Embedded newform 289.3.e.l.249.1

• Label: 289.3.e.l.249.1
• Level: $289$
• Weight: $3$
• Character: 289.249
• 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:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 17)
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			249.1
Root			\(-0.923880 + 0.382683i\) of defining polynomial
Character	\(\chi\)	\(=\)	289.249
Dual form			289.3.e.l.65.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.79690 + 1.15851i) q^{2} +(0.158513 + 0.796897i) q^{3} +(3.65205 + 3.65205i) q^{4} +(6.67619 + 4.46088i) q^{5} +(-0.479872 + 2.41248i) q^{6} +(-6.53073 + 4.36370i) q^{7} +(1.34942 + 3.25778i) q^{8} +(7.70500 - 3.19151i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 16 q^{5} + 8 q^{6} - 40 q^{7} + 40 q^{8} - 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{7}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.79690	+	1.15851i	1.39845	+	0.579256i	0.949348	−	0.314225i	\(-0.101745\pi\)
							0.449100	+	0.893481i	\(0.351745\pi\)
\(3\)	0.158513	+	0.796897i	0.0528376	+	0.265632i	0.998170	−	0.0604771i	\(-0.0192622\pi\)
							−0.945332	+	0.326109i	\(0.894262\pi\)
\(5\)	6.67619	+	4.46088i	1.33524	+	0.892177i	0.998773	−	0.0495193i	\(-0.0157689\pi\)
							0.336464	+	0.941696i	\(0.390769\pi\)
\(7\)	−6.53073	+	4.36370i	−0.932962	+	0.623385i	−0.926379	−	0.376593i	\(-0.877096\pi\)
							−0.00658318	+	0.999978i	\(0.502096\pi\)
\(11\)	−7.92030	−	1.57545i	−0.720027	−	0.143222i	−0.178542	−	0.983932i	\(-0.557138\pi\)
							−0.541486	+	0.840710i	\(0.682138\pi\)
\(13\)	0.798835	−	0.798835i	0.0614489	−	0.0614489i	−0.675715	−	0.737163i	\(-0.736165\pi\)
							0.737163	+	0.675715i	\(0.236165\pi\)
\(17\)		0			0
							\(19\)	−2.59882	−	1.07647i	−0.136780	−	0.0566561i	0.313244	−	0.949673i	\(-0.398584\pi\)
−0.450023	+	0.893017i	\(0.648584\pi\)
\(23\)	1.67393	−	8.41543i	0.0727797	−	0.365888i	−0.927182	−	0.374611i	\(-0.877776\pi\)
							0.999962	+	0.00872222i	\(0.00277640\pi\)
\(29\)	1.70587	−	2.55301i	0.0588230	−	0.0880348i	−0.800892	−	0.598809i	\(-0.795641\pi\)
							0.859715	+	0.510774i	\(0.170641\pi\)
\(31\)	12.5915	−	2.50461i	0.406179	−	0.0807940i	0.0122273	−	0.999925i	\(-0.496108\pi\)
							0.393951	+	0.919131i	\(0.371108\pi\)
\(37\)	−4.61798	−	23.2161i	−0.124810	−	0.627463i	−0.991659	−	0.128892i	\(-0.958858\pi\)
							0.866848	−	0.498572i	\(-0.166142\pi\)
\(41\)	18.1595	−	12.1338i	0.442914	−	0.295946i	−0.314045	−	0.949408i	\(-0.601684\pi\)
							0.756959	+	0.653462i	\(0.226684\pi\)
\(43\)	21.4671	−	8.89197i	0.499235	−	0.206790i	−0.118833	−	0.992914i	\(-0.537915\pi\)
							0.618069	+	0.786124i	\(0.287915\pi\)
\(47\)	−55.6597	+	55.6597i	−1.18425	+	1.18425i	−0.205617	+	0.978632i	\(0.565920\pi\)
							−0.978632	+	0.205617i	\(0.934080\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	289.3.e.l.249.1		8
17.2	even	8		289.3.e.c.214.1		8
17.3	odd	16		289.3.e.k.65.1		8
17.4	even	4		289.3.e.b.40.1		8
17.5	odd	16		289.3.e.d.224.1		8
17.6	odd	16		289.3.e.m.158.1		8
17.7	odd	16		289.3.e.c.131.1		8
17.8	even	8		289.3.e.m.75.1		8
17.9	even	8		289.3.e.i.75.1		8
17.10	odd	16		17.3.e.a.12.1	yes	8
17.11	odd	16		289.3.e.i.158.1		8
17.12	odd	16		289.3.e.b.224.1		8
17.13	even	4		289.3.e.d.40.1		8
17.14	odd	16	inner	289.3.e.l.65.1		8
17.15	even	8		17.3.e.a.10.1	✓	8
17.16	even	2		289.3.e.k.249.1		8
51.32	odd	8		153.3.p.b.10.1		8
51.44	even	16		153.3.p.b.46.1		8
68.15	odd	8		272.3.bh.c.129.1		8
68.27	even	16		272.3.bh.c.97.1		8
85.27	even	16		425.3.t.c.199.1		8
85.32	odd	8		425.3.t.a.299.1		8
85.44	odd	16		425.3.u.b.301.1		8
85.49	even	8		425.3.u.b.401.1		8
85.78	even	16		425.3.t.a.199.1		8
85.83	odd	8		425.3.t.c.299.1		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.3.e.a.10.1	✓	8	17.15	even	8
17.3.e.a.12.1	yes	8	17.10	odd	16
153.3.p.b.10.1		8	51.32	odd	8
153.3.p.b.46.1		8	51.44	even	16
272.3.bh.c.97.1		8	68.27	even	16
272.3.bh.c.129.1		8	68.15	odd	8
289.3.e.b.40.1		8	17.4	even	4
289.3.e.b.224.1		8	17.12	odd	16
289.3.e.c.131.1		8	17.7	odd	16
289.3.e.c.214.1		8	17.2	even	8
289.3.e.d.40.1		8	17.13	even	4
289.3.e.d.224.1		8	17.5	odd	16
289.3.e.i.75.1		8	17.9	even	8
289.3.e.i.158.1		8	17.11	odd	16
289.3.e.k.65.1		8	17.3	odd	16
289.3.e.k.249.1		8	17.16	even	2
289.3.e.l.65.1		8	17.14	odd	16	inner
289.3.e.l.249.1		8	1.1	even	1	trivial
289.3.e.m.75.1		8	17.8	even	8
289.3.e.m.158.1		8	17.6	odd	16
425.3.t.a.199.1		8	85.78	even	16
425.3.t.a.299.1		8	85.32	odd	8
425.3.t.c.199.1		8	85.27	even	16
425.3.t.c.299.1		8	85.83	odd	8
425.3.u.b.301.1		8	85.44	odd	16
425.3.u.b.401.1		8	85.49	even	8