Embedded newform 169.2.h.a.12.10

• Label: 169.2.h.a.12.10
• Level: $169$
• Weight: $2$
• Character: 169.12
• 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			12.10
Character	\(\chi\)	\(=\)	169.12
Dual form			169.2.h.a.155.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.651938 - 0.450000i) q^{2} +(-1.14686 - 0.601918i) q^{3} +(-0.486687 + 1.28329i) q^{4} +(-2.65258 - 2.99414i) q^{5} +(-1.01854 + 0.123674i) q^{6} +(-0.815660 - 3.30926i) q^{7} +(0.639345 + 2.59392i) q^{8} +(-0.751216 - 1.08832i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q - 13 q^{2} - 13 q^{3} + q^{4} - 13 q^{5} - 13 q^{6} - 13 q^{7} - 13 q^{8} - 27 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{21}{26}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.651938	−	0.450000i	0.460990	−	0.318198i	−0.314835	−	0.949146i	\(-0.601949\pi\)
							0.775825	+	0.630948i	\(0.217334\pi\)
\(3\)	−1.14686	−	0.601918i	−0.662139	−	0.347517i	0.0999475	−	0.994993i	\(-0.468133\pi\)
							−0.762086	+	0.647475i	\(0.775825\pi\)
\(5\)	−2.65258	−	2.99414i	−1.18627	−	1.33902i	−0.927835	−	0.372991i	\(-0.878332\pi\)
							−0.258433	−	0.966029i	\(-0.583206\pi\)
\(7\)	−0.815660	−	3.30926i	−0.308290	−	1.25078i	−0.896852	−	0.442331i	\(-0.854152\pi\)
							0.588562	−	0.808452i	\(-0.299694\pi\)
\(11\)	−0.0864794	−	0.0596924i	−0.0260745	−	0.0179979i	0.554956	−	0.831880i	\(-0.312735\pi\)
							−0.581031	+	0.813882i	\(0.697350\pi\)
\(13\)	3.60323	+	0.129485i	0.999355	+	0.0359127i
							\(17\)	0.528057	−	0.130154i	0.128073	−	0.0315670i	−0.174758	−	0.984611i	\(-0.555914\pi\)
0.302831	+	0.953044i	\(0.402068\pi\)
\(19\)			1.09996i			0.252348i	0.992008	+	0.126174i	\(0.0402697\pi\)
							−0.992008	+	0.126174i	\(0.959730\pi\)
\(23\)	−0.747912			−0.155950			−0.0779752	−	0.996955i	\(-0.524846\pi\)
							−0.0779752	+	0.996955i	\(0.524846\pi\)
\(29\)	−5.92507	−	8.58394i	−1.10026	−	1.59400i	−0.755299	−	0.655380i	\(-0.772509\pi\)
							−0.344958	−	0.938618i	\(-0.612107\pi\)
\(31\)	4.48092	−	0.544083i	0.804798	−	0.0977201i	0.292209	−	0.956354i	\(-0.405610\pi\)
							0.512589	+	0.858634i	\(0.328687\pi\)
\(37\)	6.81322	−	0.827274i	1.12009	−	0.136003i	0.460520	−	0.887649i	\(-0.347663\pi\)
							0.659566	+	0.751646i	\(0.270740\pi\)
\(41\)	5.07205	−	9.66399i	0.792122	−	1.50926i	−0.0671506	−	0.997743i	\(-0.521391\pi\)
							0.859272	−	0.511519i	\(-0.170917\pi\)
\(43\)	0.481709	−	3.96723i	0.0734599	−	0.604997i	−0.908768	−	0.417303i	\(-0.862975\pi\)
							0.982227	−	0.187694i	\(-0.0601014\pi\)
\(47\)	7.74511	−	2.93733i	1.12974	−	0.428454i	0.282290	−	0.959329i	\(-0.408906\pi\)
							0.847450	+	0.530875i	\(0.178137\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.2.h.a.12.10	✓	168
169.155	even	26	inner	169.2.h.a.155.10	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.h.a.12.10	✓	168	1.1	even	1	trivial
169.2.h.a.155.10	yes	168	169.155	even	26	inner