Embedded newform 17.3.e.b.14.1

• Label: 17.3.e.b.14.1
• Level: $17$
• Weight: $3$
• Character: 17.14
• 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			14.1
Root			\(-0.923880 - 0.382683i\) of defining polynomial
Character	\(\chi\)	\(=\)	17.14
Dual form			17.3.e.b.11.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.216773 - 0.0897902i) q^{2} +(0.273561 - 1.37529i) q^{3} +(-2.78950 + 2.78950i) q^{4} +(-0.286621 + 0.191514i) q^{5} +(-0.0641865 - 0.322688i) q^{6} +(-5.96908 - 3.98841i) q^{7} +(-0.713379 + 1.72225i) q^{8} +(6.49834 + 2.69170i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} - 24 q^{5} - 16 q^{7} + 16 q^{8} + 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{9}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.216773	−	0.0897902i	0.108386	−	0.0448951i	−0.327831	−	0.944736i	\(-0.606318\pi\)
							0.436218	+	0.899841i	\(0.356318\pi\)
\(3\)	0.273561	−	1.37529i	0.0911871	−	0.458428i	−0.908031	−	0.418902i	\(-0.862415\pi\)
							0.999219	−	0.0395264i	\(-0.0125849\pi\)
\(5\)	−0.286621	+	0.191514i	−0.0573243	+	0.0383029i	−0.583902	−	0.811824i	\(-0.698475\pi\)
							0.526578	+	0.850127i	\(0.323475\pi\)
\(7\)	−5.96908	−	3.98841i	−0.852726	−	0.569773i	0.0506046	−	0.998719i	\(-0.483885\pi\)
							−0.903330	+	0.428946i	\(0.858885\pi\)
\(11\)	12.8888	−	2.56374i	1.17171	−	0.233067i	0.429392	−	0.903118i	\(-0.358728\pi\)
							0.742315	+	0.670052i	\(0.233728\pi\)
\(13\)	−5.20259	−	5.20259i	−0.400199	−	0.400199i	0.478104	−	0.878303i	\(-0.341324\pi\)
							−0.878303	+	0.478104i	\(0.841324\pi\)
\(17\)	0.525274	+	16.9919i	0.0308985	+	0.999523i
							\(19\)	−22.2860	+	9.23114i	−1.17295	+	0.485850i	−0.882165	−	0.470940i	\(-0.843915\pi\)
−0.290780	+	0.956790i	\(0.593915\pi\)
\(23\)	−6.25790	−	31.4606i	−0.272083	−	1.36785i	−0.839024	−	0.544094i	\(-0.816873\pi\)
							0.566941	−	0.823758i	\(-0.308127\pi\)
\(29\)	21.4682	+	32.1294i	0.740282	+	1.10791i	0.990202	+	0.139644i	\(0.0445957\pi\)
							−0.249920	+	0.968266i	\(0.580404\pi\)
\(31\)	10.5189	+	2.09235i	0.339321	+	0.0674951i	0.361810	−	0.932252i	\(-0.382159\pi\)
							−0.0224889	+	0.999747i	\(0.507159\pi\)
\(37\)	2.37648	−	11.9474i	0.0642291	−	0.322901i	−0.935291	−	0.353880i	\(-0.884862\pi\)
							0.999520	+	0.0309782i	\(0.00986225\pi\)
\(41\)	−30.0989	−	20.1114i	−0.734119	−	0.490522i	0.131444	−	0.991324i	\(-0.458039\pi\)
							−0.865562	+	0.500801i	\(0.833039\pi\)
\(43\)	29.2698	+	12.1240i	0.680693	+	0.281952i	0.696117	−	0.717929i	\(-0.254910\pi\)
							−0.0154234	+	0.999881i	\(0.504910\pi\)
\(47\)	−28.8242	−	28.8242i	−0.613281	−	0.613281i	0.330518	−	0.943800i	\(-0.392776\pi\)
							−0.943800	+	0.330518i	\(0.892776\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.3.e.b.14.1	yes	8
3.2	odd	2		153.3.p.a.82.1		8
4.3	odd	2		272.3.bh.b.65.1		8
5.2	odd	4		425.3.t.b.99.1		8
5.3	odd	4		425.3.t.d.99.1		8
5.4	even	2		425.3.u.a.201.1		8
17.2	even	8		289.3.e.e.158.1		8
17.3	odd	16		289.3.e.a.75.1		8
17.4	even	4		289.3.e.f.224.1		8
17.5	odd	16		289.3.e.n.214.1		8
17.6	odd	16		289.3.e.g.249.1		8
17.7	odd	16		289.3.e.f.40.1		8
17.8	even	8		289.3.e.j.131.1		8
17.9	even	8		289.3.e.n.131.1		8
17.10	odd	16		289.3.e.h.40.1		8
17.11	odd	16	inner	17.3.e.b.11.1	✓	8
17.12	odd	16		289.3.e.j.214.1		8
17.13	even	4		289.3.e.h.224.1		8
17.14	odd	16		289.3.e.e.75.1		8
17.15	even	8		289.3.e.a.158.1		8
17.16	even	2		289.3.e.g.65.1		8
51.11	even	16		153.3.p.a.28.1		8
68.11	even	16		272.3.bh.b.113.1		8
85.28	even	16		425.3.t.b.249.1		8
85.62	even	16		425.3.t.d.249.1		8
85.79	odd	16		425.3.u.a.351.1		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.3.e.b.11.1	✓	8	17.11	odd	16	inner
17.3.e.b.14.1	yes	8	1.1	even	1	trivial
153.3.p.a.28.1		8	51.11	even	16
153.3.p.a.82.1		8	3.2	odd	2
272.3.bh.b.65.1		8	4.3	odd	2
272.3.bh.b.113.1		8	68.11	even	16
289.3.e.a.75.1		8	17.3	odd	16
289.3.e.a.158.1		8	17.15	even	8
289.3.e.e.75.1		8	17.14	odd	16
289.3.e.e.158.1		8	17.2	even	8
289.3.e.f.40.1		8	17.7	odd	16
289.3.e.f.224.1		8	17.4	even	4
289.3.e.g.65.1		8	17.16	even	2
289.3.e.g.249.1		8	17.6	odd	16
289.3.e.h.40.1		8	17.10	odd	16
289.3.e.h.224.1		8	17.13	even	4
289.3.e.j.131.1		8	17.8	even	8
289.3.e.j.214.1		8	17.12	odd	16
289.3.e.n.131.1		8	17.9	even	8
289.3.e.n.214.1		8	17.5	odd	16
425.3.t.b.99.1		8	5.2	odd	4
425.3.t.b.249.1		8	85.28	even	16
425.3.t.d.99.1		8	5.3	odd	4
425.3.t.d.249.1		8	85.62	even	16
425.3.u.a.201.1		8	5.4	even	2
425.3.u.a.351.1		8	85.79	odd	16