Embedded newform 273.2.bf.b.152.8

• Label: 273.2.bf.b.152.8
• Level: $273$
• Weight: $2$
• Character: 273.152
• 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.bf (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			152.8
Character	\(\chi\)	\(=\)	273.152
Dual form			273.2.bf.b.185.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.60258 + 0.925247i) q^{2} +(0.929062 + 1.46179i) q^{3} +(0.712166 - 1.23351i) q^{4} +(1.21731 - 2.10844i) q^{5} +(-2.84141 - 1.48302i) q^{6} +(-0.567699 + 2.58413i) q^{7} -1.06527i q^{8} +(-1.27369 + 2.71620i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 32 q^{4} - 12 q^{6} - 4 q^{7} + 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(92\)	\(106\)	\(157\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.60258	+	0.925247i	−1.13319	+	0.654249i	−0.944736	−	0.327833i	\(-0.893682\pi\)
							−0.188456	+	0.982082i	\(0.560348\pi\)
\(3\)	0.929062	+	1.46179i	0.536394	+	0.843968i
							\(5\)	1.21731	−	2.10844i	0.544398	−	0.942924i	−0.454247	−	0.890876i	\(-0.650092\pi\)
0.998645	−	0.0520486i	\(-0.0165751\pi\)
\(7\)	−0.567699	+	2.58413i	−0.214570	+	0.976709i
							\(11\)			2.91044i			0.877530i	0.898602	+	0.438765i	\(0.144584\pi\)
−0.898602	+	0.438765i	\(0.855416\pi\)
\(13\)	2.41102	+	2.68085i	0.668697	+	0.743535i
							\(17\)	−0.467925	+	0.810471i	−0.113489	+	0.196568i	−0.917175	−	0.398486i	\(-0.869536\pi\)
0.803686	+	0.595054i	\(0.202869\pi\)
\(19\)		−	2.21042i		−	0.507105i	−0.967322	−	0.253552i	\(-0.918401\pi\)
							0.967322	−	0.253552i	\(-0.0815990\pi\)
\(23\)	−5.40559	+	3.12092i	−1.12714	+	0.650756i	−0.943215	−	0.332183i	\(-0.892215\pi\)
							−0.183928	+	0.982940i	\(0.558881\pi\)
\(29\)	1.34529	+	0.776702i	0.249814	+	0.144230i	0.619679	−	0.784855i	\(-0.287263\pi\)
							−0.369865	+	0.929085i	\(0.620596\pi\)
\(31\)	2.50768	−	1.44781i	0.450392	−	0.260034i	−0.257604	−	0.966251i	\(-0.582933\pi\)
							0.707996	+	0.706217i	\(0.249600\pi\)
\(37\)	−4.07926	−	7.06548i	−0.670626	−	1.16156i	−0.977727	−	0.209882i	\(-0.932692\pi\)
							0.307101	−	0.951677i	\(-0.400641\pi\)
\(41\)	4.41024	−	7.63876i	0.688764	−	1.19297i	−0.283475	−	0.958980i	\(-0.591487\pi\)
							0.972238	−	0.233994i	\(-0.0751795\pi\)
\(43\)	5.40029	+	9.35358i	0.823537	+	1.42641i	0.903033	+	0.429572i	\(0.141336\pi\)
							−0.0794962	+	0.996835i	\(0.525331\pi\)
\(47\)	−3.63232	+	6.29136i	−0.529828	+	0.917689i	0.469567	+	0.882897i	\(0.344410\pi\)
							−0.999395	+	0.0347919i	\(0.988923\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.bf.b.152.8	yes	64
3.2	odd	2	inner	273.2.bf.b.152.25	yes	64
7.3	odd	6		273.2.r.b.269.8	yes	64
13.3	even	3		273.2.r.b.68.8	✓	64
21.17	even	6		273.2.r.b.269.25	yes	64
39.29	odd	6		273.2.r.b.68.25	yes	64
91.3	odd	6	inner	273.2.bf.b.185.25	yes	64
273.185	even	6	inner	273.2.bf.b.185.8	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.r.b.68.8	✓	64	13.3	even	3
273.2.r.b.68.25	yes	64	39.29	odd	6
273.2.r.b.269.8	yes	64	7.3	odd	6
273.2.r.b.269.25	yes	64	21.17	even	6
273.2.bf.b.152.8	yes	64	1.1	even	1	trivial
273.2.bf.b.152.25	yes	64	3.2	odd	2	inner
273.2.bf.b.185.8	yes	64	273.185	even	6	inner
273.2.bf.b.185.25	yes	64	91.3	odd	6	inner