Embedded newform 17.6.d.a.8.7

• Label: 17.6.d.a.8.7
• Level: $17$
• Weight: $6$
• Character: 17.8
• Analytic conductor: $2.727$
• Analytic rank: $0$
• Dimension: $28$
• 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			8.7
Character	\(\chi\)	\(=\)	17.8
Dual form			17.6.d.a.15.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(7.21923 - 7.21923i) q^{2} +(-2.36967 + 5.72090i) q^{3} -72.2346i q^{4} +(34.6952 + 14.3712i) q^{5} +(24.1933 + 58.4077i) q^{6} +(-123.701 + 51.2388i) q^{7} +(-290.463 - 290.463i) q^{8} +(144.714 + 144.714i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 4 q^{2} - 4 q^{3} + 40 q^{5} - 364 q^{6} - 4 q^{7} + 124 q^{8} - 136 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	7.21923	−	7.21923i	1.27619	−	1.27619i	0.333410	−	0.942782i	\(-0.391801\pi\)
							0.942782	−	0.333410i	\(-0.108199\pi\)
\(3\)	−2.36967	+	5.72090i	−0.152015	+	0.366996i	−0.981481	−	0.191561i	\(-0.938645\pi\)
							0.829466	+	0.558557i	\(0.188645\pi\)
\(5\)	34.6952	+	14.3712i	0.620647	+	0.257080i	0.670773	−	0.741663i	\(-0.265962\pi\)
							−0.0501263	+	0.998743i	\(0.515962\pi\)
\(7\)	−123.701	+	51.2388i	−0.954178	+	0.395233i	−0.804800	−	0.593547i	\(-0.797727\pi\)
							−0.149378	+	0.988780i	\(0.547727\pi\)
\(11\)	205.821	+	496.897i	0.512872	+	1.23818i	0.942205	+	0.335036i	\(0.108748\pi\)
							−0.429334	+	0.903146i	\(0.641252\pi\)
\(13\)		−	864.042i		−	1.41800i	−0.705208	−	0.709000i	\(-0.749147\pi\)
							0.705208	−	0.709000i	\(-0.250853\pi\)
\(17\)	−1175.68	+	194.013i	−0.986656	+	0.162821i
							\(19\)	−719.193	+	719.193i	−0.457048	+	0.457048i	−0.897685	−	0.440637i	\(-0.854752\pi\)
0.440637	+	0.897685i	\(0.354752\pi\)
\(23\)	−1355.59	−	3272.69i	−0.534330	−	1.28999i	−0.928631	−	0.371005i	\(-0.879013\pi\)
							0.394301	−	0.918981i	\(-0.370987\pi\)
\(29\)	4124.49	+	1708.42i	0.910701	+	0.377225i	0.788325	−	0.615259i	\(-0.210949\pi\)
							0.122376	+	0.992484i	\(0.460949\pi\)
\(31\)	758.937	−	1832.24i	0.141841	−	0.342434i	−0.836955	−	0.547272i	\(-0.815667\pi\)
							0.978796	+	0.204837i	\(0.0656665\pi\)
\(37\)	881.005	−	2126.93i	0.105797	−	0.255417i	−0.862112	−	0.506718i	\(-0.830858\pi\)
							0.967909	+	0.251301i	\(0.0808585\pi\)
\(41\)	3648.20	−	1511.13i	0.338937	−	0.140392i	−0.206722	−	0.978400i	\(-0.566279\pi\)
							0.545659	+	0.838007i	\(0.316279\pi\)
\(43\)	11468.9	+	11468.9i	0.945911	+	0.945911i	0.998610	−	0.0526996i	\(-0.0167826\pi\)
							−0.0526996	+	0.998610i	\(0.516783\pi\)
\(47\)			13301.3i			0.878311i	0.898411	+	0.439155i	\(0.144722\pi\)
							−0.898411	+	0.439155i	\(0.855278\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.6.d.a.8.7	✓	28
17.7	odd	16		289.6.a.j.1.2		28
17.10	odd	16		289.6.a.j.1.1		28
17.15	even	8	inner	17.6.d.a.15.7	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.6.d.a.8.7	✓	28	1.1	even	1	trivial
17.6.d.a.15.7	yes	28	17.15	even	8	inner
289.6.a.j.1.1		28	17.10	odd	16
289.6.a.j.1.2		28	17.7	odd	16