Embedded newform 273.2.y.b.101.5

• Label: 273.2.y.b.101.5
• Level: $273$
• Weight: $2$
• Character: 273.101
• 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.y (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			101.5
Character	\(\chi\)	\(=\)	273.101
Dual form			273.2.y.b.173.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.06779 + 1.84947i) q^{2} +(1.37855 - 1.04862i) q^{3} +(-1.28036 - 2.21764i) q^{4} +(3.07271 - 1.77403i) q^{5} +(0.467389 + 3.66929i) q^{6} +(-2.57290 - 0.616608i) q^{7} +1.19745 q^{8} +(0.800790 - 2.89115i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 28 q^{4} - 12 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(92\)	\(106\)	\(157\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.06779	+	1.84947i	−0.755042	+	1.30777i	0.190311	+	0.981724i	\(0.439050\pi\)
							−0.945353	+	0.326048i	\(0.894283\pi\)
\(3\)	1.37855	−	1.04862i	0.795905	−	0.605421i
							\(5\)	3.07271	−	1.77403i	1.37416	−	0.793370i	0.382709	−	0.923869i	\(-0.374991\pi\)
0.991448	+	0.130499i	\(0.0416579\pi\)
\(7\)	−2.57290	−	0.616608i	−0.972463	−	0.233056i
							\(11\)	1.68728			0.508733			0.254367	−	0.967108i	\(-0.418133\pi\)
0.254367	+	0.967108i	\(0.418133\pi\)
\(13\)	0.905061	−	3.49011i	0.251019	−	0.967982i
							\(17\)	1.25399	+	2.17197i	0.304136	+	0.526780i	0.977069	−	0.212924i	\(-0.0682988\pi\)
−0.672932	+	0.739704i	\(0.734965\pi\)
\(19\)	−3.33372			−0.764808			−0.382404	−	0.923995i	\(-0.624904\pi\)
							−0.382404	+	0.923995i	\(0.624904\pi\)
\(23\)	6.61106	+	3.81690i	1.37850	+	0.795878i	0.991979	−	0.126402i	\(-0.0403428\pi\)
							0.386522	+	0.922280i	\(0.373676\pi\)
\(29\)	−3.85641	+	2.22650i	−0.716117	+	0.413450i	−0.813322	−	0.581814i	\(-0.802343\pi\)
							0.0972050	+	0.995264i	\(0.469010\pi\)
\(31\)	−5.26838	+	9.12510i	−0.946229	+	1.63892i	−0.192958	+	0.981207i	\(0.561808\pi\)
							−0.753271	+	0.657710i	\(0.771525\pi\)
\(37\)	4.80954	+	2.77679i	0.790683	+	0.456501i	0.840203	−	0.542272i	\(-0.182436\pi\)
							−0.0495197	+	0.998773i	\(0.515769\pi\)
\(41\)	−2.89982	+	1.67421i	−0.452875	+	0.261468i	−0.709044	−	0.705165i	\(-0.750873\pi\)
							0.256169	+	0.966632i	\(0.417540\pi\)
\(43\)	−0.303610	+	0.525869i	−0.0463002	+	0.0801942i	−0.888247	−	0.459367i	\(-0.848076\pi\)
							0.841947	+	0.539561i	\(0.181410\pi\)
\(47\)	−1.95815	+	1.13054i	−0.285626	+	0.164906i	−0.635968	−	0.771716i	\(-0.719399\pi\)
							0.350342	+	0.936622i	\(0.386065\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.y.b.101.5	✓	64
3.2	odd	2	inner	273.2.y.b.101.28	yes	64
7.5	odd	6		273.2.br.b.257.28	yes	64
13.4	even	6		273.2.br.b.17.5	yes	64
21.5	even	6		273.2.br.b.257.5	yes	64
39.17	odd	6		273.2.br.b.17.28	yes	64
91.82	odd	6	inner	273.2.y.b.173.28	yes	64
273.173	even	6	inner	273.2.y.b.173.5	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.y.b.101.5	✓	64	1.1	even	1	trivial
273.2.y.b.101.28	yes	64	3.2	odd	2	inner
273.2.y.b.173.5	yes	64	273.173	even	6	inner
273.2.y.b.173.28	yes	64	91.82	odd	6	inner
273.2.br.b.17.5	yes	64	13.4	even	6
273.2.br.b.17.28	yes	64	39.17	odd	6
273.2.br.b.257.5	yes	64	21.5	even	6
273.2.br.b.257.28	yes	64	7.5	odd	6