Embedded newform 17.3.e.b.3.1

• Label: 17.3.e.b.3.1
• Level: $17$
• Weight: $3$
• Character: 17.3
• 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			3.1
Root			\(0.923880 + 0.382683i\) of defining polynomial
Character	\(\chi\)	\(=\)	17.3
Dual form			17.3.e.b.6.1

$q$-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 \(\left(\mathbb{Z}/17\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{16}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.3.e.b.3.1	✓	8	1.1	even	1	trivial
17.3.e.b.6.1	yes	8	17.6	odd	16	inner
153.3.p.a.37.1		8	3.2	odd	2
153.3.p.a.91.1		8	51.23	even	16
272.3.bh.b.193.1		8	68.23	even	16
272.3.bh.b.241.1		8	4.3	odd	2
289.3.e.a.131.1		8	17.15	even	8
289.3.e.a.214.1		8	17.14	odd	16
289.3.e.e.131.1		8	17.2	even	8
289.3.e.e.214.1		8	17.3	odd	16
289.3.e.f.65.1		8	17.4	even	4
289.3.e.f.249.1		8	17.10	odd	16
289.3.e.g.40.1		8	17.11	odd	16
289.3.e.g.224.1		8	17.16	even	2
289.3.e.h.65.1		8	17.13	even	4
289.3.e.h.249.1		8	17.7	odd	16
289.3.e.j.75.1		8	17.5	odd	16
289.3.e.j.158.1		8	17.8	even	8
289.3.e.n.75.1		8	17.12	odd	16
289.3.e.n.158.1		8	17.9	even	8
425.3.t.b.74.1		8	85.23	even	16
425.3.t.b.224.1		8	5.2	odd	4
425.3.t.d.74.1		8	85.57	even	16
425.3.t.d.224.1		8	5.3	odd	4
425.3.u.a.176.1		8	85.74	odd	16
425.3.u.a.326.1		8	5.4	even	2