Embedded newform 289.3.e.m.158.1

• Label: 289.3.e.m.158.1
• Level: $289$
• Weight: $3$
• Character: 289.158
• Analytic conductor: $7.875$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 289 = 17^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	289.e (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.87467964001\)
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:	no (minimal twist has level 17)
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			158.1
Root			\(-0.382683 + 0.923880i\) of defining polynomial
Character	\(\chi\)	\(=\)	289.158
Dual form			289.3.e.m.75.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.15851 + 2.79690i) q^{2} +(0.451406 + 0.675577i) q^{3} +(-3.65205 + 3.65205i) q^{4} +(1.56645 - 7.87510i) q^{5} +(-1.36656 + 2.04520i) q^{6} +(-1.53233 - 7.70353i) q^{7} +(-3.25778 - 1.34942i) q^{8} +(3.19151 - 7.70500i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 8 q^{3} + 16 q^{5} + 24 q^{6} - 8 q^{7} + 24 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/289\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.15851	+	2.79690i	0.579256	+	1.39845i	0.893481	+	0.449100i	\(0.148255\pi\)
							−0.314225	+	0.949348i	\(0.601745\pi\)
\(3\)	0.451406	+	0.675577i	0.150469	+	0.225192i	0.899045	−	0.437857i	\(-0.144262\pi\)
							−0.748576	+	0.663049i	\(0.769262\pi\)
\(5\)	1.56645	−	7.87510i	0.313291	−	1.57502i	−0.427964	−	0.903796i	\(-0.640769\pi\)
							0.741255	−	0.671224i	\(-0.234231\pi\)
\(7\)	−1.53233	−	7.70353i	−0.218904	−	1.10050i	−0.921340	−	0.388757i	\(-0.872904\pi\)
							0.702436	−	0.711746i	\(-0.252096\pi\)
\(11\)	6.71450	+	4.48649i	0.610410	+	0.407863i	0.821993	−	0.569497i	\(-0.192862\pi\)
							−0.211584	+	0.977360i	\(0.567862\pi\)
\(13\)	−0.798835	−	0.798835i	−0.0614489	−	0.0614489i	0.675715	−	0.737163i	\(-0.263835\pi\)
							−0.737163	+	0.675715i	\(0.763835\pi\)
\(17\)		0			0
							\(19\)	−1.07647	−	2.59882i	−0.0566561	−	0.136780i	0.893017	−	0.450023i	\(-0.148584\pi\)
−0.949673	+	0.313244i	\(0.898584\pi\)
\(23\)	4.76696	−	7.13426i	0.207259	−	0.310185i	−0.713247	−	0.700912i	\(-0.752776\pi\)
							0.920507	+	0.390727i	\(0.127776\pi\)
\(29\)	3.01148	+	0.599020i	0.103844	+	0.0206559i	0.246739	−	0.969082i	\(-0.420641\pi\)
							−0.142895	+	0.989738i	\(0.545641\pi\)
\(31\)	−10.6746	+	7.13254i	−0.344342	+	0.230082i	−0.715700	−	0.698408i	\(-0.753892\pi\)
							0.371358	+	0.928490i	\(0.378892\pi\)
\(37\)	−13.1509	−	19.6817i	−0.355429	−	0.531938i	0.610068	−	0.792349i	\(-0.291142\pi\)
							−0.965498	+	0.260411i	\(0.916142\pi\)
\(41\)	4.26082	+	21.4206i	0.103922	+	0.522453i	0.997319	+	0.0731833i	\(0.0233158\pi\)
							−0.893396	+	0.449270i	\(0.851684\pi\)
\(43\)	8.89197	−	21.4671i	0.206790	−	0.499235i	−0.786124	−	0.618069i	\(-0.787915\pi\)
							0.992914	+	0.118833i	\(0.0379154\pi\)
\(47\)	55.6597	+	55.6597i	1.18425	+	1.18425i	0.978632	+	0.205617i	\(0.0659202\pi\)
							0.205617	+	0.978632i	\(0.434080\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	289.3.e.m.158.1		8
17.2	even	8		289.3.e.b.224.1		8
17.3	odd	16		289.3.e.l.249.1		8
17.4	even	4		289.3.e.c.131.1		8
17.5	odd	16		289.3.e.d.40.1		8
17.6	odd	16		289.3.e.c.214.1		8
17.7	odd	16	inner	289.3.e.m.75.1		8
17.8	even	8		289.3.e.l.65.1		8
17.9	even	8		289.3.e.k.65.1		8
17.10	odd	16		289.3.e.i.75.1		8
17.11	odd	16		17.3.e.a.10.1	✓	8
17.12	odd	16		289.3.e.b.40.1		8
17.13	even	4		17.3.e.a.12.1	yes	8
17.14	odd	16		289.3.e.k.249.1		8
17.15	even	8		289.3.e.d.224.1		8
17.16	even	2		289.3.e.i.158.1		8
51.11	even	16		153.3.p.b.10.1		8
51.47	odd	4		153.3.p.b.46.1		8
68.11	even	16		272.3.bh.c.129.1		8
68.47	odd	4		272.3.bh.c.97.1		8
85.13	odd	4		425.3.t.a.199.1		8
85.28	even	16		425.3.t.c.299.1		8
85.47	odd	4		425.3.t.c.199.1		8
85.62	even	16		425.3.t.a.299.1		8
85.64	even	4		425.3.u.b.301.1		8
85.79	odd	16		425.3.u.b.401.1		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.3.e.a.10.1	✓	8	17.11	odd	16
17.3.e.a.12.1	yes	8	17.13	even	4
153.3.p.b.10.1		8	51.11	even	16
153.3.p.b.46.1		8	51.47	odd	4
272.3.bh.c.97.1		8	68.47	odd	4
272.3.bh.c.129.1		8	68.11	even	16
289.3.e.b.40.1		8	17.12	odd	16
289.3.e.b.224.1		8	17.2	even	8
289.3.e.c.131.1		8	17.4	even	4
289.3.e.c.214.1		8	17.6	odd	16
289.3.e.d.40.1		8	17.5	odd	16
289.3.e.d.224.1		8	17.15	even	8
289.3.e.i.75.1		8	17.10	odd	16
289.3.e.i.158.1		8	17.16	even	2
289.3.e.k.65.1		8	17.9	even	8
289.3.e.k.249.1		8	17.14	odd	16
289.3.e.l.65.1		8	17.8	even	8
289.3.e.l.249.1		8	17.3	odd	16
289.3.e.m.75.1		8	17.7	odd	16	inner
289.3.e.m.158.1		8	1.1	even	1	trivial
425.3.t.a.199.1		8	85.13	odd	4
425.3.t.a.299.1		8	85.62	even	16
425.3.t.c.199.1		8	85.47	odd	4
425.3.t.c.299.1		8	85.28	even	16
425.3.u.b.301.1		8	85.64	even	4
425.3.u.b.401.1		8	85.79	odd	16