Embedded newform 273.2.bf.b.152.15

• Label: 273.2.bf.b.152.15
• 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			152.15
Character	\(\chi\)	\(=\)	273.152
Dual form			273.2.bf.b.185.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.473030 + 0.273104i) q^{2} +(0.966120 + 1.43757i) q^{3} +(-0.850828 + 1.47368i) q^{4} +(-1.25657 + 2.17644i) q^{5} +(-0.849611 - 0.416164i) q^{6} +(2.53535 + 0.756296i) q^{7} -2.02188i q^{8} +(-1.13323 + 2.77773i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 32 q^{4} - 12 q^{6} - 4 q^{7} + 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.473030	+	0.273104i	−0.334483	+	0.193114i	−0.657830	−	0.753167i	\(-0.728525\pi\)
							0.323347	+	0.946281i	\(0.395192\pi\)
\(3\)	0.966120	+	1.43757i	0.557789	+	0.829982i
							\(5\)	−1.25657	+	2.17644i	−0.561955	+	0.973335i	0.435371	+	0.900251i	\(0.356617\pi\)
−0.997326	+	0.0730836i	\(0.976716\pi\)
\(7\)	2.53535	+	0.756296i	0.958273	+	0.285853i
							\(11\)		−	6.02763i		−	1.81740i	−0.417450	−	0.908700i	\(-0.637076\pi\)
0.417450	−	0.908700i	\(-0.362924\pi\)
\(13\)	−1.89716	+	3.06607i	−0.526176	+	0.850376i
							\(17\)	−0.323651	+	0.560581i	−0.0784970	+	0.135961i	−0.902602	−	0.430477i	\(-0.858345\pi\)
0.824105	+	0.566438i	\(0.191679\pi\)
\(19\)			2.29555i			0.526636i	0.964709	+	0.263318i	\(0.0848168\pi\)
							−0.964709	+	0.263318i	\(0.915183\pi\)
\(23\)	−0.894874	+	0.516656i	−0.186594	+	0.107730i	−0.590387	−	0.807120i	\(-0.701025\pi\)
							0.403793	+	0.914850i	\(0.367692\pi\)
\(29\)	4.46255	+	2.57646i	0.828676	+	0.478436i	0.853399	−	0.521258i	\(-0.174537\pi\)
							−0.0247234	+	0.999694i	\(0.507871\pi\)
\(31\)	9.25745	−	5.34479i	1.66269	−	0.959953i	0.691264	−	0.722602i	\(-0.257054\pi\)
							0.971424	−	0.237351i	\(-0.0762792\pi\)
\(37\)	1.97766	+	3.42541i	0.325126	+	0.563134i	0.981538	−	0.191268i	\(-0.0612601\pi\)
							−0.656412	+	0.754402i	\(0.727927\pi\)
\(41\)	−4.48689	+	7.77153i	−0.700735	+	1.21371i	0.267474	+	0.963565i	\(0.413811\pi\)
							−0.968209	+	0.250143i	\(0.919522\pi\)
\(43\)	−4.09951	−	7.10056i	−0.625169	−	1.08282i	−0.988508	−	0.151168i	\(-0.951697\pi\)
							0.363339	−	0.931657i	\(-0.381637\pi\)
\(47\)	−2.09646	+	3.63118i	−0.305800	+	0.529662i	−0.977439	−	0.211217i	\(-0.932257\pi\)
							0.671639	+	0.740879i	\(0.265591\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.bf.b.152.15	yes	64
3.2	odd	2	inner	273.2.bf.b.152.18	yes	64
7.3	odd	6		273.2.r.b.269.15	yes	64
13.3	even	3		273.2.r.b.68.15	✓	64
21.17	even	6		273.2.r.b.269.18	yes	64
39.29	odd	6		273.2.r.b.68.18	yes	64
91.3	odd	6	inner	273.2.bf.b.185.18	yes	64
273.185	even	6	inner	273.2.bf.b.185.15	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.r.b.68.15	✓	64	13.3	even	3
273.2.r.b.68.18	yes	64	39.29	odd	6
273.2.r.b.269.15	yes	64	7.3	odd	6
273.2.r.b.269.18	yes	64	21.17	even	6
273.2.bf.b.152.15	yes	64	1.1	even	1	trivial
273.2.bf.b.152.18	yes	64	3.2	odd	2	inner
273.2.bf.b.185.15	yes	64	273.185	even	6	inner
273.2.bf.b.185.18	yes	64	91.3	odd	6	inner