Embedded newform 959.2.o.b.50.15

• Label: 959.2.o.b.50.15
• 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			50.15
Character	\(\chi\)	\(=\)	959.50
Dual form			959.2.o.b.211.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.291083 + 0.265358i) q^{2} +(2.53433 + 1.56919i) q^{3} +(-0.170222 + 1.83699i) q^{4} +(0.715567 + 0.652325i) q^{5} +(-1.15410 + 0.215738i) q^{6} +(-0.273663 - 0.961826i) q^{7} +(-0.912645 - 1.20854i) q^{8} +(2.62325 + 5.26820i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 576 q - 6 q^{2} - 4 q^{3} - 14 q^{4} - 6 q^{5} - 16 q^{6} - 36 q^{7} - 24 q^{8} - 54 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.291083	+	0.265358i	−0.205827	+	0.187636i	−0.770045	−	0.637990i	\(-0.779766\pi\)
							0.564218	+	0.825626i	\(0.309178\pi\)
\(3\)	2.53433	+	1.56919i	1.46320	+	0.905972i	0.999804	+	0.0198186i	\(0.00630886\pi\)
							0.463392	+	0.886154i	\(0.346632\pi\)
\(5\)	0.715567	+	0.652325i	0.320011	+	0.291729i	0.817721	−	0.575615i	\(-0.195237\pi\)
							−0.497710	+	0.867344i	\(0.665825\pi\)
\(7\)	−0.273663	−	0.961826i	−0.103435	−	0.363536i
							\(11\)	−0.479177	+	5.17114i	−0.144477	+	1.55916i	0.547232	+	0.836981i	\(0.315681\pi\)
−0.691709	+	0.722176i	\(0.743142\pi\)
\(13\)	−1.28120	−	4.50295i	−0.355341	−	1.24889i	−0.909227	−	0.416301i	\(-0.863326\pi\)
							0.553886	−	0.832593i	\(-0.313144\pi\)
\(17\)	−2.41213	+	3.19417i	−0.585027	+	0.774701i	−0.990218	−	0.139531i	\(-0.955440\pi\)
							0.405191	+	0.914232i	\(0.367205\pi\)
\(19\)	1.79463	+	0.695242i	0.411716	+	0.159500i	0.558187	−	0.829715i	\(-0.311497\pi\)
							−0.146471	+	0.989215i	\(0.546791\pi\)
\(23\)	−6.81943	−	1.27477i	−1.42195	−	0.265808i	−0.584087	−	0.811691i	\(-0.698547\pi\)
							−0.837862	+	0.545883i	\(0.816194\pi\)
\(29\)	8.05747	+	1.50620i	1.49624	+	0.279695i	0.867309	−	0.497770i	\(-0.165848\pi\)
							0.628926	+	0.777465i	\(0.283495\pi\)
\(31\)	1.36518	−	0.528872i	0.245193	−	0.0949882i	−0.235510	−	0.971872i	\(-0.575676\pi\)
							0.480702	+	0.876884i	\(0.340382\pi\)
\(37\)	5.12445			0.842454			0.421227	−	0.906955i	\(-0.361599\pi\)
							0.421227	+	0.906955i	\(0.361599\pi\)
\(41\)	4.45694			0.696058			0.348029	−	0.937484i	\(-0.386851\pi\)
							0.348029	+	0.937484i	\(0.386851\pi\)
\(43\)	5.18066	+	2.00700i	0.790043	+	0.306064i	0.722247	−	0.691635i	\(-0.243109\pi\)
							0.0677955	+	0.997699i	\(0.478403\pi\)
\(47\)	4.12994	+	8.29403i	0.602413	+	1.20981i	0.961061	+	0.276336i	\(0.0891201\pi\)
							−0.358648	+	0.933473i	\(0.616762\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	959.2.o.b.50.15	✓	576
137.74	even	17	inner	959.2.o.b.211.15	yes	576

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
959.2.o.b.50.15	✓	576	1.1	even	1	trivial
959.2.o.b.211.15	yes	576	137.74	even	17	inner