Embedded newform 169.2.i.a.3.8

• Label: 169.2.i.a.3.8
• Level: $169$
• Weight: $2$
• Character: 169.3
• Analytic conductor: $1.349$
• Analytic rank: $0$
• Dimension: $336$
• CM: no
• 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			3.8
Character	\(\chi\)	\(=\)	169.3
Dual form			169.2.i.a.113.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0796839 + 0.0598785i) q^{2} +(-0.377738 + 1.85028i) q^{3} +(-0.553671 + 1.91149i) q^{4} +(-2.61855 - 1.37432i) q^{5} +(-0.0806927 - 0.170056i) q^{6} +(-1.55454 - 0.252638i) q^{7} +(-0.141029 - 0.371862i) q^{8} +(-0.520928 - 0.221947i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 26 q^{2} - 26 q^{3} - 12 q^{4} - 26 q^{5} - 26 q^{6} - 26 q^{7} - 26 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{31}{39}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0796839	+	0.0598785i	−0.0563450	+	0.0423405i	−0.628542	−	0.777776i	\(-0.716348\pi\)
							0.572197	+	0.820116i	\(0.306091\pi\)
\(3\)	−0.377738	+	1.85028i	−0.218087	+	1.06826i	0.712354	+	0.701821i	\(0.247629\pi\)
							−0.930441	+	0.366442i	\(0.880576\pi\)
\(5\)	−2.61855	−	1.37432i	−1.17105	−	0.614616i	−0.236916	−	0.971530i	\(-0.576137\pi\)
							−0.934137	+	0.356914i	\(0.883829\pi\)
\(7\)	−1.55454	−	0.252638i	−0.587562	−	0.0954881i	−0.140640	−	0.990061i	\(-0.544916\pi\)
							−0.446922	+	0.894573i	\(0.647480\pi\)
\(11\)	−0.353037	+	0.150415i	−0.106445	+	0.0453519i	−0.444533	−	0.895763i	\(-0.646630\pi\)
							0.338088	+	0.941115i	\(0.390220\pi\)
\(13\)	−3.28093	+	1.49516i	−0.909966	+	0.414683i
							\(17\)	7.06592	+	1.14832i	1.71374	+	0.278509i	0.936492	−	0.350689i	\(-0.114052\pi\)
0.777245	+	0.629198i	\(0.216617\pi\)
\(19\)	−3.51136	+	6.08185i	−0.805561	+	1.39527i	0.110351	+	0.993893i	\(0.464803\pi\)
							−0.915912	+	0.401380i	\(0.868531\pi\)
\(23\)	3.65819	+	6.33617i	0.762785	+	1.32118i	0.941410	+	0.337265i	\(0.109502\pi\)
							−0.178625	+	0.983917i	\(0.557165\pi\)
\(29\)	−1.86861	+	1.40417i	−0.346992	+	0.260747i	−0.759921	−	0.650016i	\(-0.774762\pi\)
							0.412929	+	0.910763i	\(0.364506\pi\)
\(31\)	0.855210	−	1.23899i	0.153600	−	0.222528i	−0.738628	−	0.674113i	\(-0.764526\pi\)
							0.892228	+	0.451585i	\(0.149141\pi\)
\(37\)	1.79074	+	0.144564i	0.294396	+	0.0237661i	0.226782	−	0.973945i	\(-0.427179\pi\)
							0.0676137	+	0.997712i	\(0.478461\pi\)
\(41\)	1.60439	−	7.85884i	0.250564	−	1.22734i	−0.638522	−	0.769604i	\(-0.720454\pi\)
							0.889086	−	0.457740i	\(-0.151341\pi\)
\(43\)	10.5096	−	0.848424i	1.60270	−	0.129383i	0.753498	−	0.657450i	\(-0.228365\pi\)
							0.849203	+	0.528067i	\(0.177083\pi\)
\(47\)	−3.87145	−	0.954226i	−0.564708	−	0.139188i	−0.0533811	−	0.998574i	\(-0.517000\pi\)
							−0.511327	+	0.859386i	\(0.670846\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.2.i.a.3.8	✓	336
169.113	even	39	inner	169.2.i.a.113.8	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.i.a.3.8	✓	336	1.1	even	1	trivial
169.2.i.a.113.8	yes	336	169.113	even	39	inner