Embedded newform 169.2.h.a.25.6

• Label: 169.2.h.a.25.6
• 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			25.6
Character	\(\chi\)	\(=\)	169.25
Dual form			169.2.h.a.142.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.622921 - 0.236243i) q^{2} +(-1.59376 - 2.30896i) q^{3} +(-1.16480 - 1.03192i) q^{4} +(3.47190 + 0.421565i) q^{5} +(0.447311 + 1.81481i) q^{6} +(-1.63679 - 3.11863i) q^{7} +(1.10100 + 2.09779i) q^{8} +(-1.72740 + 4.55479i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q - 13 q^{2} - 13 q^{3} + q^{4} - 13 q^{5} - 13 q^{6} - 13 q^{7} - 13 q^{8} - 27 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.622921	−	0.236243i	−0.440472	−	0.167049i	0.124392	−	0.992233i	\(-0.460302\pi\)
							−0.564864	+	0.825184i	\(0.691071\pi\)
\(3\)	−1.59376	−	2.30896i	−0.920157	−	1.33308i	−0.942809	−	0.333334i	\(-0.891826\pi\)
							0.0226520	−	0.999743i	\(-0.492789\pi\)
\(5\)	3.47190	+	0.421565i	1.55268	+	0.188530i	0.851463	−	0.524415i	\(-0.175716\pi\)
							0.701220	+	0.712945i	\(0.252639\pi\)
\(7\)	−1.63679	−	3.11863i	−0.618647	−	1.17873i	−0.970141	−	0.242543i	\(-0.922018\pi\)
							0.351494	−	0.936190i	\(-0.385674\pi\)
\(11\)	−4.16859	+	1.58094i	−1.25688	+	0.476671i	−0.891005	−	0.453994i	\(-0.849999\pi\)
							−0.365873	+	0.930665i	\(0.619230\pi\)
\(13\)	−1.62040	−	3.22092i	−0.449419	−	0.893321i
							\(17\)	0.291510	−	0.152996i	0.0707016	−	0.0371070i	−0.429002	−	0.903304i	\(-0.641135\pi\)
0.499703	+	0.866197i	\(0.333442\pi\)
\(19\)			1.89876i			0.435605i	0.975993	+	0.217802i	\(0.0698888\pi\)
							−0.975993	+	0.217802i	\(0.930111\pi\)
\(23\)	5.94018			1.23861			0.619307	−	0.785149i	\(-0.287414\pi\)
							0.619307	+	0.785149i	\(0.287414\pi\)
\(29\)	2.40046	−	6.32950i	0.445755	−	1.17536i	−0.504369	−	0.863488i	\(-0.668275\pi\)
							0.950124	−	0.311871i	\(-0.100956\pi\)
\(31\)	−1.59005	−	6.45110i	−0.285582	−	1.15865i	−0.921963	−	0.387277i	\(-0.873416\pi\)
							0.636381	−	0.771375i	\(-0.280431\pi\)
\(37\)	−0.381270	−	1.54687i	−0.0626805	−	0.254305i	0.931240	−	0.364407i	\(-0.118728\pi\)
							−0.993920	+	0.110102i	\(0.964882\pi\)
\(41\)	−2.97928	+	2.05645i	−0.465286	+	0.321164i	−0.777538	−	0.628835i	\(-0.783532\pi\)
							0.312252	+	0.949999i	\(0.398917\pi\)
\(43\)	4.78503	+	1.17940i	0.729710	+	0.179857i	0.586632	−	0.809854i	\(-0.300454\pi\)
							0.143079	+	0.989711i	\(0.454300\pi\)
\(47\)	−6.93071	−	7.82316i	−1.01095	−	1.14112i	−0.989960	−	0.141345i	\(-0.954857\pi\)
							−0.0209880	−	0.999780i	\(-0.506681\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.2.h.a.25.6	✓	168
169.142	even	26	inner	169.2.h.a.142.6	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.h.a.25.6	✓	168	1.1	even	1	trivial
169.2.h.a.142.6	yes	168	169.142	even	26	inner