Embedded newform 169.2.i.a.3.2

• Label: 169.2.i.a.3.2
• 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			3.2
Character	\(\chi\)	\(=\)	169.3
Dual form			169.2.i.a.113.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.03393 + 1.52840i) q^{2} +(0.603355 - 2.95543i) q^{3} +(1.24444 - 4.29630i) q^{4} +(-1.41359 - 0.741908i) q^{5} +(3.28990 + 6.93331i) q^{6} +(-0.790704 - 0.128502i) q^{7} +(2.23100 + 5.88267i) q^{8} +(-5.61059 - 2.39045i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 26 q^{2} - 26 q^{3} - 12 q^{4} - 26 q^{5} - 26 q^{6} - 26 q^{7} - 26 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.03393	+	1.52840i	−1.43821	+	1.08074i	−0.456635	+	0.889654i	\(0.650946\pi\)
							−0.981573	+	0.191089i	\(0.938798\pi\)
\(3\)	0.603355	−	2.95543i	0.348347	−	1.70632i	−0.307604	−	0.951515i	\(-0.599527\pi\)
							0.655951	−	0.754803i	\(-0.272268\pi\)
\(5\)	−1.41359	−	0.741908i	−0.632175	−	0.331791i	0.118016	−	0.993012i	\(-0.462346\pi\)
							−0.750192	+	0.661220i	\(0.770039\pi\)
\(7\)	−0.790704	−	0.128502i	−0.298858	−	0.0485691i	0.00913350	−	0.999958i	\(-0.497093\pi\)
							−0.307991	+	0.951389i	\(0.599657\pi\)
\(11\)	−4.48369	+	1.91032i	−1.35188	+	0.575984i	−0.941834	−	0.336079i	\(-0.890899\pi\)
							−0.410050	+	0.912063i	\(0.634489\pi\)
\(13\)	−1.97473	+	3.01669i	−0.547692	+	0.836680i
							\(17\)	1.50267	+	0.244207i	0.364450	+	0.0592289i	0.339879	−	0.940469i	\(-0.389614\pi\)
0.0245712	+	0.999698i	\(0.492178\pi\)
\(19\)	3.81115	−	6.60110i	0.874338	−	1.51440i	0.0168708	−	0.999858i	\(-0.494630\pi\)
							0.857467	−	0.514539i	\(-0.172037\pi\)
\(23\)	−0.299267	−	0.518346i	−0.0624015	−	0.108083i	0.833137	−	0.553067i	\(-0.186543\pi\)
							−0.895538	+	0.444984i	\(0.853209\pi\)
\(29\)	−0.922905	+	0.693518i	−0.171379	+	0.128783i	−0.682971	−	0.730446i	\(-0.739312\pi\)
							0.511592	+	0.859229i	\(0.329056\pi\)
\(31\)	2.99621	−	4.34076i	0.538135	−	0.779623i	−0.455742	−	0.890112i	\(-0.650626\pi\)
							0.993877	+	0.110488i	\(0.0352415\pi\)
\(37\)	−8.12264	−	0.655727i	−1.33535	−	0.107801i	−0.607930	−	0.793990i	\(-0.708000\pi\)
							−0.727423	+	0.686190i	\(0.759282\pi\)
\(41\)	2.29000	−	11.2172i	0.357639	−	1.75183i	−0.260229	−	0.965547i	\(-0.583798\pi\)
							0.617867	−	0.786282i	\(-0.287997\pi\)
\(43\)	4.09804	−	0.330828i	0.624945	−	0.0504508i	0.236062	−	0.971738i	\(-0.424143\pi\)
							0.388883	+	0.921287i	\(0.372861\pi\)
\(47\)	−5.45078	−	1.34350i	−0.795079	−	0.195969i	−0.179202	−	0.983812i	\(-0.557352\pi\)
							−0.615876	+	0.787843i	\(0.711198\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.2.i.a.3.2	✓	336
169.113	even	39	inner	169.2.i.a.113.2	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.i.a.3.2	✓	336	1.1	even	1	trivial
169.2.i.a.113.2	yes	336	169.113	even	39	inner