Embedded newform 273.2.br.b.17.1

• Label: 273.2.br.b.17.1
• 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			17.1
Character	\(\chi\)	\(=\)	273.17
Dual form			273.2.br.b.257.1

$q$-expansion

\(f(q)\)	\(=\)	\(q-2.61237 q^{2} +(1.05832 + 1.37112i) q^{3} +4.82446 q^{4} +(-0.648305 + 0.374299i) q^{5} +(-2.76471 - 3.58186i) q^{6} +(-2.10140 + 1.60752i) q^{7} -7.37851 q^{8} +(-0.759929 + 2.90216i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 6 q^{3} + 56 q^{4} - 6 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.61237			−1.84722			−0.923611	−	0.383331i	\(-0.874777\pi\)
							−0.923611	+	0.383331i	\(0.874777\pi\)
\(3\)	1.05832	+	1.37112i	0.611020	+	0.791615i
							\(5\)	−0.648305	+	0.374299i	−0.289931	+	0.167392i	−0.637911	−	0.770110i	\(-0.720201\pi\)
0.347980	+	0.937502i	\(0.386868\pi\)
\(7\)	−2.10140	+	1.60752i	−0.794253	+	0.607587i
							\(11\)	−0.0727716	−	0.126044i	−0.0219415	−	0.0380037i	0.854846	−	0.518881i	\(-0.173651\pi\)
−0.876788	+	0.480878i	\(0.840318\pi\)
\(13\)	−3.45217	+	1.04045i	−0.957459	+	0.288569i
							\(17\)	0.900163			0.218321			0.109161	−	0.994024i	\(-0.465184\pi\)
0.109161	+	0.994024i	\(0.465184\pi\)
\(19\)	2.39024	−	4.14001i	0.548358	−	0.949785i	−0.450029	−	0.893014i	\(-0.648586\pi\)
							0.998387	−	0.0567705i	\(-0.0180803\pi\)
\(23\)			2.97434i			0.620192i	0.950705	+	0.310096i	\(0.100361\pi\)
							−0.950705	+	0.310096i	\(0.899639\pi\)
\(29\)	−7.14470	−	4.12499i	−1.32674	−	0.765992i	−0.341944	−	0.939721i	\(-0.611085\pi\)
							−0.984794	+	0.173728i	\(0.944418\pi\)
\(31\)	−3.66655	+	6.35065i	−0.658531	+	1.14061i	0.322465	+	0.946582i	\(0.395489\pi\)
							−0.980996	+	0.194028i	\(0.937845\pi\)
\(37\)			0.938700i			0.154321i	0.997019	+	0.0771606i	\(0.0245854\pi\)
							−0.997019	+	0.0771606i	\(0.975415\pi\)
\(41\)	5.16704	+	2.98319i	0.806955	+	0.465896i	0.845897	−	0.533346i	\(-0.179065\pi\)
							−0.0389421	+	0.999241i	\(0.512399\pi\)
\(43\)	−3.68427	−	6.38135i	−0.561846	−	0.973147i	−0.997335	−	0.0729525i	\(-0.976758\pi\)
							0.435489	−	0.900194i	\(-0.356575\pi\)
\(47\)	−9.96413	+	5.75279i	−1.45342	+	0.839131i	−0.998673	−	0.0514915i	\(-0.983602\pi\)
							−0.454744	+	0.890622i	\(0.650269\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.br.b.17.1	yes	64
3.2	odd	2	inner	273.2.br.b.17.32	yes	64
7.5	odd	6		273.2.y.b.173.32	yes	64
13.10	even	6		273.2.y.b.101.1	✓	64
21.5	even	6		273.2.y.b.173.1	yes	64
39.23	odd	6		273.2.y.b.101.32	yes	64
91.75	odd	6	inner	273.2.br.b.257.32	yes	64
273.257	even	6	inner	273.2.br.b.257.1	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.y.b.101.1	✓	64	13.10	even	6
273.2.y.b.101.32	yes	64	39.23	odd	6
273.2.y.b.173.1	yes	64	21.5	even	6
273.2.y.b.173.32	yes	64	7.5	odd	6
273.2.br.b.17.1	yes	64	1.1	even	1	trivial
273.2.br.b.17.32	yes	64	3.2	odd	2	inner
273.2.br.b.257.1	yes	64	273.257	even	6	inner
273.2.br.b.257.32	yes	64	91.75	odd	6	inner