Embedded newform 273.2.y.b.101.20

• Label: 273.2.y.b.101.20
• 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			101.20
Character	\(\chi\)	\(=\)	273.101
Dual form			273.2.y.b.173.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.366273 - 0.634404i) q^{2} +(1.53905 + 0.794556i) q^{3} +(0.731688 + 1.26732i) q^{4} +(2.08780 - 1.20539i) q^{5} +(1.06778 - 0.685356i) q^{6} +(-2.55085 - 0.702242i) q^{7} +2.53708 q^{8} +(1.73736 + 2.44573i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 28 q^{4} - 12 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.366273	−	0.634404i	0.258994	−	0.448591i	−0.706978	−	0.707235i	\(-0.749942\pi\)
							0.965973	+	0.258644i	\(0.0832755\pi\)
\(3\)	1.53905	+	0.794556i	0.888572	+	0.458737i
							\(5\)	2.08780	−	1.20539i	0.933691	−	0.539067i	0.0457138	−	0.998955i	\(-0.485444\pi\)
0.887977	+	0.459888i	\(0.152110\pi\)
\(7\)	−2.55085	−	0.702242i	−0.964132	−	0.265422i
							\(11\)	−3.75986			−1.13364			−0.566820	−	0.823842i	\(-0.691826\pi\)
−0.566820	+	0.823842i	\(0.691826\pi\)
\(13\)	−2.09909	−	2.93152i	−0.582183	−	0.813057i
							\(17\)	0.481637	+	0.834220i	0.116814	+	0.202328i	0.918504	−	0.395413i	\(-0.129398\pi\)
−0.801689	+	0.597741i	\(0.796065\pi\)
\(19\)	−2.27773			−0.522546			−0.261273	−	0.965265i	\(-0.584142\pi\)
							−0.261273	+	0.965265i	\(0.584142\pi\)
\(23\)	−0.499449	−	0.288357i	−0.104142	−	0.0601266i	0.447024	−	0.894522i	\(-0.352484\pi\)
							−0.551167	+	0.834395i	\(0.685817\pi\)
\(29\)	5.91477	−	3.41489i	1.09834	−	0.634130i	0.162559	−	0.986699i	\(-0.448025\pi\)
							0.935786	+	0.352569i	\(0.114692\pi\)
\(31\)	0.843529	−	1.46103i	0.151502	−	0.262410i	−0.780278	−	0.625433i	\(-0.784922\pi\)
							0.931780	+	0.363024i	\(0.118256\pi\)
\(37\)	−9.75612	−	5.63270i	−1.60390	−	0.926010i	−0.990698	−	0.136081i	\(-0.956549\pi\)
							−0.613198	−	0.789929i	\(-0.710117\pi\)
\(41\)	8.39746	−	4.84828i	1.31146	−	0.757174i	0.329125	−	0.944286i	\(-0.393246\pi\)
							0.982338	+	0.187113i	\(0.0599130\pi\)
\(43\)	−1.10299	+	1.91043i	−0.168204	+	0.291338i	−0.937789	−	0.347207i	\(-0.887130\pi\)
							0.769584	+	0.638545i	\(0.220463\pi\)
\(47\)	1.59967	−	0.923572i	0.233336	−	0.134717i	−0.378774	−	0.925489i	\(-0.623654\pi\)
							0.612110	+	0.790772i	\(0.290321\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.y.b.101.20	yes	64
3.2	odd	2	inner	273.2.y.b.101.13	✓	64
7.5	odd	6		273.2.br.b.257.13	yes	64
13.4	even	6		273.2.br.b.17.20	yes	64
21.5	even	6		273.2.br.b.257.20	yes	64
39.17	odd	6		273.2.br.b.17.13	yes	64
91.82	odd	6	inner	273.2.y.b.173.13	yes	64
273.173	even	6	inner	273.2.y.b.173.20	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.y.b.101.13	✓	64	3.2	odd	2	inner
273.2.y.b.101.20	yes	64	1.1	even	1	trivial
273.2.y.b.173.13	yes	64	91.82	odd	6	inner
273.2.y.b.173.20	yes	64	273.173	even	6	inner
273.2.br.b.17.13	yes	64	39.17	odd	6
273.2.br.b.17.20	yes	64	13.4	even	6
273.2.br.b.257.13	yes	64	7.5	odd	6
273.2.br.b.257.20	yes	64	21.5	even	6