Embedded newform 289.2.f.a.137.9

• Label: 289.2.f.a.137.9
• Level: $289$
• Weight: $2$
• Character: 289.137
• 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			137.9
Character	\(\chi\)	\(=\)	289.137
Dual form			289.2.f.a.154.9

$q$-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 \(\left(\mathbb{Z}/289\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{11}{17}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
289.2.f.a.137.9	✓	384	1.1	even	1	trivial
289.2.f.a.154.9	yes	384	289.154	even	17	inner