Embedded newform 169.2.g.a.14.8

• Label: 169.2.g.a.14.8
• 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			14.8
Character	\(\chi\)	\(=\)	169.14
Dual form			169.2.g.a.157.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0291084 - 0.0421708i) q^{2} +(2.28283 - 1.19812i) q^{3} +(0.708279 + 1.86758i) q^{4} +(-2.44658 - 2.16748i) q^{5} +(0.0159238 - 0.131144i) q^{6} +(1.95389 + 0.481590i) q^{7} +(0.198879 + 0.0490192i) q^{8} +(2.07162 - 3.00127i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 10 q^{2} - 9 q^{3} - 20 q^{4} - 7 q^{5} - q^{6} - 5 q^{7} + 2 q^{8} - 14 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0291084	−	0.0421708i	0.0205827	−	0.0298192i	−0.812557	−	0.582881i	\(-0.801925\pi\)
							0.833140	+	0.553062i	\(0.186541\pi\)
\(3\)	2.28283	−	1.19812i	1.31799	−	0.691736i	0.348605	−	0.937270i	\(-0.386656\pi\)
							0.969388	+	0.245534i	\(0.0789632\pi\)
\(5\)	−2.44658	−	2.16748i	−1.09414	−	0.969326i	−0.0944861	−	0.995526i	\(-0.530121\pi\)
							−0.999657	+	0.0262002i	\(0.991659\pi\)
\(7\)	1.95389	+	0.481590i	0.738500	+	0.182024i	0.590584	−	0.806976i	\(-0.298897\pi\)
							0.147916	+	0.989000i	\(0.452744\pi\)
\(11\)	−0.973026	−	1.40967i	−0.293378	−	0.425032i	0.648383	−	0.761315i	\(-0.275446\pi\)
							−0.941761	+	0.336283i	\(0.890830\pi\)
\(13\)	0.791320	+	3.51764i	0.219473	+	0.975619i
							\(17\)	−3.06904	−	0.756451i	−0.744352	−	0.183466i	−0.151140	−	0.988512i	\(-0.548294\pi\)
−0.593212	+	0.805046i	\(0.702141\pi\)
\(19\)	−3.47634			−0.797528			−0.398764	−	0.917054i	\(-0.630561\pi\)
							−0.398764	+	0.917054i	\(0.630561\pi\)
\(23\)	5.13219			1.07014			0.535068	−	0.844809i	\(-0.320286\pi\)
							0.535068	+	0.844809i	\(0.320286\pi\)
\(29\)	−4.23710	+	6.13850i	−0.786810	+	1.13989i	0.200283	+	0.979738i	\(0.435814\pi\)
							−0.987092	+	0.160153i	\(0.948801\pi\)
\(31\)	0.953821	−	7.85543i	0.171311	−	1.41088i	−0.612791	−	0.790245i	\(-0.709953\pi\)
							0.784103	−	0.620631i	\(-0.213124\pi\)
\(37\)	−0.254397	+	2.09515i	−0.0418226	+	0.344440i	0.956745	+	0.290929i	\(0.0939643\pi\)
							−0.998567	+	0.0535111i	\(0.982959\pi\)
\(41\)	−6.52287	+	3.42347i	−1.01870	+	0.534656i	−0.889487	−	0.456961i	\(-0.848938\pi\)
							−0.129215	+	0.991617i	\(0.541246\pi\)
\(43\)	−0.847945	−	6.98346i	−0.129310	−	1.06497i	−0.902469	−	0.430755i	\(-0.858247\pi\)
							0.773159	−	0.634213i	\(-0.218676\pi\)
\(47\)	−1.14272	+	3.01309i	−0.166682	+	0.439505i	−0.992167	−	0.124920i	\(-0.960133\pi\)
							0.825485	+	0.564425i	\(0.190902\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.2.g.a.14.8	✓	156
169.157	even	13	inner	169.2.g.a.157.8	yes	156

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.g.a.14.8	✓	156	1.1	even	1	trivial
169.2.g.a.157.8	yes	156	169.157	even	13	inner