Embedded newform 169.3.j.a.5.14

• Label: 169.3.j.a.5.14
• Level: $169$
• Weight: $3$
• Character: 169.5
• Analytic conductor: $4.605$
• Analytic rank: $0$
• Dimension: $720$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	169.j (of order \(52\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.60491646769\)
Analytic rank:	\(0\)
Dimension:	\(720\)
Relative dimension:	\(30\) over \(\Q(\zeta_{52})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{52}]$

Embedding invariants

Embedding label			5.14
Character	\(\chi\)	\(=\)	169.5
Dual form			169.3.j.a.34.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.701607 - 0.128574i) q^{2} +(1.48553 + 0.779668i) q^{3} +(-3.26434 - 1.23800i) q^{4} +(-0.293027 + 4.84432i) q^{5} +(-0.942016 - 0.738022i) q^{6} +(3.36050 + 2.03150i) q^{7} +(4.57280 + 2.76435i) q^{8} +(-3.51366 - 5.09041i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 720 q - 22 q^{2} - 22 q^{3} - 26 q^{4} - 34 q^{5} - 10 q^{6} - 14 q^{7} - 62 q^{8} - 202 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{3}{52}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.701607	−	0.128574i	−0.350804	−	0.0642872i	0.00196407	−	0.999998i	\(-0.499375\pi\)
							−0.352768	+	0.935711i	\(0.614759\pi\)
\(3\)	1.48553	+	0.779668i	0.495178	+	0.259889i	0.693775	−	0.720192i	\(-0.255946\pi\)
							−0.198597	+	0.980081i	\(0.563639\pi\)
\(5\)	−0.293027	+	4.84432i	−0.0586055	+	0.968864i	0.843440	+	0.537223i	\(0.180527\pi\)
							−0.902045	+	0.431641i	\(0.857935\pi\)
\(7\)	3.36050	+	2.03150i	0.480072	+	0.290214i	0.736607	−	0.676321i	\(-0.236427\pi\)
							−0.256535	+	0.966535i	\(0.582581\pi\)
\(11\)	−17.5755	+	3.22083i	−1.59777	+	0.292803i	−0.904046	−	0.427434i	\(-0.859418\pi\)
							−0.693728	+	0.720237i	\(0.744033\pi\)
\(13\)	−1.53130	+	12.9095i	−0.117792	+	0.993038i
							\(17\)	1.68631	+	6.84161i	0.0991945	+	0.402448i	0.999459	−	0.0328959i	\(-0.0104730\pi\)
−0.900264	+	0.435344i	\(0.856627\pi\)
\(19\)	−25.4605	+	25.4605i	−1.34003	+	1.34003i	−0.444000	+	0.896027i	\(0.646441\pi\)
							−0.896027	+	0.444000i	\(0.853559\pi\)
\(23\)		−	7.44723i		−	0.323793i	−0.986808	−	0.161896i	\(-0.948239\pi\)
							0.986808	−	0.161896i	\(-0.0517610\pi\)
\(29\)	3.36239	+	4.87126i	0.115944	+	0.167974i	0.876659	−	0.481112i	\(-0.159767\pi\)
							−0.760715	+	0.649086i	\(0.775151\pi\)
\(31\)	33.5955	+	26.3204i	1.08373	+	0.849045i	0.989214	−	0.146479i	\(-0.0467941\pi\)
							0.0945118	+	0.995524i	\(0.469871\pi\)
\(37\)	6.19797	−	7.91113i	0.167513	−	0.213814i	−0.696134	−	0.717912i	\(-0.745098\pi\)
							0.863647	+	0.504098i	\(0.168175\pi\)
\(41\)	−65.9309	+	20.5449i	−1.60807	+	0.501095i	−0.964453	−	0.264253i	\(-0.914875\pi\)
							−0.643617	+	0.765348i	\(0.722567\pi\)
\(43\)	4.89400	+	0.594239i	0.113814	+	0.0138195i	0.177245	−	0.984167i	\(-0.443281\pi\)
							−0.0634315	+	0.997986i	\(0.520204\pi\)
\(47\)	26.8151	−	59.5808i	0.570535	−	1.26768i	−0.370332	−	0.928899i	\(-0.620756\pi\)
							0.940867	−	0.338777i	\(-0.110013\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.3.j.a.5.14	✓	720
169.34	odd	52	inner	169.3.j.a.34.14	yes	720

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.3.j.a.5.14	✓	720	1.1	even	1	trivial
169.3.j.a.34.14	yes	720	169.34	odd	52	inner