Embedded newform 169.2.i.a.113.3

• Label: 169.2.i.a.113.3
• Level: $169$
• Weight: $2$
• Character: 169.113
• Analytic conductor: $1.349$
• Analytic rank: $0$
• Dimension: $336$
• 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			113.3
Character	\(\chi\)	\(=\)	169.113
Dual form			169.2.i.a.3.3

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

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.i.a.3.3	✓	336	169.3	even	39	inner
169.2.i.a.113.3	yes	336	1.1	even	1	trivial