Embedded newform 169.2.i.a.3.3

• Label: 169.2.i.a.3.3
• 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			3.3
Character	\(\chi\)	\(=\)	169.3
Dual form			169.2.i.a.113.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.59194 + 1.19627i) q^{2} +(-0.271241 + 1.32863i) q^{3} +(0.546788 - 1.88773i) q^{4} +(-0.978405 - 0.513506i) q^{5} +(-1.15759 - 2.43957i) q^{6} +(-3.69364 - 0.600275i) q^{7} +(-0.0244856 - 0.0645632i) q^{8} +(1.06827 + 0.455145i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 26 q^{2} - 26 q^{3} - 12 q^{4} - 26 q^{5} - 26 q^{6} - 26 q^{7} - 26 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.59194	+	1.19627i	−1.12567	+	0.845887i	−0.989462	−	0.144793i	\(-0.953748\pi\)
							−0.136210	+	0.990680i	\(0.543492\pi\)
\(3\)	−0.271241	+	1.32863i	−0.156601	+	0.767082i	0.823301	+	0.567604i	\(0.192130\pi\)
							−0.979902	+	0.199478i	\(0.936075\pi\)
\(5\)	−0.978405	−	0.513506i	−0.437556	−	0.229647i	0.231537	−	0.972826i	\(-0.425625\pi\)
							−0.669092	+	0.743179i	\(0.733317\pi\)
\(7\)	−3.69364	−	0.600275i	−1.39606	−	0.226883i	−0.584618	−	0.811309i	\(-0.698755\pi\)
							−0.811447	+	0.584426i	\(0.801320\pi\)
\(11\)	−3.85597	+	1.64287i	−1.16262	+	0.495345i	−0.885112	−	0.465378i	\(-0.845918\pi\)
							−0.277507	+	0.960724i	\(0.589508\pi\)
\(13\)	1.14755	−	3.41806i	0.318274	−	0.947999i
							\(17\)	−3.76067	−	0.611169i	−0.912098	−	0.148230i	−0.313786	−	0.949494i	\(-0.601597\pi\)
−0.598312	+	0.801263i	\(0.704162\pi\)
\(19\)	1.11991	−	1.93974i	0.256925	−	0.445007i	−0.708491	−	0.705719i	\(-0.750624\pi\)
							0.965417	+	0.260712i	\(0.0839573\pi\)
\(23\)	−0.105568	−	0.182848i	−0.0220124	−	0.0381265i	0.854809	−	0.518942i	\(-0.173674\pi\)
							−0.876822	+	0.480816i	\(0.840341\pi\)
\(29\)	−3.73062	+	2.80338i	−0.692758	+	0.520574i	−0.887805	−	0.460220i	\(-0.847771\pi\)
							0.195047	+	0.980794i	\(0.437514\pi\)
\(31\)	−2.91561	+	4.22400i	−0.523660	+	0.758652i	−0.992182	−	0.124800i	\(-0.960171\pi\)
							0.468522	+	0.883452i	\(0.344787\pi\)
\(37\)	7.40227	+	0.597573i	1.21693	+	0.0982404i	0.672197	−	0.740372i	\(-0.265351\pi\)
							0.544728	+	0.838613i	\(0.316633\pi\)
\(41\)	−2.04116	+	9.99825i	−0.318775	+	1.56146i	0.432598	+	0.901587i	\(0.357597\pi\)
							−0.751373	+	0.659877i	\(0.770608\pi\)
\(43\)	−11.3849	+	0.919084i	−1.73618	+	0.140159i	−0.907635	−	0.419761i	\(-0.862114\pi\)
							−0.828547	+	0.559920i	\(0.810832\pi\)
\(47\)	−6.89125	−	1.69854i	−1.00519	−	0.247757i	−0.297837	−	0.954617i	\(-0.596265\pi\)
							−0.707354	+	0.706859i	\(0.750111\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.2.i.a.3.3	✓	336
169.113	even	39	inner	169.2.i.a.113.3	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.i.a.3.3	✓	336	1.1	even	1	trivial
169.2.i.a.113.3	yes	336	169.113	even	39	inner