Embedded newform 169.2.i.a.3.7

• Label: 169.2.i.a.3.7
• 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			3.7
Character	\(\chi\)	\(=\)	169.3
Dual form			169.2.i.a.113.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.108893 + 0.0818278i) q^{2} +(0.662843 - 3.24682i) q^{3} +(-0.551273 + 1.90322i) q^{4} +(2.52336 + 1.32436i) q^{5} +(0.193501 + 0.407795i) q^{6} +(3.39567 + 0.551850i) q^{7} +(-0.192308 - 0.507076i) q^{8} +(-7.34253 - 3.12836i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 26 q^{2} - 26 q^{3} - 12 q^{4} - 26 q^{5} - 26 q^{6} - 26 q^{7} - 26 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.108893	+	0.0818278i	−0.0769990	+	0.0578610i	−0.638545	−	0.769585i	\(-0.720463\pi\)
							0.561546	+	0.827446i	\(0.310207\pi\)
\(3\)	0.662843	−	3.24682i	0.382692	−	1.87455i	−0.0941426	−	0.995559i	\(-0.530011\pi\)
							0.476835	−	0.878993i	\(-0.341784\pi\)
\(5\)	2.52336	+	1.32436i	1.12848	+	0.592272i	0.922390	−	0.386261i	\(-0.126234\pi\)
							0.206091	+	0.978533i	\(0.433926\pi\)
\(7\)	3.39567	+	0.551850i	1.28344	+	0.208580i	0.763604	−	0.645685i	\(-0.223428\pi\)
							0.519838	+	0.854265i	\(0.325992\pi\)
\(11\)	−1.83426	+	0.781505i	−0.553050	+	0.235633i	−0.650366	−	0.759621i	\(-0.725385\pi\)
							0.0973159	+	0.995254i	\(0.468974\pi\)
\(13\)	−0.870786	−	3.49882i	−0.241513	−	0.970398i
							\(17\)	−2.11974	−	0.344491i	−0.514112	−	0.0835514i	−0.102180	−	0.994766i	\(-0.532582\pi\)
−0.411932	+	0.911215i	\(0.635146\pi\)
\(19\)	−1.46859	+	2.54367i	−0.336918	+	0.583559i	−0.983851	−	0.178988i	\(-0.942718\pi\)
							0.646934	+	0.762546i	\(0.276051\pi\)
\(23\)	2.42675	+	4.20325i	0.506012	+	0.876438i	0.999976	+	0.00695598i	\(0.00221418\pi\)
							−0.493964	+	0.869482i	\(0.664452\pi\)
\(29\)	−2.63839	+	1.98262i	−0.489937	+	0.368164i	−0.816812	−	0.576904i	\(-0.804261\pi\)
							0.326875	+	0.945068i	\(0.394004\pi\)
\(31\)	0.275651	−	0.399349i	0.0495084	−	0.0717252i	−0.797448	−	0.603388i	\(-0.793817\pi\)
							0.846956	+	0.531663i	\(0.178433\pi\)
\(37\)	7.48602	+	0.604334i	1.23069	+	0.0993519i	0.678649	−	0.734462i	\(-0.262566\pi\)
							0.552045	+	0.833814i	\(0.313848\pi\)
\(41\)	−0.185978	+	0.910980i	−0.0290449	+	0.142271i	−0.991727	−	0.128362i	\(-0.959028\pi\)
							0.962682	+	0.270633i	\(0.0872331\pi\)
\(43\)	2.62549	−	0.211951i	0.400383	−	0.0323222i	0.121364	−	0.992608i	\(-0.461273\pi\)
							0.279018	+	0.960286i	\(0.409991\pi\)
\(47\)	−10.0113	−	2.46757i	−1.46030	−	0.359932i	−0.572403	−	0.819973i	\(-0.693989\pi\)
							−0.887898	+	0.460041i	\(0.847835\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.2.i.a.3.7	✓	336
169.113	even	39	inner	169.2.i.a.113.7	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.i.a.3.7	✓	336	1.1	even	1	trivial
169.2.i.a.113.7	yes	336	169.113	even	39	inner