Embedded newform 273.2.br.b.17.13

• Label: 273.2.br.b.17.13
• 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			17.13
Character	\(\chi\)	\(=\)	273.17
Dual form			273.2.br.b.257.13

$q$-expansion

\(f(q)\)	\(=\)	\(q-0.732547 q^{2} +(1.45763 + 0.935580i) q^{3} -1.46338 q^{4} +(2.08780 - 1.20539i) q^{5} +(-1.06778 - 0.685356i) q^{6} +(-1.88359 + 1.85798i) q^{7} +2.53708 q^{8} +(1.24938 + 2.72746i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 6 q^{3} + 56 q^{4} - 6 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.732547			−0.517989			−0.258994	−	0.965879i	\(-0.583391\pi\)
							−0.258994	+	0.965879i	\(0.583391\pi\)
\(3\)	1.45763	+	0.935580i	0.841564	+	0.540157i
							\(5\)	2.08780	−	1.20539i	0.933691	−	0.539067i	0.0457138	−	0.998955i	\(-0.485444\pi\)
0.887977	+	0.459888i	\(0.152110\pi\)
\(7\)	−1.88359	+	1.85798i	−0.711929	+	0.702252i
							\(11\)	1.87993	+	3.25613i	0.566820	+	0.981760i	0.996878	+	0.0789593i	\(0.0251597\pi\)
−0.430058	+	0.902801i	\(0.641507\pi\)
\(13\)	2.09909	−	2.93152i	0.582183	−	0.813057i
							\(17\)	0.963275			0.233628			0.116814	−	0.993154i	\(-0.462732\pi\)
0.116814	+	0.993154i	\(0.462732\pi\)
\(19\)	−1.13886	+	1.97257i	−0.261273	+	0.452538i	−0.966581	−	0.256363i	\(-0.917476\pi\)
							0.705307	+	0.708902i	\(0.250809\pi\)
\(23\)		−	0.576714i		−	0.120253i	−0.998191	−	0.0601266i	\(-0.980850\pi\)
							0.998191	−	0.0601266i	\(-0.0191504\pi\)
\(29\)	5.91477	+	3.41489i	1.09834	+	0.634130i	0.935786	−	0.352569i	\(-0.114692\pi\)
							0.162559	+	0.986699i	\(0.448025\pi\)
\(31\)	−0.843529	+	1.46103i	−0.151502	+	0.262410i	−0.931780	−	0.363024i	\(-0.881744\pi\)
							0.780278	+	0.625433i	\(0.215078\pi\)
\(37\)		−	11.2654i		−	1.85202i	−0.377500	−	0.926010i	\(-0.623216\pi\)
							0.377500	−	0.926010i	\(-0.376784\pi\)
\(41\)	−8.39746	−	4.84828i	−1.31146	−	0.757174i	−0.329125	−	0.944286i	\(-0.606754\pi\)
							−0.982338	+	0.187113i	\(0.940087\pi\)
\(43\)	−1.10299	−	1.91043i	−0.168204	−	0.291338i	0.769584	−	0.638545i	\(-0.220463\pi\)
							−0.937789	+	0.347207i	\(0.887130\pi\)
\(47\)	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.br.b.17.13	yes	64
3.2	odd	2	inner	273.2.br.b.17.20	yes	64
7.5	odd	6		273.2.y.b.173.20	yes	64
13.10	even	6		273.2.y.b.101.13	✓	64
21.5	even	6		273.2.y.b.173.13	yes	64
39.23	odd	6		273.2.y.b.101.20	yes	64
91.75	odd	6	inner	273.2.br.b.257.20	yes	64
273.257	even	6	inner	273.2.br.b.257.13	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.y.b.101.13	✓	64	13.10	even	6
273.2.y.b.101.20	yes	64	39.23	odd	6
273.2.y.b.173.13	yes	64	21.5	even	6
273.2.y.b.173.20	yes	64	7.5	odd	6
273.2.br.b.17.13	yes	64	1.1	even	1	trivial
273.2.br.b.17.20	yes	64	3.2	odd	2	inner
273.2.br.b.257.13	yes	64	273.257	even	6	inner
273.2.br.b.257.20	yes	64	91.75	odd	6	inner