Embedded newform 169.2.g.a.14.5

• Label: 169.2.g.a.14.5
• Level: $169$
• Weight: $2$
• Character: 169.14
• Analytic conductor: $1.349$
• Analytic rank: $0$
• Dimension: $156$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.34947179416\)
Analytic rank:	\(0\)
Dimension:	\(156\)
Relative dimension:	\(13\) over \(\Q(\zeta_{13})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{13}]$

Embedding invariants

Embedding label			14.5
Character	\(\chi\)	\(=\)	169.14
Dual form			169.2.g.a.157.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.707334 + 1.02475i) q^{2} +(2.32028 - 1.21778i) q^{3} +(0.159418 + 0.420350i) q^{4} +(1.61135 + 1.42753i) q^{5} +(-0.393297 + 3.23909i) q^{6} +(-1.03213 - 0.254398i) q^{7} +(-2.96148 - 0.729939i) q^{8} +(2.19654 - 3.18224i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 10 q^{2} - 9 q^{3} - 20 q^{4} - 7 q^{5} - q^{6} - 5 q^{7} + 2 q^{8} - 14 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{9}{13}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.707334	+	1.02475i	−0.500161	+	0.724608i	−0.989024	−	0.147752i	\(-0.952796\pi\)
							0.488864	+	0.872360i	\(0.337412\pi\)
\(3\)	2.32028	−	1.21778i	1.33962	−	0.703085i	0.365859	−	0.930670i	\(-0.380775\pi\)
							0.973758	+	0.227585i	\(0.0730830\pi\)
\(5\)	1.61135	+	1.42753i	0.720618	+	0.638412i	0.941548	−	0.336878i	\(-0.109371\pi\)
							−0.220931	+	0.975290i	\(0.570909\pi\)
\(7\)	−1.03213	−	0.254398i	−0.390109	−	0.0961533i	0.0393822	−	0.999224i	\(-0.487461\pi\)
							−0.429491	+	0.903071i	\(0.641307\pi\)
\(11\)	−0.173539	−	0.251415i	−0.0523240	−	0.0758044i	0.795949	−	0.605364i	\(-0.206973\pi\)
							−0.848273	+	0.529560i	\(0.822357\pi\)
\(13\)	−2.66602	−	2.42741i	−0.739422	−	0.673242i
							\(17\)	3.99680	+	0.985122i	0.969366	+	0.238927i	0.692049	−	0.721851i	\(-0.256708\pi\)
0.277318	+	0.960778i	\(0.410555\pi\)
\(19\)	−4.43794			−1.01813			−0.509067	−	0.860727i	\(-0.670009\pi\)
							−0.509067	+	0.860727i	\(0.670009\pi\)
\(23\)	0.658914			0.137393			0.0686966	−	0.997638i	\(-0.478116\pi\)
							0.0686966	+	0.997638i	\(0.478116\pi\)
\(29\)	−0.846759	+	1.22674i	−0.157239	+	0.227800i	−0.893684	−	0.448698i	\(-0.851888\pi\)
							0.736444	+	0.676498i	\(0.236503\pi\)
\(31\)	−0.816771	+	6.72671i	−0.146696	+	1.20815i	0.714410	+	0.699728i	\(0.246695\pi\)
							−0.861106	+	0.508425i	\(0.830228\pi\)
\(37\)	1.23739	−	10.1908i	0.203426	−	1.67536i	−0.430981	−	0.902361i	\(-0.641832\pi\)
							0.634406	−	0.773000i	\(-0.281245\pi\)
\(41\)	6.43378	−	3.37671i	1.00479	−	0.527353i	0.119716	−	0.992808i	\(-0.461802\pi\)
							0.885072	+	0.465455i	\(0.154109\pi\)
\(43\)	−0.964779	−	7.94567i	−0.147127	−	1.21170i	−0.859957	−	0.510366i	\(-0.829510\pi\)
							0.712830	−	0.701337i	\(-0.247413\pi\)
\(47\)	−3.81052	+	10.0475i	−0.555822	+	1.46558i	0.302842	+	0.953041i	\(0.402065\pi\)
							−0.858663	+	0.512540i	\(0.828705\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.2.g.a.14.5	✓	156
169.157	even	13	inner	169.2.g.a.157.5	yes	156

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.g.a.14.5	✓	156	1.1	even	1	trivial
169.2.g.a.157.5	yes	156	169.157	even	13	inner