\)	\(=\)	959.50
Dual form			959.2.o.b.211.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.186070 + 0.169625i) q^{2} +(0.157810 + 0.0977116i) q^{3} +(-0.178687 + 1.92834i) q^{4} +(0.754055 + 0.687412i) q^{5} +(-0.0459380 + 0.00858729i) q^{6} +(-0.273663 - 0.961826i) q^{7} +(-0.597314 - 0.790971i) q^{8} +(-1.32186 - 2.65465i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 576 q - 6 q^{2} - 4 q^{3} - 14 q^{4} - 6 q^{5} - 16 q^{6} - 36 q^{7} - 24 q^{8} - 54 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/959\mathbb{Z}\right)^\times\).

\(n\)	\(414\)	\(549\)
\(\chi(n)\)	\(e\left(\frac{3}{17}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.186070	+	0.169625i	−0.131571	+	0.119943i	−0.736807	−	0.676103i	\(-0.763667\pi\)
							0.605235	+	0.796047i	\(0.293079\pi\)
\(3\)	0.157810	+	0.0977116i	0.0911114	+	0.0564138i	0.571232	−	0.820789i	\(-0.306466\pi\)
							−0.480120	+	0.877203i	\(0.659407\pi\)
\(5\)	0.754055	+	0.687412i	0.337224	+	0.307420i	0.824538	−	0.565807i	\(-0.191435\pi\)
							−0.487314	+	0.873227i	\(0.662023\pi\)
\(7\)	−0.273663	−	0.961826i	−0.103435	−	0.363536i
							\(11\)	0.541161	−	5.84006i	0.163166	−	1.76084i	−0.383544	−	0.923523i	\(-0.625297\pi\)
0.546710	−	0.837322i	\(-0.315880\pi\)
\(13\)	−0.885684	−	3.11286i	−0.245645	−	0.863351i	−0.982337	−	0.187123i	\(-0.940084\pi\)
							0.736692	−	0.676228i	\(-0.236387\pi\)
\(17\)	−3.87236	+	5.12783i	−0.939185	+	1.24368i	0.0305531	+	0.999533i	\(0.490273\pi\)
							−0.969738	+	0.244149i	\(0.921492\pi\)
\(19\)	0.321333	+	0.124485i	0.0737189	+	0.0285588i	0.397819	−	0.917464i	\(-0.369767\pi\)
							−0.324100	+	0.946023i	\(0.605061\pi\)
\(23\)	−6.71817	−	1.25584i	−1.40083	−	0.261861i	−0.571495	−	0.820606i	\(-0.693636\pi\)
							−0.829340	+	0.558745i	\(0.811283\pi\)
\(29\)	−6.50297	−	1.21562i	−1.20757	−	0.225734i	−0.458786	−	0.888547i	\(-0.651716\pi\)
							−0.748785	+	0.662812i	\(0.769363\pi\)
\(31\)	−3.07973	+	1.19309i	−0.553135	+	0.214286i	−0.621554	−	0.783371i	\(-0.713499\pi\)
							0.0684194	+	0.997657i	\(0.478204\pi\)
\(37\)	3.07961			0.506284			0.253142	−	0.967429i	\(-0.418536\pi\)
							0.253142	+	0.967429i	\(0.418536\pi\)
\(41\)	10.4787			1.63649			0.818247	−	0.574866i	\(-0.194946\pi\)
							0.818247	+	0.574866i	\(0.194946\pi\)
\(43\)	−3.06727	−	1.18827i	−0.467754	−	0.181209i	0.115817	−	0.993271i	\(-0.463051\pi\)
							−0.583572	+	0.812061i	\(0.698345\pi\)
\(47\)	−3.13915	−	6.30426i	−0.457892	−	0.919571i	−0.997005	−	0.0773377i	\(-0.975358\pi\)
							0.539113	−	0.842234i	\(-0.318760\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	959.2.o.b.50.16	✓	576
137.74	even	17	inner	959.2.o.b.211.16	yes	576

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
959.2.o.b.50.16	✓	576	1.1	even	1	trivial
959.2.o.b.211.16	yes	576	137.74	even	17	inner