Embedded newform 169.2.i.a.113.8

• Label: 169.2.i.a.113.8
• Level: $169$
• Weight: $2$
• Character: 169.113
• 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			113.8
Character	\(\chi\)	\(=\)	169.113
Dual form			169.2.i.a.3.8

$q$-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 \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{8}{39}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.i.a.3.8	✓	336	169.3	even	39	inner
169.2.i.a.113.8	yes	336	1.1	even	1	trivial