Embedded newform 17.6.d.a.9.5

• Label: 17.6.d.a.9.5
• 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			9.5
Character	\(\chi\)	\(=\)	17.9
Dual form			17.6.d.a.2.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.65830 - 2.65830i) q^{2} +(15.7332 + 6.51691i) q^{3} +17.8669i q^{4} +(13.8556 - 33.4503i) q^{5} +(59.1474 - 24.4997i) q^{6} +(-40.7626 - 98.4095i) q^{7} +(132.561 + 132.561i) q^{8} +(33.2366 + 33.2366i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 4 q^{2} - 4 q^{3} + 40 q^{5} - 364 q^{6} - 4 q^{7} + 124 q^{8} - 136 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.65830	−	2.65830i	0.469925	−	0.469925i	−0.431965	−	0.901890i	\(-0.642180\pi\)
							0.901890	+	0.431965i	\(0.142180\pi\)
\(3\)	15.7332	+	6.51691i	1.00929	+	0.418060i	0.825194	−	0.564849i	\(-0.191066\pi\)
							0.184091	+	0.982909i	\(0.441066\pi\)
\(5\)	13.8556	−	33.4503i	0.247856	−	0.598378i	−0.750165	−	0.661250i	\(-0.770026\pi\)
							0.998022	+	0.0628726i	\(0.0200262\pi\)
\(7\)	−40.7626	−	98.4095i	−0.314425	−	0.759088i	−0.999530	−	0.0306444i	\(-0.990244\pi\)
							0.685106	−	0.728444i	\(-0.259756\pi\)
\(11\)	−521.428	+	215.982i	−1.29931	+	0.538191i	−0.921747	−	0.387793i	\(-0.873238\pi\)
							−0.377562	+	0.925984i	\(0.623238\pi\)
\(13\)		−	117.540i		−	0.192898i	−0.995338	−	0.0964488i	\(-0.969252\pi\)
							0.995338	−	0.0964488i	\(-0.0307484\pi\)
\(17\)	−991.208	+	661.335i	−0.831845	+	0.555008i
							\(19\)	771.155	−	771.155i	0.490069	−	0.490069i	−0.418259	−	0.908328i	\(-0.637359\pi\)
0.908328	+	0.418259i	\(0.137359\pi\)
\(23\)	3093.11	−	1281.21i	1.21920	−	0.505011i	0.322048	−	0.946723i	\(-0.395629\pi\)
							0.897157	+	0.441713i	\(0.145629\pi\)
\(29\)	46.9095	−	113.250i	0.0103578	−	0.0250058i	−0.918616	−	0.395151i	\(-0.870692\pi\)
							0.928974	+	0.370145i	\(0.120692\pi\)
\(31\)	−4155.68	−	1721.34i	−0.776672	−	0.321708i	−0.0411001	−	0.999155i	\(-0.513086\pi\)
							−0.735571	+	0.677447i	\(0.763086\pi\)
\(37\)	9694.90	+	4015.76i	1.16423	+	0.482240i	0.879282	−	0.476302i	\(-0.158023\pi\)
							0.284949	+	0.958543i	\(0.408023\pi\)
\(41\)	30.9488	+	74.7169i	0.00287530	+	0.00694160i	0.925311	−	0.379210i	\(-0.123804\pi\)
							−0.922435	+	0.386152i	\(0.873804\pi\)
\(43\)	15708.1	+	15708.1i	1.29555	+	1.29555i	0.931303	+	0.364245i	\(0.118673\pi\)
							0.364245	+	0.931303i	\(0.381327\pi\)
\(47\)			11939.2i			0.788368i	0.919032	+	0.394184i	\(0.128973\pi\)
							−0.919032	+	0.394184i	\(0.871027\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.6.d.a.9.5	yes	28
17.2	even	8	inner	17.6.d.a.2.5	✓	28
17.6	odd	16		289.6.a.j.1.18		28
17.11	odd	16		289.6.a.j.1.17		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.6.d.a.2.5	✓	28	17.2	even	8	inner
17.6.d.a.9.5	yes	28	1.1	even	1	trivial
289.6.a.j.1.17		28	17.11	odd	16
289.6.a.j.1.18		28	17.6	odd	16