Embedded newform 17.3.e.b.7.1

• Label: 17.3.e.b.7.1
• Level: $17$
• Weight: $3$
• Character: 17.7
• 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			7.1
Root			\(-0.382683 - 0.923880i\) of defining polynomial
Character	\(\chi\)	\(=\)	17.7
Dual form			17.3.e.b.5.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.08979 - 2.63099i) q^{2} +(-2.88669 + 4.32023i) q^{3} +(-2.90602 - 2.90602i) q^{4} +(-0.711297 - 3.57593i) q^{5} +(8.22059 + 12.3030i) q^{6} +(-0.644047 + 3.23784i) q^{7} +(-0.288703 + 0.119585i) q^{8} +(-6.88730 - 16.6274i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} - 24 q^{5} - 16 q^{7} + 16 q^{8} + 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{11}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.08979	−	2.63099i	0.544895	−	1.31549i	−0.376338	−	0.926482i	\(-0.622817\pi\)
							0.921233	−	0.389011i	\(-0.127183\pi\)
\(3\)	−2.88669	+	4.32023i	−0.962229	+	1.44008i	−0.0653191	+	0.997864i	\(0.520807\pi\)
							−0.896910	+	0.442213i	\(0.854193\pi\)
\(5\)	−0.711297	−	3.57593i	−0.142259	−	0.715187i	−0.984402	−	0.175932i	\(-0.943706\pi\)
							0.842143	−	0.539255i	\(-0.181294\pi\)
\(7\)	−0.644047	+	3.23784i	−0.0920066	+	0.462549i	0.907124	+	0.420863i	\(0.138273\pi\)
							−0.999131	+	0.0416855i	\(0.986727\pi\)
\(11\)	2.02046	−	1.35003i	0.183678	−	0.122730i	−0.460332	−	0.887747i	\(-0.652270\pi\)
							0.644010	+	0.765017i	\(0.277270\pi\)
\(13\)	−7.73173	+	7.73173i	−0.594748	+	0.594748i	−0.938910	−	0.344162i	\(-0.888163\pi\)
							0.344162	+	0.938910i	\(0.388163\pi\)
\(17\)	15.0347	+	7.93458i	0.884395	+	0.466740i
							\(19\)	−3.50319	+	8.45745i	−0.184379	+	0.445129i	−0.988860	−	0.148849i	\(-0.952443\pi\)
0.804481	+	0.593978i	\(0.202443\pi\)
\(23\)	−1.91942	−	2.87262i	−0.0834532	−	0.124897i	0.787395	−	0.616449i	\(-0.211429\pi\)
							−0.870848	+	0.491553i	\(0.836429\pi\)
\(29\)	−23.5861	+	4.69157i	−0.813314	+	0.161778i	−0.584188	−	0.811618i	\(-0.698587\pi\)
							−0.229126	+	0.973397i	\(0.573587\pi\)
\(31\)	−23.0300	−	15.3882i	−0.742903	−	0.496392i	0.125597	−	0.992081i	\(-0.459915\pi\)
							−0.868500	+	0.495689i	\(0.834915\pi\)
\(37\)	−21.3995	+	32.0267i	−0.578366	+	0.865586i	−0.999135	−	0.0415820i	\(-0.986760\pi\)
							0.420769	+	0.907168i	\(0.361760\pi\)
\(41\)	15.0157	−	75.4890i	0.366237	−	1.84120i	−0.155171	−	0.987888i	\(-0.549593\pi\)
							0.521407	−	0.853308i	\(-0.325407\pi\)
\(43\)	20.7301	+	50.0470i	0.482096	+	1.16388i	0.958612	+	0.284717i	\(0.0918996\pi\)
							−0.476515	+	0.879166i	\(0.658100\pi\)
\(47\)	−5.13810	+	5.13810i	−0.109321	+	0.109321i	−0.759652	−	0.650330i	\(-0.774631\pi\)
							0.650330	+	0.759652i	\(0.274631\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.3.e.b.7.1	yes	8
3.2	odd	2		153.3.p.a.109.1		8
4.3	odd	2		272.3.bh.b.177.1		8
5.2	odd	4		425.3.t.d.24.1		8
5.3	odd	4		425.3.t.b.24.1		8
5.4	even	2		425.3.u.a.126.1		8
17.2	even	8		289.3.e.j.249.1		8
17.3	odd	16		289.3.e.f.131.1		8
17.4	even	4		289.3.e.h.214.1		8
17.5	odd	16	inner	17.3.e.b.5.1	✓	8
17.6	odd	16		289.3.e.n.65.1		8
17.7	odd	16		289.3.e.e.224.1		8
17.8	even	8		289.3.e.e.40.1		8
17.9	even	8		289.3.e.a.40.1		8
17.10	odd	16		289.3.e.a.224.1		8
17.11	odd	16		289.3.e.j.65.1		8
17.12	odd	16		289.3.e.g.158.1		8
17.13	even	4		289.3.e.f.214.1		8
17.14	odd	16		289.3.e.h.131.1		8
17.15	even	8		289.3.e.n.249.1		8
17.16	even	2		289.3.e.g.75.1		8
51.5	even	16		153.3.p.a.73.1		8
68.39	even	16		272.3.bh.b.209.1		8
85.22	even	16		425.3.t.b.124.1		8
85.39	odd	16		425.3.u.a.226.1		8
85.73	even	16		425.3.t.d.124.1		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.3.e.b.5.1	✓	8	17.5	odd	16	inner
17.3.e.b.7.1	yes	8	1.1	even	1	trivial
153.3.p.a.73.1		8	51.5	even	16
153.3.p.a.109.1		8	3.2	odd	2
272.3.bh.b.177.1		8	4.3	odd	2
272.3.bh.b.209.1		8	68.39	even	16
289.3.e.a.40.1		8	17.9	even	8
289.3.e.a.224.1		8	17.10	odd	16
289.3.e.e.40.1		8	17.8	even	8
289.3.e.e.224.1		8	17.7	odd	16
289.3.e.f.131.1		8	17.3	odd	16
289.3.e.f.214.1		8	17.13	even	4
289.3.e.g.75.1		8	17.16	even	2
289.3.e.g.158.1		8	17.12	odd	16
289.3.e.h.131.1		8	17.14	odd	16
289.3.e.h.214.1		8	17.4	even	4
289.3.e.j.65.1		8	17.11	odd	16
289.3.e.j.249.1		8	17.2	even	8
289.3.e.n.65.1		8	17.6	odd	16
289.3.e.n.249.1		8	17.15	even	8
425.3.t.b.24.1		8	5.3	odd	4
425.3.t.b.124.1		8	85.22	even	16
425.3.t.d.24.1		8	5.2	odd	4
425.3.t.d.124.1		8	85.73	even	16
425.3.u.a.126.1		8	5.4	even	2
425.3.u.a.226.1		8	85.39	odd	16