Embedded newform 169.4.i.a.3.1

• Label: 169.4.i.a.3.1
• 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			3.1
Character	\(\chi\)	\(=\)	169.3
Dual form			169.4.i.a.113.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.36857 + 3.28276i) q^{2} +(0.666615 - 3.26530i) q^{3} +(6.08208 - 20.9978i) q^{4} +(11.6758 + 6.12792i) q^{5} +(7.80705 + 16.4530i) q^{6} +(-30.8182 - 5.00845i) q^{7} +(26.8589 + 70.8210i) q^{8} +(14.6217 + 6.22971i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1080 q - 27 q^{2} - 19 q^{3} + 155 q^{4} - 30 q^{5} - 56 q^{6} + 9 q^{7} + 4 q^{8} + 364 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.36857	+	3.28276i	−1.54452	+	1.16063i	−0.616940	+	0.787010i	\(0.711628\pi\)
							−0.927581	+	0.373622i	\(0.878116\pi\)
\(3\)	0.666615	−	3.26530i	0.128290	−	0.628407i	−0.863899	−	0.503665i	\(-0.831985\pi\)
							0.992189	−	0.124742i	\(-0.0398102\pi\)
\(5\)	11.6758	+	6.12792i	1.04431	+	0.548097i	0.897476	−	0.441063i	\(-0.145399\pi\)
							0.146836	+	0.989161i	\(0.453091\pi\)
\(7\)	−30.8182	−	5.00845i	−1.66403	−	0.270431i	−0.745709	−	0.666272i	\(-0.767889\pi\)
							−0.918319	+	0.395841i	\(0.870453\pi\)
\(11\)	−37.1613	+	15.8329i	−1.01859	+	0.433983i	−0.835721	−	0.549154i	\(-0.814950\pi\)
							−0.182873	+	0.983136i	\(0.558540\pi\)
\(13\)	30.1739	−	35.8683i	0.643749	−	0.765236i
							\(17\)	119.281	+	19.3850i	1.70175	+	0.276562i	0.932180	−	0.361995i	\(-0.117904\pi\)
0.769574	+	0.638557i	\(0.220468\pi\)
\(19\)	3.98682	−	6.90538i	0.0481389	−	0.0833790i	−0.840952	−	0.541110i	\(-0.818004\pi\)
							0.889091	+	0.457731i	\(0.151338\pi\)
\(23\)	−13.9276	−	24.1233i	−0.126265	−	0.218698i	0.795962	−	0.605347i	\(-0.206966\pi\)
							−0.922227	+	0.386649i	\(0.873632\pi\)
\(29\)	10.8164	−	8.12800i	0.0692605	−	0.0520459i	−0.565548	−	0.824715i	\(-0.691335\pi\)
							0.634809	+	0.772669i	\(0.281079\pi\)
\(31\)	30.0999	−	43.6072i	0.174390	−	0.252648i	−0.726036	−	0.687657i	\(-0.758639\pi\)
							0.900426	+	0.435009i	\(0.143255\pi\)
\(37\)	288.792	+	23.3137i	1.28317	+	0.103588i	0.703164	−	0.711027i	\(-0.251770\pi\)
							0.580002	+	0.814615i	\(0.303052\pi\)
\(41\)	−28.6631	+	140.401i	−0.109181	+	0.534805i	0.888005	+	0.459835i	\(0.152091\pi\)
							−0.997186	+	0.0749703i	\(0.976114\pi\)
\(43\)	448.492	−	36.2060i	1.59057	−	0.128404i	0.746776	−	0.665075i	\(-0.231601\pi\)
							0.843791	+	0.536671i	\(0.180319\pi\)
\(47\)	414.953	+	102.277i	1.28781	+	0.317417i	0.823024	−	0.568006i	\(-0.192285\pi\)
							0.464786	+	0.885423i	\(0.346131\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.4.i.a.3.1	✓	1080
169.113	even	39	inner	169.4.i.a.113.1	yes	1080

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.4.i.a.3.1	✓	1080	1.1	even	1	trivial
169.4.i.a.113.1	yes	1080	169.113	even	39	inner