Embedded newform 169.2.g.a.14.13

• Label: 169.2.g.a.14.13
• 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			14.13
Character	\(\chi\)	\(=\)	169.14
Dual form			169.2.g.a.157.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.39315 - 2.01832i) q^{2} +(0.422612 - 0.221804i) q^{3} +(-1.42355 - 3.75360i) q^{4} +(1.09172 + 0.967184i) q^{5} +(0.141089 - 1.16197i) q^{6} +(0.0247635 + 0.00610365i) q^{7} +(-4.79682 - 1.18231i) q^{8} +(-1.57479 + 2.28148i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 10 q^{2} - 9 q^{3} - 20 q^{4} - 7 q^{5} - q^{6} - 5 q^{7} + 2 q^{8} - 14 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.39315	−	2.01832i	0.985103	−	1.42717i	0.0825511	−	0.996587i	\(-0.473693\pi\)
							0.902552	−	0.430581i	\(-0.141691\pi\)
\(3\)	0.422612	−	0.221804i	0.243995	−	0.128058i	−0.338294	−	0.941041i	\(-0.609850\pi\)
							0.582289	+	0.812982i	\(0.302157\pi\)
\(5\)	1.09172	+	0.967184i	0.488234	+	0.432538i	0.870982	−	0.491315i	\(-0.163484\pi\)
							−0.382748	+	0.923853i	\(0.625022\pi\)
\(7\)	0.0247635	+	0.00610365i	0.00935971	+	0.00230696i	0.243993	−	0.969777i	\(-0.421543\pi\)
							−0.234633	+	0.972084i	\(0.575389\pi\)
\(11\)	−0.525155	−	0.760819i	−0.158340	−	0.229396i	0.735782	−	0.677219i	\(-0.236815\pi\)
							−0.894122	+	0.447823i	\(0.852199\pi\)
\(13\)	−1.67528	+	3.19272i	−0.464639	+	0.885500i
							\(17\)	−3.00961	−	0.741802i	−0.729938	−	0.179914i	−0.143204	−	0.989693i	\(-0.545740\pi\)
−0.586734	+	0.809780i	\(0.699587\pi\)
\(19\)	0.108296			0.0248447			0.0124224	−	0.999923i	\(-0.496046\pi\)
							0.0124224	+	0.999923i	\(0.496046\pi\)
\(23\)	7.06888			1.47396			0.736981	−	0.675913i	\(-0.236251\pi\)
							0.736981	+	0.675913i	\(0.236251\pi\)
\(29\)	−2.26582	+	3.28261i	−0.420752	+	0.609565i	−0.974843	−	0.222891i	\(-0.928451\pi\)
							0.554091	+	0.832456i	\(0.313066\pi\)
\(31\)	−0.280618	+	2.31110i	−0.0504006	+	0.415086i	0.945571	+	0.325416i	\(0.105504\pi\)
							−0.995972	+	0.0896700i	\(0.971419\pi\)
\(37\)	0.673803	−	5.54927i	0.110773	−	0.912294i	−0.826317	−	0.563205i	\(-0.809568\pi\)
							0.937090	−	0.349089i	\(-0.113509\pi\)
\(41\)	4.78352	−	2.51058i	0.747060	−	0.392087i	−0.0478183	−	0.998856i	\(-0.515227\pi\)
							0.794879	+	0.606769i	\(0.207535\pi\)
\(43\)	−0.311275	−	2.56358i	−0.0474690	−	0.390942i	−0.997008	−	0.0772944i	\(-0.975372\pi\)
							0.949539	−	0.313648i	\(-0.101551\pi\)
\(47\)	−1.52449	+	4.01975i	−0.222370	+	0.586340i	−0.999051	−	0.0435530i	\(-0.986132\pi\)
							0.776681	+	0.629894i	\(0.216901\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.2.g.a.14.13	✓	156
169.157	even	13	inner	169.2.g.a.157.13	yes	156

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.g.a.14.13	✓	156	1.1	even	1	trivial
169.2.g.a.157.13	yes	156	169.157	even	13	inner