Embedded newform 169.2.k.a.4.3

• Label: 169.2.k.a.4.3
• 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			4.3
Character	\(\chi\)	\(=\)	169.4
Dual form			169.2.k.a.127.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.00062 - 0.0806224i) q^{2} +(-0.731230 + 2.52450i) q^{3} +(2.00248 + 0.161657i) q^{4} +(3.39858 + 1.28891i) q^{5} +(1.66645 - 4.99162i) q^{6} +(0.132610 + 0.311247i) q^{7} +(-0.0178867 - 0.00217183i) q^{8} +(-3.30283 - 2.08858i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 360 q - 23 q^{2} - 24 q^{3} - 41 q^{4} - 26 q^{5} - 32 q^{6} - 26 q^{7} - 26 q^{8} - 11 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.00062	−	0.0806224i	−1.41466	−	0.0570087i	−0.678887	−	0.734242i	\(-0.737538\pi\)
							−0.735768	+	0.677234i	\(0.763179\pi\)
\(3\)	−0.731230	+	2.52450i	−0.422176	+	1.45752i	0.414614	+	0.909997i	\(0.363917\pi\)
							−0.836790	+	0.547524i	\(0.815571\pi\)
\(5\)	3.39858	+	1.28891i	1.51989	+	0.576419i	0.966537	−	0.256526i	\(-0.0825780\pi\)
							0.553355	+	0.832945i	\(0.313347\pi\)
\(7\)	0.132610	+	0.311247i	0.0501218	+	0.117640i	0.943158	−	0.332345i	\(-0.107840\pi\)
							−0.893036	+	0.449985i	\(0.851429\pi\)
\(11\)	0.00238191	+	0.00376669i	0.000718173	+	0.00113570i	0.845549	−	0.533898i	\(-0.179273\pi\)
							−0.844831	+	0.535034i	\(0.820299\pi\)
\(13\)	2.37696	+	2.71110i	0.659251	+	0.751923i
							\(17\)	0.00809655	−	0.00344962i	0.00196370	−	0.000836655i	−0.390985	−	0.920397i	\(-0.627865\pi\)
0.392948	+	0.919561i	\(0.371455\pi\)
\(19\)	−6.79345	+	3.92220i	−1.55852	+	0.899814i	−0.561125	+	0.827731i	\(0.689631\pi\)
							−0.997399	+	0.0720830i	\(0.977035\pi\)
\(23\)	3.78890	−	6.56257i	0.790040	−	1.36839i	−0.135901	−	0.990722i	\(-0.543393\pi\)
							0.925941	−	0.377667i	\(-0.123274\pi\)
\(29\)	0.0827532	−	2.05350i	0.0153669	−	0.381325i	−0.973146	−	0.230189i	\(-0.926066\pi\)
							0.988513	−	0.151136i	\(-0.0482933\pi\)
\(31\)	−4.27826	−	4.82916i	−0.768399	−	0.867343i	0.225881	−	0.974155i	\(-0.427474\pi\)
							−0.994279	+	0.106812i	\(0.965936\pi\)
\(37\)	−0.936614	+	0.191211i	−0.153978	+	0.0314349i	−0.276396	−	0.961044i	\(-0.589140\pi\)
							0.122418	+	0.992479i	\(0.460935\pi\)
\(41\)	−0.964930	−	0.279495i	−0.150697	−	0.0436498i	0.202012	−	0.979383i	\(-0.435252\pi\)
							−0.352708	+	0.935733i	\(0.614739\pi\)
\(43\)	−0.978870	+	4.79482i	−0.149276	+	0.731204i	0.834386	+	0.551180i	\(0.185822\pi\)
							−0.983662	+	0.180023i	\(0.942383\pi\)
\(47\)	6.92404	−	4.77932i	1.00997	−	0.697135i	0.0564359	−	0.998406i	\(-0.482026\pi\)
							0.953538	+	0.301271i	\(0.0974110\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.2.k.a.4.3	✓	360
169.127	even	78	inner	169.2.k.a.127.3	yes	360

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.k.a.4.3	✓	360	1.1	even	1	trivial
169.2.k.a.127.3	yes	360	169.127	even	78	inner