Embedded newform 273.2.br.b.17.19

• Label: 273.2.br.b.17.19
• Level: $273$
• Weight: $2$
• Character: 273.17
• Analytic conductor: $2.180$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 273 = 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	273.br (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.17991597518\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(32\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			17.19
Character	\(\chi\)	\(=\)	273.17
Dual form			273.2.br.b.257.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+0.419567 q^{2} +(-0.967375 + 1.43673i) q^{3} -1.82396 q^{4} +(3.21070 - 1.85370i) q^{5} +(-0.405879 + 0.602804i) q^{6} +(2.36714 - 1.18181i) q^{7} -1.60441 q^{8} +(-1.12837 - 2.77971i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 6 q^{3} + 56 q^{4} - 6 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/273\mathbb{Z}\right)^\times\).

\(n\)	\(92\)	\(106\)	\(157\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.419567			0.296679			0.148339	−	0.988937i	\(-0.452607\pi\)
							0.148339	+	0.988937i	\(0.452607\pi\)
\(3\)	−0.967375	+	1.43673i	−0.558514	+	0.829495i
							\(5\)	3.21070	−	1.85370i	1.43587	−	0.828998i	0.438308	−	0.898825i	\(-0.355578\pi\)
0.997559	+	0.0698271i	\(0.0222447\pi\)
\(7\)	2.36714	−	1.18181i	0.894693	−	0.446681i
							\(11\)	2.25261	+	3.90164i	0.679188	+	1.17639i	0.975226	+	0.221212i	\(0.0710012\pi\)
−0.296038	+	0.955176i	\(0.595665\pi\)
\(13\)	2.65882	+	2.43530i	0.737423	+	0.675431i
							\(17\)	−1.86863			−0.453209			−0.226604	−	0.973987i	\(-0.572762\pi\)
−0.226604	+	0.973987i	\(0.572762\pi\)
\(19\)	1.01445	−	1.75707i	0.232730	−	0.403100i	−0.725881	−	0.687821i	\(-0.758568\pi\)
							0.958611	+	0.284721i	\(0.0919009\pi\)
\(23\)			1.71381i			0.357353i	0.983908	+	0.178677i	\(0.0571816\pi\)
							−0.983908	+	0.178677i	\(0.942818\pi\)
\(29\)	−0.535895	−	0.309399i	−0.0995133	−	0.0574540i	0.449417	−	0.893322i	\(-0.351632\pi\)
							−0.548931	+	0.835868i	\(0.684965\pi\)
\(31\)	−0.476920	+	0.826049i	−0.0856573	+	0.148363i	−0.905671	−	0.423981i	\(-0.860632\pi\)
							0.820014	+	0.572344i	\(0.193966\pi\)
\(37\)		−	5.89590i		−	0.969280i	−0.874714	−	0.484640i	\(-0.838951\pi\)
							0.874714	−	0.484640i	\(-0.161049\pi\)
\(41\)	−5.98390	−	3.45481i	−0.934528	−	0.539550i	−0.0462874	−	0.998928i	\(-0.514739\pi\)
							−0.888241	+	0.459378i	\(0.848072\pi\)
\(43\)	−1.32592	−	2.29655i	−0.202200	−	0.350221i	0.747037	−	0.664783i	\(-0.231476\pi\)
							−0.949237	+	0.314562i	\(0.898142\pi\)
\(47\)	−10.1715	+	5.87250i	−1.48366	+	0.856593i	−0.999828	−	0.0185665i	\(-0.994090\pi\)
							−0.483835	+	0.875159i	\(0.660756\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.br.b.17.19	yes	64
3.2	odd	2	inner	273.2.br.b.17.14	yes	64
7.5	odd	6		273.2.y.b.173.14	yes	64
13.10	even	6		273.2.y.b.101.19	yes	64
21.5	even	6		273.2.y.b.173.19	yes	64
39.23	odd	6		273.2.y.b.101.14	✓	64
91.75	odd	6	inner	273.2.br.b.257.14	yes	64
273.257	even	6	inner	273.2.br.b.257.19	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.y.b.101.14	✓	64	39.23	odd	6
273.2.y.b.101.19	yes	64	13.10	even	6
273.2.y.b.173.14	yes	64	7.5	odd	6
273.2.y.b.173.19	yes	64	21.5	even	6
273.2.br.b.17.14	yes	64	3.2	odd	2	inner
273.2.br.b.17.19	yes	64	1.1	even	1	trivial
273.2.br.b.257.14	yes	64	91.75	odd	6	inner
273.2.br.b.257.19	yes	64	273.257	even	6	inner