Embedded newform 17.6.d.a.9.2

• Label: 17.6.d.a.9.2
• Level: $17$
• Weight: $6$
• Character: 17.9
• 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			9.2
Character	\(\chi\)	\(=\)	17.9
Dual form			17.6.d.a.2.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.57898 + 5.57898i) q^{2} +(-9.69582 - 4.01614i) q^{3} -30.2500i q^{4} +(21.4113 - 51.6915i) q^{5} +(76.4987 - 31.6868i) q^{6} +(-20.6531 - 49.8610i) q^{7} +(-9.76339 - 9.76339i) q^{8} +(-93.9474 - 93.9474i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 4 q^{2} - 4 q^{3} + 40 q^{5} - 364 q^{6} - 4 q^{7} + 124 q^{8} - 136 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.57898	+	5.57898i	−0.986233	+	0.986233i	−0.999907	−	0.0136734i	\(-0.995647\pi\)
							0.0136734	+	0.999907i	\(0.495647\pi\)
\(3\)	−9.69582	−	4.01614i	−0.621987	−	0.257635i	0.0493571	−	0.998781i	\(-0.484283\pi\)
							−0.671344	+	0.741146i	\(0.734283\pi\)
\(5\)	21.4113	−	51.6915i	0.383017	−	0.924685i	−0.608362	−	0.793660i	\(-0.708173\pi\)
							0.991379	−	0.131025i	\(-0.0418270\pi\)
\(7\)	−20.6531	−	49.8610i	−0.159309	−	0.384606i	0.823990	−	0.566605i	\(-0.191743\pi\)
							−0.983299	+	0.181999i	\(0.941743\pi\)
\(11\)	−383.787	+	158.970i	−0.956331	+	0.396125i	−0.805607	−	0.592450i	\(-0.798161\pi\)
							−0.150724	+	0.988576i	\(0.548161\pi\)
\(13\)		−	14.1387i		−	0.0232033i	−0.999933	−	0.0116016i	\(-0.996307\pi\)
							0.999933	−	0.0116016i	\(-0.00369300\pi\)
\(17\)	−829.534	−	855.412i	−0.696164	−	0.717882i
							\(19\)	960.097	−	960.097i	0.610143	−	0.610143i	−0.332841	−	0.942983i	\(-0.608007\pi\)
0.942983	+	0.332841i	\(0.108007\pi\)
\(23\)	−3748.46	+	1552.66i	−1.47752	+	0.612008i	−0.968560	−	0.248782i	\(-0.919970\pi\)
							−0.508960	+	0.860790i	\(0.669970\pi\)
\(29\)	1423.91	−	3437.63i	0.314404	−	0.759039i	−0.685127	−	0.728424i	\(-0.740253\pi\)
							0.999531	−	0.0306155i	\(-0.00974674\pi\)
\(31\)	5244.26	+	2172.24i	0.980121	+	0.405979i	0.814470	−	0.580205i	\(-0.197028\pi\)
							0.165651	+	0.986185i	\(0.447028\pi\)
\(37\)	−13612.5	−	5638.49i	−1.63468	−	0.677108i	−0.638939	−	0.769258i	\(-0.720626\pi\)
							−0.995745	+	0.0921492i	\(0.970626\pi\)
\(41\)	1701.86	+	4108.64i	0.158111	+	0.381715i	0.983006	−	0.183571i	\(-0.0587658\pi\)
							−0.824895	+	0.565286i	\(0.808766\pi\)
\(43\)	3710.06	+	3710.06i	0.305992	+	0.305992i	0.843353	−	0.537361i	\(-0.180579\pi\)
							−0.537361	+	0.843353i	\(0.680579\pi\)
\(47\)		−	24613.8i		−	1.62530i	−0.582750	−	0.812652i	\(-0.698023\pi\)
							0.582750	−	0.812652i	\(-0.301977\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.6.d.a.9.2	yes	28
17.2	even	8	inner	17.6.d.a.2.2	✓	28
17.6	odd	16		289.6.a.j.1.5		28
17.11	odd	16		289.6.a.j.1.6		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.6.d.a.2.2	✓	28	17.2	even	8	inner
17.6.d.a.9.2	yes	28	1.1	even	1	trivial
289.6.a.j.1.5		28	17.6	odd	16
289.6.a.j.1.6		28	17.11	odd	16