Embedded newform 169.2.i.a.9.5

• Label: 169.2.i.a.9.5
• Level: $169$
• Weight: $2$
• Character: 169.9
• Analytic conductor: $1.349$
• Analytic rank: $0$
• Dimension: $336$
• 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			9.5
Character	\(\chi\)	\(=\)	169.9
Dual form			169.2.i.a.94.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.386628 + 1.33479i) q^{2} +(1.08858 + 0.463801i) q^{3} +(0.0581875 + 0.0367956i) q^{4} +(0.442938 + 0.641707i) q^{5} +(-1.03995 + 1.27371i) q^{6} +(0.633960 + 0.211647i) q^{7} +(-2.15196 + 1.90647i) q^{8} +(-1.10828 - 1.15384i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 26 q^{2} - 26 q^{3} - 12 q^{4} - 26 q^{5} - 26 q^{6} - 26 q^{7} - 26 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{23}{39}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.386628	+	1.33479i	−0.273387	+	0.943842i	0.699898	+	0.714243i	\(0.253229\pi\)
							−0.973285	+	0.229599i	\(0.926259\pi\)
\(3\)	1.08858	+	0.463801i	0.628493	+	0.267776i	0.682637	−	0.730757i	\(-0.260833\pi\)
							−0.0541446	+	0.998533i	\(0.517243\pi\)
\(5\)	0.442938	+	0.641707i	0.198088	+	0.286980i	0.909422	−	0.415874i	\(-0.136524\pi\)
							−0.711334	+	0.702854i	\(0.751909\pi\)
\(7\)	0.633960	+	0.211647i	0.239614	+	0.0799950i	0.433939	−	0.900942i	\(-0.357123\pi\)
							−0.194325	+	0.980937i	\(0.562252\pi\)
\(11\)	3.50667	−	3.65083i	1.05730	−	1.10077i	0.0625911	−	0.998039i	\(-0.480064\pi\)
							0.994710	−	0.102727i	\(-0.0327569\pi\)
\(13\)	0.739519	+	3.52890i	0.205106	+	0.978740i
							\(17\)	−7.25608	−	2.42244i	−1.75986	−	0.587527i	−0.761685	−	0.647947i	\(-0.775628\pi\)
−0.998173	+	0.0604205i	\(0.980756\pi\)
\(19\)	1.50539	+	2.60741i	0.345359	+	0.598180i	0.985419	−	0.170145i	\(-0.0544237\pi\)
							−0.640060	+	0.768325i	\(0.721090\pi\)
\(23\)	−1.64108	+	2.84243i	−0.342188	+	0.592687i	−0.984839	−	0.173472i	\(-0.944501\pi\)
							0.642651	+	0.766159i	\(0.277835\pi\)
\(29\)	1.41390	−	4.88135i	0.262555	−	0.906445i	−0.715590	−	0.698521i	\(-0.753842\pi\)
							0.978145	−	0.207924i	\(-0.0666708\pi\)
\(31\)	−0.276439	−	0.728910i	−0.0496499	−	0.130916i	0.907897	−	0.419193i	\(-0.137687\pi\)
							−0.957547	+	0.288277i	\(0.906918\pi\)
\(37\)	8.61878	+	1.40069i	1.41692	+	0.230272i	0.820153	−	0.572144i	\(-0.193888\pi\)
							0.596765	+	0.802416i	\(0.296452\pi\)
\(41\)	11.1079	+	4.73264i	1.73476	+	0.739114i	0.998017	+	0.0629458i	\(0.0200495\pi\)
							0.736747	+	0.676168i	\(0.236361\pi\)
\(43\)	5.38638	−	0.875372i	0.821415	−	0.133493i	0.264842	−	0.964292i	\(-0.414680\pi\)
							0.556573	+	0.830799i	\(0.312116\pi\)
\(47\)	−11.8034	−	6.19492i	−1.72171	−	0.903622i	−0.969641	−	0.244534i	\(-0.921365\pi\)
							−0.752066	−	0.659088i	\(-0.770943\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.2.i.a.9.5	✓	336
169.94	even	39	inner	169.2.i.a.94.5	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.i.a.9.5	✓	336	1.1	even	1	trivial
169.2.i.a.94.5	yes	336	169.94	even	39	inner