Embedded newform 169.2.i.a.3.11

• Label: 169.2.i.a.3.11
• Level: $169$
• Weight: $2$
• Character: 169.3
• Analytic conductor: $1.349$
• Analytic rank: $0$
• Dimension: $336$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.34947179416\)
Analytic rank:	\(0\)
Dimension:	\(336\)
Relative dimension:	\(14\) over \(\Q(\zeta_{39})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{39}]$

Embedding invariants

Embedding label			3.11
Character	\(\chi\)	\(=\)	169.3
Dual form			169.2.i.a.113.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.18774 - 0.892529i) q^{2} +(0.500302 - 2.45064i) q^{3} +(0.0576840 - 0.199148i) q^{4} +(-1.46041 - 0.766482i) q^{5} +(-1.59304 - 3.35726i) q^{6} +(-0.935624 - 0.152054i) q^{7} +(0.944448 + 2.49030i) q^{8} +(-2.99541 - 1.27623i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 26 q^{2} - 26 q^{3} - 12 q^{4} - 26 q^{5} - 26 q^{6} - 26 q^{7} - 26 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.18774	−	0.892529i	0.839859	−	0.631113i	−0.0912605	−	0.995827i	\(-0.529090\pi\)
							0.931120	+	0.364714i	\(0.118833\pi\)
\(3\)	0.500302	−	2.45064i	0.288850	−	1.41488i	−0.533453	−	0.845830i	\(-0.679106\pi\)
							0.822303	−	0.569050i	\(-0.192689\pi\)
\(5\)	−1.46041	−	0.766482i	−0.653115	−	0.342781i	0.105407	−	0.994429i	\(-0.466386\pi\)
							−0.758521	+	0.651648i	\(0.774078\pi\)
\(7\)	−0.935624	−	0.152054i	−0.353633	−	0.0574709i	−0.0189999	−	0.999819i	\(-0.506048\pi\)
							−0.334633	+	0.942349i	\(0.608612\pi\)
\(11\)	4.83932	−	2.06184i	1.45911	−	0.621668i	0.490766	−	0.871291i	\(-0.336717\pi\)
							0.968343	+	0.249623i	\(0.0803067\pi\)
\(13\)	−1.03714	+	3.45316i	−0.287651	+	0.957735i
							\(17\)	0.773864	+	0.125765i	0.187689	+	0.0305025i	0.253529	−	0.967328i	\(-0.418409\pi\)
−0.0658399	+	0.997830i	\(0.520973\pi\)
\(19\)	0.141822	−	0.245643i	0.0325362	−	0.0563543i	−0.849299	−	0.527912i	\(-0.822975\pi\)
							0.881835	+	0.471558i	\(0.156308\pi\)
\(23\)	0.567784	+	0.983430i	0.118391	+	0.205059i	0.919130	−	0.393954i	\(-0.128893\pi\)
							−0.800739	+	0.599013i	\(0.795560\pi\)
\(29\)	−5.27151	+	3.96129i	−0.978896	+	0.735592i	−0.964440	−	0.264303i	\(-0.914858\pi\)
							−0.0144561	+	0.999896i	\(0.504602\pi\)
\(31\)	−1.39275	+	2.01775i	−0.250146	+	0.362399i	−0.927861	−	0.372925i	\(-0.878355\pi\)
							0.677715	+	0.735324i	\(0.262970\pi\)
\(37\)	7.39917	+	0.597323i	1.21642	+	0.0981992i	0.671958	−	0.740589i	\(-0.265453\pi\)
							0.544458	+	0.838788i	\(0.316736\pi\)
\(41\)	−0.480442	+	2.35336i	−0.0750324	+	0.367533i	−0.999919	−	0.0127096i	\(-0.995954\pi\)
							0.924887	+	0.380242i	\(0.124159\pi\)
\(43\)	−8.92205	+	0.720262i	−1.36060	+	0.109839i	−0.739110	−	0.673584i	\(-0.764754\pi\)
							−0.621489	+	0.783423i	\(0.713472\pi\)
\(47\)	3.68493	+	0.908254i	0.537502	+	0.132482i	0.498717	−	0.866765i	\(-0.333805\pi\)
							0.0387858	+	0.999248i	\(0.487651\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.2.i.a.3.11	✓	336
169.113	even	39	inner	169.2.i.a.113.11	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.i.a.3.11	✓	336	1.1	even	1	trivial
169.2.i.a.113.11	yes	336	169.113	even	39	inner