Embedded newform 169.4.i.a.3.3

• Label: 169.4.i.a.3.3
• Level: $169$
• Weight: $4$
• Character: 169.3
• Analytic conductor: $9.971$
• Analytic rank: $0$
• Dimension: $1080$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	169.i (of order \(39\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.97132279097\)
Analytic rank:	\(0\)
Dimension:	\(1080\)
Relative dimension:	\(45\) over \(\Q(\zeta_{39})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{39}]$

Embedding invariants

Embedding label			3.3
Character	\(\chi\)	\(=\)	169.3
Dual form			169.4.i.a.113.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.12652 + 3.10088i) q^{2} +(0.990669 - 4.85262i) q^{3} +(5.18696 - 17.9075i) q^{4} +(-10.2304 - 5.36931i) q^{5} +(10.9593 + 23.0963i) q^{6} +(3.16043 + 0.513619i) q^{7} +(19.4817 + 51.3691i) q^{8} +(2.27298 + 0.968427i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1080 q - 27 q^{2} - 19 q^{3} + 155 q^{4} - 30 q^{5} - 56 q^{6} + 9 q^{7} + 4 q^{8} + 364 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{31}{39}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.12652	+	3.10088i	−1.45894	+	1.09632i	−0.483689	+	0.875240i	\(0.660703\pi\)
							−0.975255	+	0.221085i	\(0.929040\pi\)
\(3\)	0.990669	−	4.85262i	0.190654	−	0.933887i	−0.766040	−	0.642793i	\(-0.777776\pi\)
							0.956695	−	0.291094i	\(-0.0940192\pi\)
\(5\)	−10.2304	−	5.36931i	−0.915032	−	0.480246i	−0.0596362	−	0.998220i	\(-0.518994\pi\)
							−0.855396	+	0.517974i	\(0.826686\pi\)
\(7\)	3.16043	+	0.513619i	0.170647	+	0.0277328i	0.245134	−	0.969489i	\(-0.421168\pi\)
							−0.0744874	+	0.997222i	\(0.523732\pi\)
\(11\)	48.7066	−	20.7519i	1.33505	−	0.568813i	0.397804	−	0.917471i	\(-0.369773\pi\)
							0.937250	+	0.348657i	\(0.113362\pi\)
\(13\)	44.2568	+	15.4382i	0.944201	+	0.329369i
							\(17\)	−131.903	−	21.4363i	−1.88183	−	0.305827i	−0.892447	−	0.451153i	\(-0.851013\pi\)
−0.989384	+	0.145326i	\(0.953577\pi\)
\(19\)	21.3442	−	36.9692i	0.257720	−	0.446385i	−0.707910	−	0.706302i	\(-0.750362\pi\)
							0.965631	+	0.259917i	\(0.0836953\pi\)
\(23\)	−84.9508	−	147.139i	−0.770151	−	1.33394i	−0.937480	−	0.348039i	\(-0.886848\pi\)
							0.167329	−	0.985901i	\(-0.446486\pi\)
\(29\)	−73.7311	+	55.4053i	−0.472121	+	0.354776i	−0.810016	−	0.586408i	\(-0.800542\pi\)
							0.337895	+	0.941184i	\(0.390285\pi\)
\(31\)	16.2008	−	23.4709i	0.0938628	−	0.135984i	−0.773239	−	0.634115i	\(-0.781365\pi\)
							0.867102	+	0.498131i	\(0.165980\pi\)
\(37\)	84.8707	+	6.85147i	0.377099	+	0.0304426i	0.267563	−	0.963540i	\(-0.413782\pi\)
							0.109536	+	0.993983i	\(0.465064\pi\)
\(41\)	−71.2820	+	349.162i	−0.271521	+	1.33000i	0.583798	+	0.811899i	\(0.301566\pi\)
							−0.855319	+	0.518101i	\(0.826639\pi\)
\(43\)	−122.672	+	9.90313i	−0.435055	+	0.0351212i	−0.296054	−	0.955171i	\(-0.595671\pi\)
							−0.139000	+	0.990292i	\(0.544389\pi\)
\(47\)	−462.459	−	113.986i	−1.43525	−	0.353756i	−0.556421	−	0.830901i	\(-0.687826\pi\)
							−0.878825	+	0.477144i	\(0.841672\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.4.i.a.3.3	✓	1080
169.113	even	39	inner	169.4.i.a.113.3	yes	1080

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.4.i.a.3.3	✓	1080	1.1	even	1	trivial
169.4.i.a.113.3	yes	1080	169.113	even	39	inner