Embedded newform 17.3.e.b.6.1

• Label: 17.3.e.b.6.1
• Level: $17$
• Weight: $3$
• Character: 17.6
• Analytic conductor: $0.463$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 17 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	17.e (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.463216449413\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	\(\Q(\zeta_{16})\)
Defining polynomial:	\( x^{8} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			6.1
Root			\(0.923880 - 0.382683i\) of defining polynomial
Character	\(\chi\)	\(=\)	17.6
Dual form			17.3.e.b.3.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.63099 - 0.675577i) q^{2} +(3.96908 - 0.789499i) q^{3} +(-0.624715 - 0.624715i) q^{4} +(-4.29916 + 6.43416i) q^{5} +(-7.00688 - 1.39376i) q^{6} +(-2.27356 - 3.40262i) q^{7} +(3.29916 + 7.96489i) q^{8} +(6.81537 - 2.82302i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} - 24 q^{5} - 16 q^{7} + 16 q^{8} + 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{15}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.63099	−	0.675577i	−0.815493	−	0.337788i	−0.0643498	−	0.997927i	\(-0.520497\pi\)
							−0.751143	+	0.660139i	\(0.770497\pi\)
\(3\)	3.96908	−	0.789499i	1.32303	−	0.263166i	0.517478	−	0.855696i	\(-0.326871\pi\)
							0.805548	+	0.592530i	\(0.201871\pi\)
\(5\)	−4.29916	+	6.43416i	−0.859833	+	1.28683i	0.0967316	+	0.995311i	\(0.469161\pi\)
							−0.956565	+	0.291521i	\(0.905839\pi\)
\(7\)	−2.27356	−	3.40262i	−0.324794	−	0.486089i	0.632757	−	0.774350i	\(-0.281923\pi\)
							−0.957552	+	0.288261i	\(0.906923\pi\)
\(11\)	1.35387	−	6.80638i	0.123079	−	0.618761i	−0.869172	−	0.494511i	\(-0.835347\pi\)
							0.992251	−	0.124251i	\(-0.0396527\pi\)
\(13\)	2.37416	−	2.37416i	0.182628	−	0.182628i	−0.609872	−	0.792500i	\(-0.708779\pi\)
							0.792500	+	0.609872i	\(0.208779\pi\)
\(17\)	−8.76791	−	14.5645i	−0.515760	−	0.856733i
							\(19\)	22.7712	+	9.43215i	1.19849	+	0.496429i	0.890509	−	0.454965i	\(-0.150348\pi\)
0.307977	+	0.951394i	\(0.400348\pi\)
\(23\)	−11.2984	−	2.24740i	−0.491237	−	0.0977131i	−0.0567447	−	0.998389i	\(-0.518072\pi\)
							−0.434492	+	0.900676i	\(0.643072\pi\)
\(29\)	9.98767	+	6.67355i	0.344402	+	0.230122i	0.715726	−	0.698381i	\(-0.246096\pi\)
							−0.371324	+	0.928504i	\(0.621096\pi\)
\(31\)	7.38055	+	37.1045i	0.238082	+	1.19692i	0.896077	+	0.443898i	\(0.146405\pi\)
							−0.657995	+	0.753022i	\(0.728595\pi\)
\(37\)	−31.6486	+	6.29529i	−0.855366	+	0.170143i	−0.603248	−	0.797554i	\(-0.706127\pi\)
							−0.252118	+	0.967696i	\(0.581127\pi\)
\(41\)	18.7852	+	28.1140i	0.458175	+	0.685707i	0.986579	−	0.163286i	\(-0.0522094\pi\)
							−0.528404	+	0.848993i	\(0.677209\pi\)
\(43\)	3.21547	−	1.33189i	0.0747784	−	0.0309742i	−0.344981	−	0.938610i	\(-0.612115\pi\)
							0.419759	+	0.907636i	\(0.362115\pi\)
\(47\)	3.16735	−	3.16735i	0.0673905	−	0.0673905i	−0.672608	−	0.739999i	\(-0.734826\pi\)
							0.739999	+	0.672608i	\(0.234826\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.3.e.b.6.1	yes	8
3.2	odd	2		153.3.p.a.91.1		8
4.3	odd	2		272.3.bh.b.193.1		8
5.2	odd	4		425.3.t.d.74.1		8
5.3	odd	4		425.3.t.b.74.1		8
5.4	even	2		425.3.u.a.176.1		8
17.2	even	8		289.3.e.n.75.1		8
17.3	odd	16	inner	17.3.e.b.3.1	✓	8
17.4	even	4		289.3.e.h.249.1		8
17.5	odd	16		289.3.e.h.65.1		8
17.6	odd	16		289.3.e.e.131.1		8
17.7	odd	16		289.3.e.j.158.1		8
17.8	even	8		289.3.e.a.214.1		8
17.9	even	8		289.3.e.e.214.1		8
17.10	odd	16		289.3.e.n.158.1		8
17.11	odd	16		289.3.e.a.131.1		8
17.12	odd	16		289.3.e.f.65.1		8
17.13	even	4		289.3.e.f.249.1		8
17.14	odd	16		289.3.e.g.224.1		8
17.15	even	8		289.3.e.j.75.1		8
17.16	even	2		289.3.e.g.40.1		8
51.20	even	16		153.3.p.a.37.1		8
68.3	even	16		272.3.bh.b.241.1		8
85.3	even	16		425.3.t.d.224.1		8
85.37	even	16		425.3.t.b.224.1		8
85.54	odd	16		425.3.u.a.326.1		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.3.e.b.3.1	✓	8	17.3	odd	16	inner
17.3.e.b.6.1	yes	8	1.1	even	1	trivial
153.3.p.a.37.1		8	51.20	even	16
153.3.p.a.91.1		8	3.2	odd	2
272.3.bh.b.193.1		8	4.3	odd	2
272.3.bh.b.241.1		8	68.3	even	16
289.3.e.a.131.1		8	17.11	odd	16
289.3.e.a.214.1		8	17.8	even	8
289.3.e.e.131.1		8	17.6	odd	16
289.3.e.e.214.1		8	17.9	even	8
289.3.e.f.65.1		8	17.12	odd	16
289.3.e.f.249.1		8	17.13	even	4
289.3.e.g.40.1		8	17.16	even	2
289.3.e.g.224.1		8	17.14	odd	16
289.3.e.h.65.1		8	17.5	odd	16
289.3.e.h.249.1		8	17.4	even	4
289.3.e.j.75.1		8	17.15	even	8
289.3.e.j.158.1		8	17.7	odd	16
289.3.e.n.75.1		8	17.2	even	8
289.3.e.n.158.1		8	17.10	odd	16
425.3.t.b.74.1		8	5.3	odd	4
425.3.t.b.224.1		8	85.37	even	16
425.3.t.d.74.1		8	5.2	odd	4
425.3.t.d.224.1		8	85.3	even	16
425.3.u.a.176.1		8	5.4	even	2
425.3.u.a.326.1		8	85.54	odd	16