Embedded newform 289.3.e.i.214.1

• Label: 289.3.e.i.214.1
• Level: $289$
• Weight: $3$
• Character: 289.214
• 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			214.1
Root			\(0.382683 + 0.923880i\) of defining polynomial
Character	\(\chi\)	\(=\)	289.214
Dual form			289.3.e.i.131.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.841487 - 2.03153i) q^{2} +(-0.134381 - 0.0897902i) q^{3} +(-0.590587 - 0.590587i) q^{4} +(-5.26197 + 1.04667i) q^{5} +(-0.295491 + 0.197441i) q^{6} +(6.12453 + 1.21824i) q^{7} +(6.42935 - 2.66313i) q^{8} +(-3.43416 - 8.29078i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} - 8 q^{3} - 16 q^{5} - 24 q^{6} + 8 q^{7} + 24 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.841487	−	2.03153i	0.420744	−	1.01577i	−0.561385	−	0.827555i	\(-0.689731\pi\)
							0.982129	−	0.188210i	\(-0.0602687\pi\)
\(3\)	−0.134381	−	0.0897902i	−0.0447935	−	0.0299301i	0.532972	−	0.846133i	\(-0.321075\pi\)
							−0.577765	+	0.816203i	\(0.696075\pi\)
\(5\)	−5.26197	+	1.04667i	−1.05239	+	0.209334i	−0.690833	−	0.723014i	\(-0.742756\pi\)
							−0.361561	+	0.932348i	\(0.617756\pi\)
\(7\)	6.12453	+	1.21824i	0.874933	+	0.174035i	0.612075	−	0.790800i	\(-0.290335\pi\)
							0.262858	+	0.964835i	\(0.415335\pi\)
\(11\)	−8.11392	−	12.1433i	−0.737629	−	1.10394i	−0.990643	−	0.136478i	\(-0.956422\pi\)
							0.253014	−	0.967463i	\(-0.418578\pi\)
\(13\)	4.79884	−	4.79884i	0.369141	−	0.369141i	−0.498023	−	0.867164i	\(-0.665940\pi\)
							0.867164	+	0.498023i	\(0.165940\pi\)
\(17\)		0			0
							\(19\)	9.56175	−	23.0841i	0.503250	−	1.21495i	−0.444454	−	0.895802i	\(-0.646602\pi\)
0.947704	−	0.319151i	\(-0.103398\pi\)
\(23\)	−10.8899	+	7.27639i	−0.473474	+	0.316365i	−0.769312	−	0.638873i	\(-0.779401\pi\)
							0.295839	+	0.955238i	\(0.404401\pi\)
\(29\)	−6.44436	−	32.3980i	−0.222219	−	1.11717i	−0.917287	−	0.398228i	\(-0.869625\pi\)
							0.695067	−	0.718945i	\(-0.255375\pi\)
\(31\)	−0.674593	+	1.00960i	−0.0217611	+	0.0325678i	−0.842191	−	0.539180i	\(-0.818734\pi\)
							0.820430	+	0.571748i	\(0.193734\pi\)
\(37\)	38.6481	+	25.8238i	1.04454	+	0.697941i	0.954565	−	0.298002i	\(-0.0963203\pi\)
							0.0899781	+	0.995944i	\(0.471320\pi\)
\(41\)	30.8466	+	6.13577i	0.752356	+	0.149653i	0.556348	−	0.830949i	\(-0.312202\pi\)
							0.196008	+	0.980602i	\(0.437202\pi\)
\(43\)	27.8948	+	67.3441i	0.648717	+	1.56614i	0.814617	+	0.579999i	\(0.196947\pi\)
							−0.165901	+	0.986142i	\(0.553053\pi\)
\(47\)	−10.4882	+	10.4882i	−0.223153	+	0.223153i	−0.809825	−	0.586672i	\(-0.800438\pi\)
							0.586672	+	0.809825i	\(0.300438\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	289.3.e.i.214.1		8
17.2	even	8		289.3.e.l.40.1		8
17.3	odd	16		289.3.e.c.158.1		8
17.4	even	4		289.3.e.c.75.1		8
17.5	odd	16		289.3.e.m.131.1		8
17.6	odd	16		289.3.e.l.224.1		8
17.7	odd	16		289.3.e.d.65.1		8
17.8	even	8		289.3.e.b.249.1		8
17.9	even	8		289.3.e.d.249.1		8
17.10	odd	16		289.3.e.b.65.1		8
17.11	odd	16		289.3.e.k.224.1		8
17.12	odd	16	inner	289.3.e.i.131.1		8
17.13	even	4		17.3.e.a.7.1	yes	8
17.14	odd	16		17.3.e.a.5.1	✓	8
17.15	even	8		289.3.e.k.40.1		8
17.16	even	2		289.3.e.m.214.1		8
51.14	even	16		153.3.p.b.73.1		8
51.47	odd	4		153.3.p.b.109.1		8
68.31	even	16		272.3.bh.c.209.1		8
68.47	odd	4		272.3.bh.c.177.1		8
85.13	odd	4		425.3.t.c.24.1		8
85.14	odd	16		425.3.u.b.226.1		8
85.47	odd	4		425.3.t.a.24.1		8
85.48	even	16		425.3.t.a.124.1		8
85.64	even	4		425.3.u.b.126.1		8
85.82	even	16		425.3.t.c.124.1		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.3.e.a.5.1	✓	8	17.14	odd	16
17.3.e.a.7.1	yes	8	17.13	even	4
153.3.p.b.73.1		8	51.14	even	16
153.3.p.b.109.1		8	51.47	odd	4
272.3.bh.c.177.1		8	68.47	odd	4
272.3.bh.c.209.1		8	68.31	even	16
289.3.e.b.65.1		8	17.10	odd	16
289.3.e.b.249.1		8	17.8	even	8
289.3.e.c.75.1		8	17.4	even	4
289.3.e.c.158.1		8	17.3	odd	16
289.3.e.d.65.1		8	17.7	odd	16
289.3.e.d.249.1		8	17.9	even	8
289.3.e.i.131.1		8	17.12	odd	16	inner
289.3.e.i.214.1		8	1.1	even	1	trivial
289.3.e.k.40.1		8	17.15	even	8
289.3.e.k.224.1		8	17.11	odd	16
289.3.e.l.40.1		8	17.2	even	8
289.3.e.l.224.1		8	17.6	odd	16
289.3.e.m.131.1		8	17.5	odd	16
289.3.e.m.214.1		8	17.16	even	2
425.3.t.a.24.1		8	85.47	odd	4
425.3.t.a.124.1		8	85.48	even	16
425.3.t.c.24.1		8	85.13	odd	4
425.3.t.c.124.1		8	85.82	even	16
425.3.u.b.126.1		8	85.64	even	4
425.3.u.b.226.1		8	85.14	odd	16