Embedded newform 169.2.k.a.4.2

• Label: 169.2.k.a.4.2
• Level: $169$
• Weight: $2$
• Character: 169.4
• Analytic conductor: $1.349$
• Analytic rank: $0$
• Dimension: $360$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	169.k (of order \(78\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.34947179416\)
Analytic rank:	\(0\)
Dimension:	\(360\)
Relative dimension:	\(15\) over \(\Q(\zeta_{78})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{78}]$

Embedding invariants

Embedding label			4.2
Character	\(\chi\)	\(=\)	169.4
Dual form			169.2.k.a.127.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.41817 - 0.0974488i) q^{2} +(-0.437408 + 1.51011i) q^{3} +(3.84452 + 0.310362i) q^{4} +(-2.71784 - 1.03074i) q^{5} +(1.20488 - 3.60907i) q^{6} +(-0.958962 - 2.25077i) q^{7} +(-4.46148 - 0.541722i) q^{8} +(0.446473 + 0.282332i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 360 q - 23 q^{2} - 24 q^{3} - 41 q^{4} - 26 q^{5} - 32 q^{6} - 26 q^{7} - 26 q^{8} - 11 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{78}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.41817	−	0.0974488i	−1.70990	−	0.0689067i	−0.834111	−	0.551597i	\(-0.814019\pi\)
							−0.875791	+	0.482690i	\(0.839660\pi\)
\(3\)	−0.437408	+	1.51011i	−0.252538	+	0.871861i	0.729628	+	0.683844i	\(0.239693\pi\)
							−0.982166	+	0.188017i	\(0.939794\pi\)
\(5\)	−2.71784	−	1.03074i	−1.21545	−	0.460961i	−0.338240	−	0.941060i	\(-0.609832\pi\)
							−0.877214	+	0.480099i	\(0.840601\pi\)
\(7\)	−0.958962	−	2.25077i	−0.362453	−	0.850709i	−0.996969	−	0.0777993i	\(-0.975211\pi\)
							0.634516	−	0.772910i	\(-0.281200\pi\)
\(11\)	0.842314	+	1.33201i	0.253967	+	0.401617i	0.947972	−	0.318353i	\(-0.103130\pi\)
							−0.694005	+	0.719970i	\(0.744155\pi\)
\(13\)	2.05596	−	2.96193i	0.570222	−	0.821491i
							\(17\)	1.77732	−	0.757244i	0.431062	−	0.183659i	−0.165527	−	0.986205i	\(-0.552932\pi\)
0.596589	+	0.802547i	\(0.296522\pi\)
\(19\)	6.36630	−	3.67559i	1.46053	−	0.843237i	0.461494	−	0.887143i	\(-0.347314\pi\)
							0.999036	+	0.0439059i	\(0.0139802\pi\)
\(23\)	2.38841	−	4.13685i	0.498018	−	0.862593i	−0.501979	−	0.864880i	\(-0.667395\pi\)
							0.999997	+	0.00228656i	\(0.000727835\pi\)
\(29\)	0.291416	−	7.23142i	0.0541147	−	1.34284i	−0.712924	−	0.701241i	\(-0.752630\pi\)
							0.767039	−	0.641601i	\(-0.221729\pi\)
\(31\)	3.48709	+	3.93612i	0.626301	+	0.706947i	0.972942	−	0.231051i	\(-0.0742163\pi\)
							−0.346641	+	0.937998i	\(0.612678\pi\)
\(37\)	−9.28405	+	1.89535i	−1.52629	+	0.311594i	−0.888593	−	0.458696i	\(-0.848317\pi\)
							−0.637694	+	0.770290i	\(0.720112\pi\)
\(41\)	−1.54584	−	0.447759i	−0.241420	−	0.0699283i	0.155302	−	0.987867i	\(-0.450365\pi\)
							−0.396722	+	0.917939i	\(0.629852\pi\)
\(43\)	0.644074	−	3.15488i	0.0982204	−	0.481115i	−0.900691	−	0.434461i	\(-0.856939\pi\)
							0.998911	−	0.0466544i	\(-0.0148560\pi\)
\(47\)	−0.644340	+	0.444756i	−0.0939867	+	0.0648743i	−0.614131	−	0.789204i	\(-0.710493\pi\)
							0.520145	+	0.854078i	\(0.325878\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.2.k.a.4.2	✓	360
169.127	even	78	inner	169.2.k.a.127.2	yes	360

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.k.a.4.2	✓	360	1.1	even	1	trivial
169.2.k.a.127.2	yes	360	169.127	even	78	inner