Embedded newform 273.2.r.b.269.21

• Label: 273.2.r.b.269.21
• Level: $273$
• Weight: $2$
• Character: 273.269
• 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.r (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			269.21
Character	\(\chi\)	\(=\)	273.269
Dual form			273.2.r.b.68.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+0.883413i q^{2} +(1.41202 - 1.00309i) q^{3} +1.21958 q^{4} +(-1.72131 - 2.98140i) q^{5} +(0.886146 + 1.24740i) q^{6} +(-2.26618 - 1.36544i) q^{7} +2.84422i q^{8} +(0.987606 - 2.83278i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 64 q^{4} + 12 q^{6} - 10 q^{7} - 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(92\)	\(106\)	\(157\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)			0.883413i			0.624668i	0.949972	+	0.312334i	\(0.101111\pi\)
							−0.949972	+	0.312334i	\(0.898889\pi\)
\(3\)	1.41202	−	1.00309i	0.815231	−	0.579136i
							\(5\)	−1.72131	−	2.98140i	−0.769794	−	1.33332i	−0.937675	−	0.347514i	\(-0.887026\pi\)
0.167881	−	0.985807i	\(-0.446308\pi\)
\(7\)	−2.26618	−	1.36544i	−0.856537	−	0.516086i
							\(11\)	3.32619	−	1.92038i	1.00289	−	0.579016i	0.0937846	−	0.995593i	\(-0.470104\pi\)
0.909101	+	0.416576i	\(0.136770\pi\)
\(13\)	0.144185	+	3.60267i	0.0399897	+	0.999200i
							\(17\)	−0.161995			−0.0392895			−0.0196448	−	0.999807i	\(-0.506254\pi\)
−0.0196448	+	0.999807i	\(0.506254\pi\)
\(19\)	−1.41529	−	0.817116i	−0.324689	−	0.187459i	0.328792	−	0.944402i	\(-0.393359\pi\)
							−0.653481	+	0.756943i	\(0.726692\pi\)
\(23\)			4.54399i			0.947488i	0.880663	+	0.473744i	\(0.157098\pi\)
							−0.880663	+	0.473744i	\(0.842902\pi\)
\(29\)	8.21721	+	4.74421i	1.52590	+	0.880978i	0.999528	+	0.0307223i	\(0.00978076\pi\)
							0.526370	+	0.850255i	\(0.323553\pi\)
\(31\)	0.139734	+	0.0806754i	0.0250970	+	0.0144897i	0.512496	−	0.858690i	\(-0.328721\pi\)
							−0.487399	+	0.873179i	\(0.662054\pi\)
\(37\)	−3.98455			−0.655056			−0.327528	−	0.944841i	\(-0.606216\pi\)
							−0.327528	+	0.944841i	\(0.606216\pi\)
\(41\)	2.41336	−	4.18006i	0.376904	−	0.652816i	−0.613706	−	0.789534i	\(-0.710322\pi\)
							0.990610	+	0.136718i	\(0.0436555\pi\)
\(43\)	4.32913	+	7.49828i	0.660186	+	1.14348i	0.980566	+	0.196187i	\(0.0628561\pi\)
							−0.320380	+	0.947289i	\(0.603811\pi\)
\(47\)	3.42471	+	5.93177i	0.499545	+	0.865238i	1.00000		0.000525066i	\(-0.000167134\pi\)
							−0.500455	+	0.865763i	\(0.666834\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.r.b.269.21	yes	64
3.2	odd	2	inner	273.2.r.b.269.12	yes	64
7.5	odd	6		273.2.bf.b.152.21	yes	64
13.3	even	3		273.2.bf.b.185.12	yes	64
21.5	even	6		273.2.bf.b.152.12	yes	64
39.29	odd	6		273.2.bf.b.185.21	yes	64
91.68	odd	6	inner	273.2.r.b.68.21	yes	64
273.68	even	6	inner	273.2.r.b.68.12	✓	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.r.b.68.12	✓	64	273.68	even	6	inner
273.2.r.b.68.21	yes	64	91.68	odd	6	inner
273.2.r.b.269.12	yes	64	3.2	odd	2	inner
273.2.r.b.269.21	yes	64	1.1	even	1	trivial
273.2.bf.b.152.12	yes	64	21.5	even	6
273.2.bf.b.152.21	yes	64	7.5	odd	6
273.2.bf.b.185.12	yes	64	13.3	even	3
273.2.bf.b.185.21	yes	64	39.29	odd	6