Embedded newform 169.2.h.a.25.5

• Label: 169.2.h.a.25.5
• Level: $169$
• Weight: $2$
• Character: 169.25
• Analytic conductor: $1.349$
• Analytic rank: $0$
• Dimension: $168$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	169.h (of order \(26\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.34947179416\)
Analytic rank:	\(0\)
Dimension:	\(168\)
Relative dimension:	\(14\) over \(\Q(\zeta_{26})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{26}]$

Embedding invariants

Embedding label			25.5
Character	\(\chi\)	\(=\)	169.25
Dual form			169.2.h.a.142.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.49527 - 0.567082i) q^{2} +(1.18578 + 1.71790i) q^{3} +(0.417236 + 0.369639i) q^{4} +(-2.96326 - 0.359805i) q^{5} +(-0.798874 - 3.24116i) q^{6} +(1.24562 + 2.37332i) q^{7} +(1.07210 + 2.04271i) q^{8} +(-0.481287 + 1.26905i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q - 13 q^{2} - 13 q^{3} + q^{4} - 13 q^{5} - 13 q^{6} - 13 q^{7} - 13 q^{8} - 27 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{3}{26}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.49527	−	0.567082i	−1.05732	−	0.400988i	−0.236175	−	0.971711i	\(-0.575894\pi\)
							−0.821143	+	0.570723i	\(0.806663\pi\)
\(3\)	1.18578	+	1.71790i	0.684610	+	0.991829i	0.999094	+	0.0425661i	\(0.0135533\pi\)
							−0.314483	+	0.949263i	\(0.601831\pi\)
\(5\)	−2.96326	−	0.359805i	−1.32521	−	0.160909i	−0.572848	−	0.819661i	\(-0.694162\pi\)
							−0.752360	+	0.658752i	\(0.771085\pi\)
\(7\)	1.24562	+	2.37332i	0.470798	+	0.897031i	0.998856	+	0.0478226i	\(0.0152282\pi\)
							−0.528058	+	0.849209i	\(0.677079\pi\)
\(11\)	−4.01299	+	1.52193i	−1.20996	+	0.458878i	−0.875356	−	0.483479i	\(-0.839373\pi\)
							−0.334607	+	0.942358i	\(0.608604\pi\)
\(13\)	−1.01665	+	3.45925i	−0.281968	+	0.959424i
							\(17\)	−3.38159	+	1.77479i	−0.820155	+	0.430451i	−0.821958	−	0.569548i	\(-0.807118\pi\)
0.00180312	+	0.999998i	\(0.499426\pi\)
\(19\)			3.49861i			0.802635i	0.915939	+	0.401318i	\(0.131448\pi\)
							−0.915939	+	0.401318i	\(0.868552\pi\)
\(23\)	6.33211			1.32034			0.660168	−	0.751118i	\(-0.270485\pi\)
							0.660168	+	0.751118i	\(0.270485\pi\)
\(29\)	−1.14319	+	3.01434i	−0.212285	+	0.559749i	−0.998289	−	0.0584720i	\(-0.981377\pi\)
							0.786004	+	0.618221i	\(0.212146\pi\)
\(31\)	−1.63386	−	6.62883i	−0.293450	−	1.19057i	−0.913773	−	0.406226i	\(-0.866845\pi\)
							0.620323	−	0.784346i	\(-0.287002\pi\)
\(37\)	0.694875	+	2.81922i	0.114237	+	0.463477i	1.00000		0.000661049i	\(-0.000210419\pi\)
							−0.885763	+	0.464138i	\(0.846364\pi\)
\(41\)	6.02013	−	4.15540i	0.940187	−	0.648964i	0.00382221	−	0.999993i	\(-0.498783\pi\)
							0.936365	+	0.351028i	\(0.114168\pi\)
\(43\)	7.11200	+	1.75295i	1.08457	+	0.267323i	0.740796	−	0.671730i	\(-0.234448\pi\)
							0.343774	+	0.939052i	\(0.388295\pi\)
\(47\)	−6.61375	−	7.46538i	−0.964714	−	1.08894i	−0.995993	−	0.0894267i	\(-0.971497\pi\)
							0.0312791	−	0.999511i	\(-0.490042\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.2.h.a.25.5	✓	168
169.142	even	26	inner	169.2.h.a.142.5	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.h.a.25.5	✓	168	1.1	even	1	trivial
169.2.h.a.142.5	yes	168	169.142	even	26	inner