Embedded newform 273.2.br.b.257.9

• Label: 273.2.br.b.257.9
• Level: $273$
• Weight: $2$
• Character: 273.257
• 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			257.9
Character	\(\chi\)	\(=\)	273.257
Dual form			273.2.br.b.17.9

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.52138 q^{2} +(-1.64324 - 0.547514i) q^{3} +0.314598 q^{4} +(1.84777 + 1.06681i) q^{5} +(2.49999 + 0.832977i) q^{6} +(-2.46764 - 0.954340i) q^{7} +2.56414 q^{8} +(2.40046 + 1.79939i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 6 q^{3} + 56 q^{4} - 6 q^{7}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.52138			−1.07578			−0.537889	−	0.843016i	\(-0.680778\pi\)
							−0.537889	+	0.843016i	\(0.680778\pi\)
\(3\)	−1.64324	−	0.547514i	−0.948723	−	0.316107i
							\(5\)	1.84777	+	1.06681i	0.826350	+	0.477093i	0.852601	−	0.522562i	\(-0.175024\pi\)
−0.0262514	+	0.999655i	\(0.508357\pi\)
\(7\)	−2.46764	−	0.954340i	−0.932679	−	0.360707i
							\(11\)	−0.983176	+	1.70291i	−0.296439	+	0.513447i	−0.975319	−	0.220802i	\(-0.929132\pi\)
0.678880	+	0.734249i	\(0.262466\pi\)
\(13\)	1.00389	−	3.46298i	0.278430	−	0.960456i
							\(17\)	−2.98019			−0.722802			−0.361401	−	0.932411i	\(-0.617701\pi\)
−0.361401	+	0.932411i	\(0.617701\pi\)
\(19\)	−1.80461	−	3.12568i	−0.414006	−	0.717079i	0.581318	−	0.813677i	\(-0.302537\pi\)
							−0.995324	+	0.0965974i	\(0.969204\pi\)
\(23\)		−	8.06678i		−	1.68204i	−0.541004	−	0.841020i	\(-0.681955\pi\)
							0.541004	−	0.841020i	\(-0.318045\pi\)
\(29\)	−5.67051	+	3.27387i	−1.05299	+	0.607942i	−0.923484	−	0.383637i	\(-0.874672\pi\)
							−0.129503	+	0.991579i	\(0.541338\pi\)
\(31\)	−2.68142	−	4.64436i	−0.481598	−	0.834152i	0.518179	−	0.855272i	\(-0.326610\pi\)
							−0.999777	+	0.0211203i	\(0.993277\pi\)
\(37\)		−	3.66472i		−	0.602476i	−0.953549	−	0.301238i	\(-0.902600\pi\)
							0.953549	−	0.301238i	\(-0.0973998\pi\)
\(41\)	1.26089	−	0.727976i	0.196918	−	0.113691i	−0.398299	−	0.917256i	\(-0.630399\pi\)
							0.595217	+	0.803565i	\(0.297066\pi\)
\(43\)	0.823833	−	1.42692i	0.125633	−	0.217603i	−0.796347	−	0.604840i	\(-0.793237\pi\)
							0.921980	+	0.387237i	\(0.126570\pi\)
\(47\)	7.14926	+	4.12763i	1.04283	+	0.602076i	0.920633	−	0.390430i	\(-0.127674\pi\)
							0.122194	+	0.992506i	\(0.461007\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.br.b.257.9	yes	64
3.2	odd	2	inner	273.2.br.b.257.24	yes	64
7.3	odd	6		273.2.y.b.101.24	yes	64
13.4	even	6		273.2.y.b.173.9	yes	64
21.17	even	6		273.2.y.b.101.9	✓	64
39.17	odd	6		273.2.y.b.173.24	yes	64
91.17	odd	6	inner	273.2.br.b.17.24	yes	64
273.17	even	6	inner	273.2.br.b.17.9	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.y.b.101.9	✓	64	21.17	even	6
273.2.y.b.101.24	yes	64	7.3	odd	6
273.2.y.b.173.9	yes	64	13.4	even	6
273.2.y.b.173.24	yes	64	39.17	odd	6
273.2.br.b.17.9	yes	64	273.17	even	6	inner
273.2.br.b.17.24	yes	64	91.17	odd	6	inner
273.2.br.b.257.9	yes	64	1.1	even	1	trivial
273.2.br.b.257.24	yes	64	3.2	odd	2	inner