Embedded newform 289.3.e.b.224.1

• Label: 289.3.e.b.224.1
• Level: $289$
• Weight: $3$
• Character: 289.224
• 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			224.1
Root			\(0.923880 + 0.382683i\) of defining polynomial
Character	\(\chi\)	\(=\)	289.224
Dual form			289.3.e.b.40.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.79690 + 1.15851i) q^{2} +(0.796897 + 0.158513i) q^{3} +(3.65205 - 3.65205i) q^{4} +(4.46088 + 6.67619i) q^{5} +(-2.41248 + 0.479872i) q^{6} +(-4.36370 + 6.53073i) q^{7} +(-1.34942 + 3.25778i) q^{8} +(-7.70500 - 3.19151i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} - 8 q^{3} - 32 q^{6} - 8 q^{7} - 40 q^{8} + 8 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.79690	+	1.15851i	−1.39845	+	0.579256i	−0.949348	−	0.314225i	\(-0.898255\pi\)
							−0.449100	+	0.893481i	\(0.648255\pi\)
\(3\)	0.796897	+	0.158513i	0.265632	+	0.0528376i	0.326109	−	0.945332i	\(-0.394262\pi\)
							−0.0604771	+	0.998170i	\(0.519262\pi\)
\(5\)	4.46088	+	6.67619i	0.892177	+	1.33524i	0.941696	+	0.336464i	\(0.109231\pi\)
							−0.0495193	+	0.998773i	\(0.515769\pi\)
\(7\)	−4.36370	+	6.53073i	−0.623385	+	0.932962i	0.376593	+	0.926379i	\(0.377096\pi\)
							−0.999978	+	0.00658318i	\(0.997904\pi\)
\(11\)	1.57545	+	7.92030i	0.143222	+	0.720027i	0.983932	+	0.178542i	\(0.0571379\pi\)
							−0.840710	+	0.541486i	\(0.817862\pi\)
\(13\)	0.798835	+	0.798835i	0.0614489	+	0.0614489i	0.737163	−	0.675715i	\(-0.236165\pi\)
							−0.675715	+	0.737163i	\(0.736165\pi\)
\(17\)		0			0
							\(19\)	2.59882	−	1.07647i	0.136780	−	0.0566561i	−0.313244	−	0.949673i	\(-0.601416\pi\)
0.450023	+	0.893017i	\(0.351416\pi\)
\(23\)	8.41543	−	1.67393i	0.365888	−	0.0727797i	−0.00872222	−	0.999962i	\(-0.502776\pi\)
							0.374611	+	0.927182i	\(0.377776\pi\)
\(29\)	−2.55301	+	1.70587i	−0.0880348	+	0.0588230i	−0.598809	−	0.800892i	\(-0.704359\pi\)
							0.510774	+	0.859715i	\(0.329359\pi\)
\(31\)	−2.50461	+	12.5915i	−0.0807940	+	0.406179i	0.919131	+	0.393951i	\(0.128892\pi\)
							−0.999925	+	0.0122273i	\(0.996108\pi\)
\(37\)	−23.2161	−	4.61798i	−0.627463	−	0.124810i	−0.128892	−	0.991659i	\(-0.541142\pi\)
							−0.498572	+	0.866848i	\(0.666142\pi\)
\(41\)	12.1338	−	18.1595i	0.295946	−	0.442914i	−0.653462	−	0.756959i	\(-0.726684\pi\)
							0.949408	+	0.314045i	\(0.101684\pi\)
\(43\)	−21.4671	−	8.89197i	−0.499235	−	0.206790i	0.118833	−	0.992914i	\(-0.462085\pi\)
							−0.618069	+	0.786124i	\(0.712085\pi\)
\(47\)	−55.6597	−	55.6597i	−1.18425	−	1.18425i	−0.978632	−	0.205617i	\(-0.934080\pi\)
							−0.205617	−	0.978632i	\(-0.565920\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.3.e.a.10.1	✓	8	17.14	odd	16
17.3.e.a.12.1	yes	8	17.15	even	8
153.3.p.b.10.1		8	51.14	even	16
153.3.p.b.46.1		8	51.32	odd	8
272.3.bh.c.97.1		8	68.15	odd	8
272.3.bh.c.129.1		8	68.31	even	16
289.3.e.b.40.1		8	17.6	odd	16	inner
289.3.e.b.224.1		8	1.1	even	1	trivial
289.3.e.c.131.1		8	17.2	even	8
289.3.e.c.214.1		8	17.3	odd	16
289.3.e.d.40.1		8	17.11	odd	16
289.3.e.d.224.1		8	17.16	even	2
289.3.e.i.75.1		8	17.5	odd	16
289.3.e.i.158.1		8	17.8	even	8
289.3.e.k.65.1		8	17.13	even	4
289.3.e.k.249.1		8	17.7	odd	16
289.3.e.l.65.1		8	17.4	even	4
289.3.e.l.249.1		8	17.10	odd	16
289.3.e.m.75.1		8	17.12	odd	16
289.3.e.m.158.1		8	17.9	even	8
425.3.t.a.199.1		8	85.83	odd	8
425.3.t.a.299.1		8	85.82	even	16
425.3.t.c.199.1		8	85.32	odd	8
425.3.t.c.299.1		8	85.48	even	16
425.3.u.b.301.1		8	85.49	even	8
425.3.u.b.401.1		8	85.14	odd	16