Embedded newform 169.2.g.a.14.9

• Label: 169.2.g.a.14.9
• Level: $169$
• Weight: $2$
• Character: 169.14
• Analytic conductor: $1.349$
• Analytic rank: $0$
• Dimension: $156$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	169.g (of order \(13\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.34947179416\)
Analytic rank:	\(0\)
Dimension:	\(156\)
Relative dimension:	\(13\) over \(\Q(\zeta_{13})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{13}]$

Embedding invariants

Embedding label			14.9
Character	\(\chi\)	\(=\)	169.14
Dual form			169.2.g.a.157.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.481559 - 0.697658i) q^{2} +(1.04345 - 0.547645i) q^{3} +(0.454382 + 1.19811i) q^{4} +(0.674379 + 0.597448i) q^{5} +(0.120414 - 0.991695i) q^{6} +(-0.515253 - 0.126998i) q^{7} +(2.70085 + 0.665700i) q^{8} +(-0.915320 + 1.32607i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 10 q^{2} - 9 q^{3} - 20 q^{4} - 7 q^{5} - q^{6} - 5 q^{7} + 2 q^{8} - 14 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{9}{13}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.481559	−	0.697658i	0.340513	−	0.493319i	−0.614894	−	0.788610i	\(-0.710801\pi\)
							0.955407	+	0.295291i	\(0.0954166\pi\)
\(3\)	1.04345	−	0.547645i	0.602437	−	0.316183i	−0.135788	−	0.990738i	\(-0.543357\pi\)
							0.738225	+	0.674555i	\(0.235664\pi\)
\(5\)	0.674379	+	0.597448i	0.301591	+	0.267187i	0.800322	−	0.599570i	\(-0.204662\pi\)
							−0.498731	+	0.866757i	\(0.666200\pi\)
\(7\)	−0.515253	−	0.126998i	−0.194747	−	0.0480009i	0.140736	−	0.990047i	\(-0.455053\pi\)
							−0.335483	+	0.942046i	\(0.608899\pi\)
\(11\)	−1.47414	−	2.13566i	−0.444471	−	0.643927i	0.535156	−	0.844753i	\(-0.320253\pi\)
							−0.979627	+	0.200826i	\(0.935637\pi\)
\(13\)	−0.909072	−	3.48907i	−0.252131	−	0.967693i
							\(17\)	−5.78443	−	1.42573i	−1.40293	−	0.345791i	−0.535998	−	0.844219i	\(-0.680065\pi\)
−0.866931	+	0.498428i	\(0.833911\pi\)
\(19\)	4.91931			1.12857			0.564284	−	0.825581i	\(-0.309152\pi\)
							0.564284	+	0.825581i	\(0.309152\pi\)
\(23\)	−4.70766			−0.981616			−0.490808	−	0.871268i	\(-0.663298\pi\)
							−0.490808	+	0.871268i	\(0.663298\pi\)
\(29\)	4.17327	−	6.04603i	0.774957	−	1.12272i	−0.214334	−	0.976760i	\(-0.568758\pi\)
							0.989291	−	0.145958i	\(-0.0466266\pi\)
\(31\)	−0.345381	+	2.84446i	−0.0620322	+	0.510881i	0.928265	+	0.371918i	\(0.121300\pi\)
							−0.990298	+	0.138963i	\(0.955623\pi\)
\(37\)	−1.35384	+	11.1499i	−0.222570	+	1.83303i	0.270123	+	0.962826i	\(0.412936\pi\)
							−0.492693	+	0.870203i	\(0.663988\pi\)
\(41\)	3.38243	−	1.77523i	0.528246	−	0.277245i	−0.179444	−	0.983768i	\(-0.557430\pi\)
							0.707690	+	0.706523i	\(0.249737\pi\)
\(43\)	1.21336	+	9.99294i	0.185036	+	1.52391i	0.728833	+	0.684692i	\(0.240063\pi\)
							−0.543796	+	0.839217i	\(0.683014\pi\)
\(47\)	−0.266006	+	0.701399i	−0.0388009	+	0.102310i	−0.953016	−	0.302919i	\(-0.902039\pi\)
							0.914215	+	0.405229i	\(0.132808\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.2.g.a.14.9	✓	156
169.157	even	13	inner	169.2.g.a.157.9	yes	156

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.g.a.14.9	✓	156	1.1	even	1	trivial
169.2.g.a.157.9	yes	156	169.157	even	13	inner