Embedded newform 169.2.k.a.4.5

• Label: 169.2.k.a.4.5
• Level: $169$
• Weight: $2$
• Character: 169.4
• Analytic conductor: $1.349$
• Analytic rank: $0$
• Dimension: $360$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	169.k (of order \(78\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.34947179416\)
Analytic rank:	\(0\)
Dimension:	\(360\)
Relative dimension:	\(15\) over \(\Q(\zeta_{78})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{78}]$

Embedding invariants

Embedding label			4.5
Character	\(\chi\)	\(=\)	169.4
Dual form			169.2.k.a.127.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.55556 - 0.0626871i) q^{2} +(0.386039 - 1.33276i) q^{3} +(0.422335 + 0.0340944i) q^{4} +(1.77605 + 0.673567i) q^{5} +(-0.684055 + 2.04899i) q^{6} +(-0.0926507 - 0.217459i) q^{7} +(2.43612 + 0.295798i) q^{8} +(0.908347 + 0.574404i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 360 q - 23 q^{2} - 24 q^{3} - 41 q^{4} - 26 q^{5} - 32 q^{6} - 26 q^{7} - 26 q^{8} - 11 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{78}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.55556	−	0.0626871i	−1.09995	−	0.0443265i	−0.516357	−	0.856373i	\(-0.672712\pi\)
							−0.583592	+	0.812047i	\(0.698353\pi\)
\(3\)	0.386039	−	1.33276i	0.222879	−	0.769469i	−0.768659	−	0.639658i	\(-0.779076\pi\)
							0.991539	−	0.129811i	\(-0.0414370\pi\)
\(5\)	1.77605	+	0.673567i	0.794274	+	0.301229i	0.718167	−	0.695871i	\(-0.244982\pi\)
							0.0761076	+	0.997100i	\(0.475751\pi\)
\(7\)	−0.0926507	−	0.217459i	−0.0350187	−	0.0821918i	0.901552	−	0.432671i	\(-0.142429\pi\)
							−0.936570	+	0.350479i	\(0.886019\pi\)
\(11\)	−3.19758	−	5.05657i	−0.964108	−	1.52461i	−0.847066	−	0.531487i	\(-0.821633\pi\)
							−0.117042	−	0.993127i	\(-0.537341\pi\)
\(13\)	2.60949	−	2.48809i	0.723742	−	0.690071i
							\(17\)	−2.36370	+	1.00708i	−0.573280	+	0.244252i	−0.659096	−	0.752059i	\(-0.729061\pi\)
0.0858153	+	0.996311i	\(0.472651\pi\)
\(19\)	6.49388	−	3.74924i	1.48980	−	0.860135i	0.489866	−	0.871798i	\(-0.337046\pi\)
							0.999932	+	0.0116624i	\(0.00371234\pi\)
\(23\)	0.0926363	−	0.160451i	0.0193160	−	0.0334563i	−0.856206	−	0.516635i	\(-0.827184\pi\)
							0.875522	+	0.483179i	\(0.160518\pi\)
\(29\)	−0.382988	+	9.50376i	−0.0711191	+	1.76480i	0.433211	+	0.901292i	\(0.357380\pi\)
							−0.504330	+	0.863511i	\(0.668261\pi\)
\(31\)	0.647763	+	0.731173i	0.116342	+	0.131323i	0.803796	−	0.594905i	\(-0.202810\pi\)
							−0.687454	+	0.726228i	\(0.741272\pi\)
\(37\)	3.25171	−	0.663842i	0.534578	−	0.109135i	0.0748503	−	0.997195i	\(-0.476152\pi\)
							0.459728	+	0.888060i	\(0.347947\pi\)
\(41\)	−2.35912	−	0.683327i	−0.368433	−	0.106718i	0.0888364	−	0.996046i	\(-0.471685\pi\)
							−0.457269	+	0.889328i	\(0.651172\pi\)
\(43\)	−2.39553	+	11.7341i	−0.365314	+	1.78943i	0.216689	+	0.976241i	\(0.430474\pi\)
							−0.582003	+	0.813186i	\(0.697731\pi\)
\(47\)	−9.77929	+	6.75016i	−1.42646	+	0.984611i	−0.429798	+	0.902925i	\(0.641415\pi\)
							−0.996658	+	0.0816863i	\(0.973969\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.2.k.a.4.5	✓	360
169.127	even	78	inner	169.2.k.a.127.5	yes	360

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.k.a.4.5	✓	360	1.1	even	1	trivial
169.2.k.a.127.5	yes	360	169.127	even	78	inner