Embedded newform 169.2.h.a.25.3

• Label: 169.2.h.a.25.3
• 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			25.3
Character	\(\chi\)	\(=\)	169.25
Dual form			169.2.h.a.142.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.68720 - 0.639871i) q^{2} +(0.0407127 + 0.0589826i) q^{3} +(0.940192 + 0.832938i) q^{4} +(0.654244 + 0.0794397i) q^{5} +(-0.0309493 - 0.125566i) q^{6} +(-0.153151 - 0.291806i) q^{7} +(0.623830 + 1.18861i) q^{8} +(1.06199 - 2.80025i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q - 13 q^{2} - 13 q^{3} + q^{4} - 13 q^{5} - 13 q^{6} - 13 q^{7} - 13 q^{8} - 27 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.68720	−	0.639871i	−1.19303	−	0.452457i	−0.323449	−	0.946245i	\(-0.604843\pi\)
							−0.869582	+	0.493788i	\(0.835612\pi\)
\(3\)	0.0407127	+	0.0589826i	0.0235055	+	0.0340536i	0.834560	−	0.550916i	\(-0.185722\pi\)
							−0.811055	+	0.584970i	\(0.801106\pi\)
\(5\)	0.654244	+	0.0794397i	0.292587	+	0.0355265i	0.265514	−	0.964107i	\(-0.414458\pi\)
							0.0270729	+	0.999633i	\(0.491381\pi\)
\(7\)	−0.153151	−	0.291806i	−0.0578858	−	0.110292i	0.854794	−	0.518967i	\(-0.173683\pi\)
							−0.912680	+	0.408675i	\(0.865991\pi\)
\(11\)	3.29754	−	1.25059i	0.994247	−	0.377068i	0.196791	−	0.980445i	\(-0.436948\pi\)
							0.797456	+	0.603377i	\(0.206179\pi\)
\(13\)	−3.47540	−	0.959994i	−0.963903	−	0.266254i
							\(17\)	4.33713	−	2.27630i	1.05191	−	0.552084i	0.152098	−	0.988365i	\(-0.451397\pi\)
0.899810	+	0.436281i	\(0.143705\pi\)
\(19\)		−	4.67024i		−	1.07143i	−0.844400	−	0.535713i	\(-0.820043\pi\)
							0.844400	−	0.535713i	\(-0.179957\pi\)
\(23\)	3.76934			0.785961			0.392980	−	0.919547i	\(-0.371444\pi\)
							0.392980	+	0.919547i	\(0.371444\pi\)
\(29\)	1.37597	−	3.62813i	0.255511	−	0.673727i	−0.744470	−	0.667655i	\(-0.767298\pi\)
							0.999982	−	0.00607198i	\(-0.00193278\pi\)
\(31\)	1.62148	+	6.57861i	0.291227	+	1.18155i	0.916140	+	0.400859i	\(0.131288\pi\)
							−0.624913	+	0.780694i	\(0.714866\pi\)
\(37\)	2.40265	+	9.74795i	0.394994	+	1.60255i	0.745843	+	0.666121i	\(0.232047\pi\)
							−0.350850	+	0.936432i	\(0.614107\pi\)
\(41\)	−8.11133	+	5.59885i	−1.26678	+	0.874393i	−0.996096	−	0.0882764i	\(-0.971864\pi\)
							−0.270681	+	0.962669i	\(0.587249\pi\)
\(43\)	2.63216	+	0.648770i	0.401401	+	0.0989365i	0.434848	−	0.900504i	\(-0.356802\pi\)
							−0.0334467	+	0.999441i	\(0.510648\pi\)
\(47\)	5.36429	+	6.05503i	0.782462	+	0.883217i	0.995590	−	0.0938070i	\(-0.0299037\pi\)
							−0.213128	+	0.977024i	\(0.568365\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.2.h.a.25.3	✓	168
169.142	even	26	inner	169.2.h.a.142.3	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.h.a.25.3	✓	168	1.1	even	1	trivial
169.2.h.a.142.3	yes	168	169.142	even	26	inner