Embedded newform 169.2.g.a.27.7

• Label: 169.2.g.a.27.7
• Level: $169$
• Weight: $2$
• Character: 169.27
• 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			27.7
Character	\(\chi\)	\(=\)	169.27
Dual form			169.2.g.a.144.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.000492135 - 0.00129765i) q^{2} +(-1.38011 + 1.99943i) q^{3} +(1.49702 - 1.32624i) q^{4} +(-0.444348 - 3.65953i) q^{5} +(0.00327377 + 0.000806912i) q^{6} +(3.94802 + 2.07208i) q^{7} +(-0.00491548 - 0.00257984i) q^{8} +(-1.02922 - 2.71382i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 10 q^{2} - 9 q^{3} - 20 q^{4} - 7 q^{5} - q^{6} - 5 q^{7} + 2 q^{8} - 14 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{5}{13}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.000492135	−	0.00129765i	−0.000347992	−	0.000917579i	0.934842	−	0.355064i	\(-0.115541\pi\)
							−0.935190	+	0.354146i	\(0.884772\pi\)
\(3\)	−1.38011	+	1.99943i	−0.796806	+	1.15437i	0.188247	+	0.982122i	\(0.439719\pi\)
							−0.985053	+	0.172251i	\(0.944896\pi\)
\(5\)	−0.444348	−	3.65953i	−0.198718	−	1.63659i	−0.661376	−	0.750055i	\(-0.730027\pi\)
							0.462657	−	0.886537i	\(-0.346896\pi\)
\(7\)	3.94802	+	2.07208i	1.49221	+	0.783173i	0.996361	−	0.0852348i	\(-0.0271640\pi\)
							0.495851	+	0.868408i	\(0.334856\pi\)
\(11\)	0.542210	−	1.42969i	0.163482	−	0.431068i	−0.828094	−	0.560589i	\(-0.810575\pi\)
							0.991577	+	0.129521i	\(0.0413440\pi\)
\(13\)	−1.24348	+	3.38434i	−0.344878	+	0.938648i
							\(17\)	1.41220	+	0.741181i	0.342509	+	0.179763i	0.627203	−	0.778856i	\(-0.284200\pi\)
−0.284694	+	0.958619i	\(0.591892\pi\)
\(19\)	4.73261			1.08574			0.542868	−	0.839818i	\(-0.317338\pi\)
							0.542868	+	0.839818i	\(0.317338\pi\)
\(23\)	−1.66155			−0.346457			−0.173228	−	0.984882i	\(-0.555420\pi\)
							−0.173228	+	0.984882i	\(0.555420\pi\)
\(29\)	−1.03007	−	2.71607i	−0.191279	−	0.504362i	0.804727	−	0.593645i	\(-0.202312\pi\)
							−0.996006	+	0.0892831i	\(0.971542\pi\)
\(31\)	−7.97254	−	1.96505i	−1.43191	−	0.352934i	−0.554303	−	0.832315i	\(-0.687015\pi\)
							−0.877607	+	0.479381i	\(0.840861\pi\)
\(37\)	−2.29530	−	0.565741i	−0.377345	−	0.0930073i	0.0460812	−	0.998938i	\(-0.485327\pi\)
							−0.423427	+	0.905930i	\(0.639173\pi\)
\(41\)	−3.00968	+	4.36027i	−0.470033	+	0.680960i	−0.984267	−	0.176690i	\(-0.943461\pi\)
							0.514234	+	0.857650i	\(0.328076\pi\)
\(43\)	−4.53725	+	1.11833i	−0.691923	+	0.170544i	−0.569570	−	0.821943i	\(-0.692890\pi\)
							−0.122353	+	0.992487i	\(0.539044\pi\)
\(47\)	9.37370	+	8.30437i	1.36729	+	1.21132i	0.953665	+	0.300872i	\(0.0972776\pi\)
							0.413630	+	0.910445i	\(0.364261\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.2.g.a.27.7	✓	156
169.144	even	13	inner	169.2.g.a.144.7	yes	156

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.g.a.27.7	✓	156	1.1	even	1	trivial
169.2.g.a.144.7	yes	156	169.144	even	13	inner