Embedded newform 169.2.g.a.14.2

• Label: 169.2.g.a.14.2
• 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			14.2
Character	\(\chi\)	\(=\)	169.14
Dual form			169.2.g.a.157.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.29190 + 1.87164i) q^{2} +(1.09718 - 0.575847i) q^{3} +(-1.12482 - 2.96591i) q^{4} +(0.226698 + 0.200837i) q^{5} +(-0.339674 + 2.79747i) q^{6} +(2.40457 + 0.592674i) q^{7} +(2.58801 + 0.637888i) q^{8} +(-0.831980 + 1.20533i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 10 q^{2} - 9 q^{3} - 20 q^{4} - 7 q^{5} - q^{6} - 5 q^{7} + 2 q^{8} - 14 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.29190	+	1.87164i	−0.913511	+	1.32345i	0.0325532	+	0.999470i	\(0.489636\pi\)
							−0.946064	+	0.323979i	\(0.894979\pi\)
\(3\)	1.09718	−	0.575847i	0.633460	−	0.332465i	−0.117245	−	0.993103i	\(-0.537406\pi\)
							0.750705	+	0.660638i	\(0.229714\pi\)
\(5\)	0.226698	+	0.200837i	0.101382	+	0.0898170i	0.712291	−	0.701884i	\(-0.247657\pi\)
							−0.610909	+	0.791701i	\(0.709196\pi\)
\(7\)	2.40457	+	0.592674i	0.908844	+	0.224010i	0.665909	−	0.746033i	\(-0.268044\pi\)
							0.242934	+	0.970043i	\(0.421890\pi\)
\(11\)	2.43347	+	3.52549i	0.733720	+	1.06298i	0.995209	+	0.0977701i	\(0.0311710\pi\)
							−0.261489	+	0.965206i	\(0.584214\pi\)
\(13\)	3.44132	+	1.07579i	0.954450	+	0.298370i
							\(17\)	−3.91461	−	0.964865i	−0.949432	−	0.234014i	−0.265955	−	0.963986i	\(-0.585687\pi\)
−0.683478	+	0.729972i	\(0.739533\pi\)
\(19\)	−1.30669			−0.299774			−0.149887	−	0.988703i	\(-0.547891\pi\)
							−0.149887	+	0.988703i	\(0.547891\pi\)
\(23\)	−1.85004			−0.385760			−0.192880	−	0.981222i	\(-0.561783\pi\)
							−0.192880	+	0.981222i	\(0.561783\pi\)
\(29\)	0.975373	−	1.41307i	0.181122	−	0.262401i	−0.721897	−	0.692000i	\(-0.756730\pi\)
							0.903020	+	0.429599i	\(0.141345\pi\)
\(31\)	0.741217	−	6.10447i	0.133127	−	1.09640i	−0.761054	−	0.648689i	\(-0.775318\pi\)
							0.894180	−	0.447707i	\(-0.147759\pi\)
\(37\)	−0.287013	+	2.36376i	−0.0471846	+	0.388600i	0.949916	+	0.312505i	\(0.101168\pi\)
							−0.997101	+	0.0760947i	\(0.975755\pi\)
\(41\)	6.37425	−	3.34547i	0.995491	−	0.522474i	0.113398	−	0.993550i	\(-0.463826\pi\)
							0.882093	+	0.471076i	\(0.156134\pi\)
\(43\)	−0.437888	−	3.60633i	−0.0667773	−	0.549960i	−0.987253	−	0.159160i	\(-0.949121\pi\)
							0.920476	−	0.390800i	\(-0.127802\pi\)
\(47\)	1.48404	−	3.91309i	0.216470	−	0.570783i	−0.782162	−	0.623075i	\(-0.785883\pi\)
							0.998632	+	0.0522914i	\(0.0166525\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.2.g.a.14.2	✓	156
169.157	even	13	inner	169.2.g.a.157.2	yes	156

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.g.a.14.2	✓	156	1.1	even	1	trivial
169.2.g.a.157.2	yes	156	169.157	even	13	inner