Embedded newform 169.2.g.a.14.12

• Label: 169.2.g.a.14.12
• 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			14.12
Character	\(\chi\)	\(=\)	169.14
Dual form			169.2.g.a.157.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.13519 - 1.64460i) q^{2} +(0.811870 - 0.426102i) q^{3} +(-0.706856 - 1.86382i) q^{4} +(-2.38631 - 2.11409i) q^{5} +(0.220855 - 1.81891i) q^{6} +(-0.234049 - 0.0576880i) q^{7} +(0.0128813 + 0.00317496i) q^{8} +(-1.22662 + 1.77707i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 10 q^{2} - 9 q^{3} - 20 q^{4} - 7 q^{5} - q^{6} - 5 q^{7} + 2 q^{8} - 14 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.13519	−	1.64460i	0.802698	−	1.16291i	−0.181071	−	0.983470i	\(-0.557956\pi\)
							0.983769	−	0.179439i	\(-0.0574282\pi\)
\(3\)	0.811870	−	0.426102i	0.468733	−	0.246010i	−0.213782	−	0.976881i	\(-0.568578\pi\)
							0.682515	+	0.730871i	\(0.260886\pi\)
\(5\)	−2.38631	−	2.11409i	−1.06719	−	0.945448i	−0.0685533	−	0.997647i	\(-0.521838\pi\)
							−0.998637	+	0.0521997i	\(0.983377\pi\)
\(7\)	−0.234049	−	0.0576880i	−0.0884624	−	0.0218040i	0.194836	−	0.980836i	\(-0.437583\pi\)
							−0.283298	+	0.959032i	\(0.591429\pi\)
\(11\)	2.59601	+	3.76097i	0.782726	+	1.13397i	0.987876	+	0.155244i	\(0.0496163\pi\)
							−0.205150	+	0.978730i	\(0.565768\pi\)
\(13\)	1.81916	−	3.11298i	0.504545	−	0.863385i
							\(17\)	6.59396	+	1.62527i	1.59927	+	0.394185i	0.935380	−	0.353644i	\(-0.115058\pi\)
0.663891	+	0.747829i	\(0.268904\pi\)
\(19\)	−2.19059			−0.502556			−0.251278	−	0.967915i	\(-0.580851\pi\)
							−0.251278	+	0.967915i	\(0.580851\pi\)
\(23\)	−7.88271			−1.64366			−0.821829	−	0.569734i	\(-0.807046\pi\)
							−0.821829	+	0.569734i	\(0.807046\pi\)
\(29\)	−4.83050	+	6.99819i	−0.897002	+	1.29953i	0.0565802	+	0.998398i	\(0.481980\pi\)
							−0.953582	+	0.301133i	\(0.902635\pi\)
\(31\)	0.331059	−	2.72652i	0.0594600	−	0.489697i	−0.932310	−	0.361661i	\(-0.882210\pi\)
							0.991770	−	0.128036i	\(-0.0408673\pi\)
\(37\)	0.786133	−	6.47439i	0.129239	−	1.06438i	−0.773380	−	0.633943i	\(-0.781435\pi\)
							0.902619	−	0.430440i	\(-0.141642\pi\)
\(41\)	−3.18715	+	1.67274i	−0.497748	+	0.261239i	−0.694863	−	0.719142i	\(-0.744535\pi\)
							0.197115	+	0.980380i	\(0.436843\pi\)
\(43\)	1.00494	+	8.27644i	0.153252	+	1.26215i	0.842928	+	0.538027i	\(0.180830\pi\)
							−0.689676	+	0.724119i	\(0.742247\pi\)
\(47\)	−0.602240	+	1.58798i	−0.0878457	+	0.231630i	−0.971739	−	0.236059i	\(-0.924144\pi\)
							0.883893	+	0.467689i	\(0.154913\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.2.g.a.14.12	✓	156
169.157	even	13	inner	169.2.g.a.157.12	yes	156

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.g.a.14.12	✓	156	1.1	even	1	trivial
169.2.g.a.157.12	yes	156	169.157	even	13	inner