Embedded newform 17.6.d.a.8.6

• Label: 17.6.d.a.8.6
• Level: $17$
• Weight: $6$
• Character: 17.8
• Analytic conductor: $2.727$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 17 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	17.d (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.72652493682\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Relative dimension:	\(7\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			8.6
Character	\(\chi\)	\(=\)	17.8
Dual form			17.6.d.a.15.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.81385 - 3.81385i) q^{2} +(6.88457 - 16.6208i) q^{3} +2.90907i q^{4} +(-27.9569 - 11.5801i) q^{5} +(-37.1326 - 89.6461i) q^{6} +(62.5721 - 25.9182i) q^{7} +(133.138 + 133.138i) q^{8} +(-57.0277 - 57.0277i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 4 q^{2} - 4 q^{3} + 40 q^{5} - 364 q^{6} - 4 q^{7} + 124 q^{8} - 136 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/17\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.81385	−	3.81385i	0.674200	−	0.674200i	−0.284481	−	0.958682i	\(-0.591821\pi\)
							0.958682	+	0.284481i	\(0.0918214\pi\)
\(3\)	6.88457	−	16.6208i	0.441646	−	1.06623i	−0.533726	−	0.845658i	\(-0.679209\pi\)
							0.975371	−	0.220569i	\(-0.0707914\pi\)
\(5\)	−27.9569	−	11.5801i	−0.500108	−	0.207151i	0.118346	−	0.992972i	\(-0.462241\pi\)
							−0.618454	+	0.785821i	\(0.712241\pi\)
\(7\)	62.5721	−	25.9182i	0.482653	−	0.199922i	−0.128071	−	0.991765i	\(-0.540878\pi\)
							0.610724	+	0.791843i	\(0.290878\pi\)
\(11\)	−26.4483	−	63.8519i	−0.0659047	−	0.159108i	0.887496	−	0.460816i	\(-0.152443\pi\)
							−0.953400	+	0.301708i	\(0.902443\pi\)
\(13\)			898.659i			1.47481i	0.675450	+	0.737406i	\(0.263950\pi\)
							−0.675450	+	0.737406i	\(0.736050\pi\)
\(17\)	−1178.09	+	178.775i	−0.988681	+	0.150032i
							\(19\)	672.670	−	672.670i	0.427482	−	0.427482i	−0.460288	−	0.887770i	\(-0.652254\pi\)
0.887770	+	0.460288i	\(0.152254\pi\)
\(23\)	−800.337	−	1932.18i	−0.315466	−	0.761603i	−0.999483	−	0.0321374i	\(-0.989769\pi\)
							0.684017	−	0.729466i	\(-0.260231\pi\)
\(29\)	−5520.95	−	2286.85i	−1.21904	−	0.504944i	−0.321937	−	0.946761i	\(-0.604334\pi\)
							−0.897105	+	0.441817i	\(0.854334\pi\)
\(31\)	−2710.17	+	6542.92i	−0.506515	+	1.22283i	0.439363	+	0.898310i	\(0.355204\pi\)
							−0.945877	+	0.324525i	\(0.894796\pi\)
\(37\)	−1771.22	+	4276.10i	−0.212700	+	0.513503i	−0.993836	−	0.110857i	\(-0.964640\pi\)
							0.781136	+	0.624360i	\(0.214640\pi\)
\(41\)	18315.7	−	7586.60i	1.70162	−	0.704835i	0.701653	−	0.712519i	\(-0.252446\pi\)
							0.999970	+	0.00768363i	\(0.00244580\pi\)
\(43\)	−4630.66	−	4630.66i	−0.381919	−	0.381919i	0.489874	−	0.871793i	\(-0.337043\pi\)
							−0.871793	+	0.489874i	\(0.837043\pi\)
\(47\)		−	2022.19i		−	0.133529i	−0.997769	−	0.0667647i	\(-0.978732\pi\)
							0.997769	−	0.0667647i	\(-0.0212677\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.6.d.a.8.6	✓	28
17.7	odd	16		289.6.a.j.1.7		28
17.10	odd	16		289.6.a.j.1.8		28
17.15	even	8	inner	17.6.d.a.15.6	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.6.d.a.8.6	✓	28	1.1	even	1	trivial
17.6.d.a.15.6	yes	28	17.15	even	8	inner
289.6.a.j.1.7		28	17.7	odd	16
289.6.a.j.1.8		28	17.10	odd	16