Embedded newform 289.2.f.a.154.9

• Label: 289.2.f.a.154.9
• Level: $289$
• Weight: $2$
• Character: 289.154
• Analytic conductor: $2.308$
• Analytic rank: $0$
• Dimension: $384$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(2.30767661842\)
Analytic rank:	\(0\)
Dimension:	\(384\)
Relative dimension:	\(24\) over \(\Q(\zeta_{17})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{17}]$

Embedding invariants

Embedding label			154.9
Character	\(\chi\)	\(=\)	289.154
Dual form			289.2.f.a.137.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.824505 - 0.154127i) q^{2} +(-0.224729 - 0.451317i) q^{3} +(-1.20889 - 0.468327i) q^{4} +(-2.39340 + 1.48193i) q^{5} +(0.115730 + 0.406750i) q^{6} +(-0.383200 + 0.507439i) q^{7} +(2.35086 + 1.45559i) q^{8} +(1.65472 - 2.19120i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q - 13 q^{2} - 13 q^{3} - 33 q^{4} - 43 q^{5} - 5 q^{6} - 13 q^{7} - 5 q^{8} - 25 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{6}{17}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.824505	−	0.154127i	−0.583013	−	0.108984i	−0.116023	−	0.993247i	\(-0.537015\pi\)
							−0.466990	+	0.884262i	\(0.654662\pi\)
\(3\)	−0.224729	−	0.451317i	−0.129748	−	0.260568i	0.820758	−	0.571276i	\(-0.193551\pi\)
							−0.950505	+	0.310708i	\(0.899434\pi\)
\(5\)	−2.39340	+	1.48193i	−1.07036	+	0.662740i	−0.944235	−	0.329274i	\(-0.893196\pi\)
							−0.126128	+	0.992014i	\(0.540255\pi\)
\(7\)	−0.383200	+	0.507439i	−0.144836	+	0.191794i	−0.864652	−	0.502371i	\(-0.832461\pi\)
							0.719816	+	0.694165i	\(0.244226\pi\)
\(11\)	2.02805	+	0.785669i	0.611479	+	0.236888i	0.647011	−	0.762480i	\(-0.276019\pi\)
							−0.0355325	+	0.999369i	\(0.511313\pi\)
\(13\)	3.58475	+	2.21959i	0.994232	+	0.615602i	0.924186	−	0.381942i	\(-0.124744\pi\)
							0.0700453	+	0.997544i	\(0.477686\pi\)
\(17\)	1.56138	−	3.81603i	0.378691	−	0.925523i
							\(19\)	5.46847	−	1.02224i	1.25455	−	0.234517i	0.485805	−	0.874067i	\(-0.338527\pi\)
0.768749	+	0.639550i	\(0.220879\pi\)
\(23\)	−3.57287	+	4.73125i	−0.744996	+	0.986533i	0.254814	+	0.966990i	\(0.417986\pi\)
							−0.999809	+	0.0195433i	\(0.993779\pi\)
\(29\)	−0.635122	−	0.246048i	−0.117939	−	0.0456899i	0.301549	−	0.953451i	\(-0.402496\pi\)
							−0.419488	+	0.907761i	\(0.637790\pi\)
\(31\)	3.73169	+	2.31056i	0.670231	+	0.414989i	0.818930	−	0.573893i	\(-0.194568\pi\)
							−0.148700	+	0.988882i	\(0.547509\pi\)
\(37\)	0.0551600	−	0.595271i	0.00906825	−	0.0978620i	−0.989997	−	0.141088i	\(-0.954940\pi\)
							0.999065	+	0.0432259i	\(0.0137635\pi\)
\(41\)	4.85027	+	9.74066i	0.757485	+	1.52124i	0.850736	+	0.525593i	\(0.176156\pi\)
							−0.0932509	+	0.995643i	\(0.529726\pi\)
\(43\)	8.75384	−	7.98018i	1.33495	−	1.21697i	0.378313	−	0.925678i	\(-0.376504\pi\)
							0.956635	−	0.291288i	\(-0.0940838\pi\)
\(47\)	4.79627	+	6.35128i	0.699607	+	0.926430i	0.999622	−	0.0275049i	\(-0.00875619\pi\)
							−0.300014	+	0.953935i	\(0.596991\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	289.2.f.a.154.9	yes	384
289.137	even	17	inner	289.2.f.a.137.9	✓	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
289.2.f.a.137.9	✓	384	289.137	even	17	inner
289.2.f.a.154.9	yes	384	1.1	even	1	trivial