Embedded newform 169.4.i.a.3.8

• Label: 169.4.i.a.3.8
• Level: $169$
• Weight: $4$
• Character: 169.3
• Analytic conductor: $9.971$
• Analytic rank: $0$
• Dimension: $1080$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	169.i (of order \(39\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.97132279097\)
Analytic rank:	\(0\)
Dimension:	\(1080\)
Relative dimension:	\(45\) over \(\Q(\zeta_{39})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{39}]$

Embedding invariants

Embedding label			3.8
Character	\(\chi\)	\(=\)	169.3
Dual form			169.4.i.a.113.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.30946 + 2.48690i) q^{2} +(-0.286302 + 1.40240i) q^{3} +(2.54213 - 8.77646i) q^{4} +(3.60691 + 1.89305i) q^{5} +(-2.54012 - 5.35320i) q^{6} +(-2.89951 - 0.471216i) q^{7} +(1.66938 + 4.40180i) q^{8} +(22.9547 + 9.78008i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1080 q - 27 q^{2} - 19 q^{3} + 155 q^{4} - 30 q^{5} - 56 q^{6} + 9 q^{7} + 4 q^{8} + 364 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{31}{39}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.30946	+	2.48690i	−1.17007	+	0.879252i	−0.994450	−	0.105206i	\(-0.966450\pi\)
							−0.175621	+	0.984458i	\(0.556193\pi\)
\(3\)	−0.286302	+	1.40240i	−0.0550989	+	0.269892i	−0.998003	−	0.0631733i	\(-0.979878\pi\)
							0.942904	+	0.333066i	\(0.108083\pi\)
\(5\)	3.60691	+	1.89305i	0.322611	+	0.169320i	0.618253	−	0.785980i	\(-0.287841\pi\)
							−0.295641	+	0.955299i	\(0.595533\pi\)
\(7\)	−2.89951	−	0.471216i	−0.156559	−	0.0254433i	0.0816267	−	0.996663i	\(-0.473988\pi\)
							−0.238185	+	0.971220i	\(0.576553\pi\)
\(11\)	4.98625	−	2.12444i	0.136674	−	0.0582312i	−0.322547	−	0.946553i	\(-0.604539\pi\)
							0.459221	+	0.888322i	\(0.348129\pi\)
\(13\)	16.4865	+	43.8770i	0.351734	+	0.936100i
							\(17\)	29.4647	+	4.78849i	0.420368	+	0.0683164i	0.366916	−	0.930254i	\(-0.380414\pi\)
0.0534516	+	0.998570i	\(0.482978\pi\)
\(19\)	−5.18281	+	8.97690i	−0.0625800	+	0.108392i	−0.895618	−	0.444824i	\(-0.853266\pi\)
							0.833038	+	0.553216i	\(0.186600\pi\)
\(23\)	73.0892	+	126.594i	0.662616	+	1.14768i	0.979926	+	0.199362i	\(0.0638871\pi\)
							−0.317310	+	0.948322i	\(0.602780\pi\)
\(29\)	−236.246	+	177.527i	−1.51275	+	1.13676i	−0.561859	+	0.827233i	\(0.689914\pi\)
							−0.950891	+	0.309525i	\(0.899830\pi\)
\(31\)	115.402	−	167.189i	0.668609	−	0.968648i	−0.331077	−	0.943604i	\(-0.607412\pi\)
							0.999686	−	0.0250442i	\(-0.00797266\pi\)
\(37\)	−335.217	−	27.0615i	−1.48944	−	0.120240i	−0.691237	−	0.722628i	\(-0.742934\pi\)
							−0.798203	+	0.602388i	\(0.794216\pi\)
\(41\)	−36.5095	+	178.836i	−0.139069	+	0.681205i	0.849149	+	0.528153i	\(0.177115\pi\)
							−0.988218	+	0.153052i	\(0.951090\pi\)
\(43\)	−371.183	+	29.9650i	−1.31639	+	0.106270i	−0.718625	−	0.695398i	\(-0.755228\pi\)
							−0.597769	+	0.801668i	\(0.703946\pi\)
\(47\)	577.452	+	142.329i	1.79213	+	0.441720i	0.988221	−	0.153035i	\(-0.0489048\pi\)
							0.803907	+	0.594755i	\(0.202751\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.4.i.a.3.8	✓	1080
169.113	even	39	inner	169.4.i.a.113.8	yes	1080

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.4.i.a.3.8	✓	1080	1.1	even	1	trivial
169.4.i.a.113.8	yes	1080	169.113	even	39	inner