Embedded newform 169.2.i.a.3.10

• Label: 169.2.i.a.3.10
• 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			3.10
Character	\(\chi\)	\(=\)	169.3
Dual form			169.2.i.a.113.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.470185 - 0.353321i) q^{2} +(0.0357404 - 0.175068i) q^{3} +(-0.460197 + 1.58878i) q^{4} +(3.13432 + 1.64502i) q^{5} +(-0.0450506 - 0.0949422i) q^{6} +(-4.24484 - 0.689853i) q^{7} +(0.762089 + 2.00946i) q^{8} +(2.73057 + 1.16339i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 26 q^{2} - 26 q^{3} - 12 q^{4} - 26 q^{5} - 26 q^{6} - 26 q^{7} - 26 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.470185	−	0.353321i	0.332471	−	0.249836i	−0.421376	−	0.906886i	\(-0.638453\pi\)
							0.753847	+	0.657050i	\(0.228196\pi\)
\(3\)	0.0357404	−	0.175068i	0.0206347	−	0.101076i	−0.968169	−	0.250297i	\(-0.919472\pi\)
							0.988804	+	0.149222i	\(0.0476768\pi\)
\(5\)	3.13432	+	1.64502i	1.40171	+	0.735673i	0.984868	−	0.173306i	\(-0.0554451\pi\)
							0.416840	+	0.908980i	\(0.363137\pi\)
\(7\)	−4.24484	−	0.689853i	−1.60440	−	0.260740i	−0.708735	−	0.705474i	\(-0.750734\pi\)
							−0.895663	+	0.444734i	\(0.853298\pi\)
\(11\)	1.45878	−	0.621528i	0.439838	−	0.187398i	−0.160683	−	0.987006i	\(-0.551370\pi\)
							0.600522	+	0.799608i	\(0.294960\pi\)
\(13\)	−2.67872	−	2.41339i	−0.742942	−	0.669355i
							\(17\)	3.12343	+	0.507607i	0.757543	+	0.123113i	0.526902	−	0.849926i	\(-0.323353\pi\)
0.230641	+	0.973039i	\(0.425918\pi\)
\(19\)	3.03277	−	5.25291i	0.695765	−	1.20510i	−0.274157	−	0.961685i	\(-0.588399\pi\)
							0.969922	−	0.243416i	\(-0.0782679\pi\)
\(23\)	−2.91817	−	5.05442i	−0.608481	−	1.05392i	−0.991491	−	0.130176i	\(-0.958446\pi\)
							0.383010	−	0.923744i	\(-0.374887\pi\)
\(29\)	−0.651024	+	0.489213i	−0.120892	+	0.0908446i	−0.659469	−	0.751732i	\(-0.729219\pi\)
							0.538577	+	0.842577i	\(0.318962\pi\)
\(31\)	−3.78703	+	5.48646i	−0.680170	+	0.985397i	0.319120	+	0.947714i	\(0.396613\pi\)
							−0.999290	+	0.0376821i	\(0.988003\pi\)
\(37\)	−9.59103	−	0.774268i	−1.57676	−	0.127289i	−0.739141	−	0.673551i	\(-0.764768\pi\)
							−0.837614	+	0.546262i	\(0.816050\pi\)
\(41\)	0.758002	−	3.71294i	0.118380	−	0.579864i	−0.876721	−	0.480999i	\(-0.840274\pi\)
							0.995101	−	0.0988643i	\(-0.0315210\pi\)
\(43\)	2.73100	−	0.220469i	0.416474	−	0.0336212i	0.129546	−	0.991573i	\(-0.458648\pi\)
							0.286928	+	0.957952i	\(0.407366\pi\)
\(47\)	1.30363	+	0.321315i	0.190154	+	0.0468686i	0.333242	−	0.942841i	\(-0.391857\pi\)
							−0.143089	+	0.989710i	\(0.545703\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.2.i.a.3.10	✓	336
169.113	even	39	inner	169.2.i.a.113.10	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.i.a.3.10	✓	336	1.1	even	1	trivial
169.2.i.a.113.10	yes	336	169.113	even	39	inner