Embedded newform 17.6.d.a.9.1

• Label: 17.6.d.a.9.1
• 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			9.1
Character	\(\chi\)	\(=\)	17.9
Dual form			17.6.d.a.2.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.32994 + 6.32994i) q^{2} +(26.8751 + 11.1320i) q^{3} -48.1363i q^{4} +(-24.9886 + 60.3278i) q^{5} +(-240.583 + 99.6526i) q^{6} +(-43.4369 - 104.866i) q^{7} +(102.142 + 102.142i) q^{8} +(426.521 + 426.521i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 4 q^{2} - 4 q^{3} + 40 q^{5} - 364 q^{6} - 4 q^{7} + 124 q^{8} - 136 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.32994	+	6.32994i	−1.11899	+	1.11899i	−0.127096	+	0.991890i	\(0.540565\pi\)
							−0.991890	+	0.127096i	\(0.959435\pi\)
\(3\)	26.8751	+	11.1320i	1.72404	+	0.714120i	0.999695	+	0.0246791i	\(0.00785639\pi\)
							0.724342	+	0.689441i	\(0.242144\pi\)
\(5\)	−24.9886	+	60.3278i	−0.447010	+	1.07918i	0.526427	+	0.850221i	\(0.323531\pi\)
							−0.973437	+	0.228957i	\(0.926469\pi\)
\(7\)	−43.4369	−	104.866i	−0.335053	−	0.808890i	−0.998176	−	0.0603792i	\(-0.980769\pi\)
							0.663122	−	0.748511i	\(-0.269231\pi\)
\(11\)	65.9090	−	27.3004i	0.164234	−	0.0680279i	−0.299052	−	0.954237i	\(-0.596670\pi\)
							0.463286	+	0.886209i	\(0.346670\pi\)
\(13\)			7.95633i			0.0130573i	0.999979	+	0.00652867i	\(0.00207815\pi\)
							−0.999979	+	0.00652867i	\(0.997922\pi\)
\(17\)	1183.73	+	136.545i	0.993413	+	0.114592i
							\(19\)	741.394	−	741.394i	0.471156	−	0.471156i	−0.431132	−	0.902289i	\(-0.641886\pi\)
0.902289	+	0.431132i	\(0.141886\pi\)
\(23\)	−424.002	+	175.627i	−0.167128	+	0.0692265i	−0.464679	−	0.885479i	\(-0.653830\pi\)
							0.297551	+	0.954706i	\(0.403830\pi\)
\(29\)	1835.83	−	4432.09i	0.405357	−	0.978618i	−0.580986	−	0.813914i	\(-0.697333\pi\)
							0.986343	−	0.164705i	\(-0.0526671\pi\)
\(31\)	−4437.75	−	1838.17i	−0.829389	−	0.343544i	−0.0727281	−	0.997352i	\(-0.523171\pi\)
							−0.756661	+	0.653808i	\(0.773171\pi\)
\(37\)	2753.98	+	1140.73i	0.330717	+	0.136987i	0.541862	−	0.840467i	\(-0.317719\pi\)
							−0.211146	+	0.977455i	\(0.567719\pi\)
\(41\)	287.916	+	695.092i	0.0267490	+	0.0645777i	0.936690	−	0.350161i	\(-0.113873\pi\)
							−0.909941	+	0.414738i	\(0.863873\pi\)
\(43\)	−14059.4	−	14059.4i	−1.15957	−	1.15957i	−0.984568	−	0.175001i	\(-0.944007\pi\)
							−0.175001	−	0.984568i	\(-0.555993\pi\)
\(47\)		−	11155.7i		−	0.736636i	−0.929700	−	0.368318i	\(-0.879934\pi\)
							0.929700	−	0.368318i	\(-0.120066\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.6.d.a.9.1	yes	28
17.2	even	8	inner	17.6.d.a.2.1	✓	28
17.6	odd	16		289.6.a.j.1.4		28
17.11	odd	16		289.6.a.j.1.3		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.6.d.a.2.1	✓	28	17.2	even	8	inner
17.6.d.a.9.1	yes	28	1.1	even	1	trivial
289.6.a.j.1.3		28	17.11	odd	16
289.6.a.j.1.4		28	17.6	odd	16