Embedded newform 169.2.k.a.10.1

• Label: 169.2.k.a.10.1
• Level: $169$
• Weight: $2$
• Character: 169.10
• 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			10.1
Character	\(\chi\)	\(=\)	169.10
Dual form			169.2.k.a.17.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.60774 - 0.532374i) q^{2} +(-2.66189 + 0.432599i) q^{3} +(4.67692 + 1.99265i) q^{4} +(-0.498005 + 2.02048i) q^{5} +(7.17182 + 0.289014i) q^{6} +(1.97372 - 0.936542i) q^{7} +(-6.75456 - 4.66233i) q^{8} +(4.05291 - 1.35306i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 360 q - 23 q^{2} - 24 q^{3} - 41 q^{4} - 26 q^{5} - 32 q^{6} - 26 q^{7} - 26 q^{8} - 11 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{5}{78}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.60774	−	0.532374i	−1.84395	−	0.376445i	−0.854280	−	0.519813i	\(-0.826002\pi\)
							−0.989669	+	0.143368i	\(0.954207\pi\)
\(3\)	−2.66189	+	0.432599i	−1.53684	+	0.249761i	−0.869102	−	0.494632i	\(-0.835303\pi\)
							−0.667741	+	0.744394i	\(0.732739\pi\)
\(5\)	−0.498005	+	2.02048i	−0.222714	+	0.903588i	0.748067	+	0.663623i	\(0.230982\pi\)
							−0.970782	+	0.239965i	\(0.922864\pi\)
\(7\)	1.97372	−	0.936542i	0.745996	−	0.353979i	−0.0174530	−	0.999848i	\(-0.505556\pi\)
							0.763449	+	0.645868i	\(0.223504\pi\)
\(11\)	−1.04466	+	3.12913i	−0.314976	+	0.943467i	0.665427	+	0.746463i	\(0.268250\pi\)
							−0.980403	+	0.197004i	\(0.936879\pi\)
\(13\)	−3.56877	+	0.513691i	−0.989799	+	0.142472i
							\(17\)	−0.00965072	−	0.0203385i	−0.00234064	−	0.00493280i	0.902277	−	0.431157i	\(-0.141895\pi\)
−0.904617	+	0.426225i	\(0.859843\pi\)
\(19\)	−5.30097	−	3.06052i	−1.21613	−	0.702131i	−0.252040	−	0.967717i	\(-0.581101\pi\)
							−0.964087	+	0.265586i	\(0.914435\pi\)
\(23\)	−3.51581	−	6.08956i	−0.733096	−	1.26976i	−0.955554	−	0.294818i	\(-0.904741\pi\)
							0.222457	−	0.974942i	\(-0.428592\pi\)
\(29\)	0.687458	−	3.36739i	0.127658	−	0.625309i	−0.864738	−	0.502223i	\(-0.832516\pi\)
							0.992396	−	0.123086i	\(-0.0392791\pi\)
\(31\)	−3.76670	−	7.17686i	−0.676520	−	1.28900i	−0.945427	−	0.325835i	\(-0.894355\pi\)
							0.268906	−	0.963166i	\(-0.413338\pi\)
\(37\)	−6.26268	+	9.90364i	−1.02958	+	1.62815i	−0.287568	+	0.957760i	\(0.592847\pi\)
							−0.742011	+	0.670388i	\(0.766128\pi\)
\(41\)	−0.137662	−	0.847071i	−0.0214993	−	0.132290i	0.974082	−	0.226196i	\(-0.0726289\pi\)
							−0.995581	+	0.0939055i	\(0.970065\pi\)
\(43\)	2.04832	−	1.29528i	0.312366	−	0.197528i	−0.369078	−	0.929398i	\(-0.620327\pi\)
							0.681444	+	0.731870i	\(0.261352\pi\)
\(47\)	−6.60753	−	0.802300i	−0.963808	−	0.117027i	−0.376522	−	0.926407i	\(-0.622880\pi\)
							−0.587285	+	0.809380i	\(0.699803\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.2.k.a.10.1	✓	360
169.17	even	78	inner	169.2.k.a.17.1	yes	360

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.k.a.10.1	✓	360	1.1	even	1	trivial
169.2.k.a.17.1	yes	360	169.17	even	78	inner