Embedded newform 273.2.bf.b.152.19

• Label: 273.2.bf.b.152.19
• 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			152.19
Character	\(\chi\)	\(=\)	273.152
Dual form			273.2.bf.b.185.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.667801 - 0.385555i) q^{2} +(-1.00894 - 1.40785i) q^{3} +(-0.702694 + 1.21710i) q^{4} +(-0.907647 + 1.57209i) q^{5} +(-1.21657 - 0.551162i) q^{6} +(1.94952 + 1.78868i) q^{7} +2.62593i q^{8} +(-0.964085 + 2.84087i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 32 q^{4} - 12 q^{6} - 4 q^{7} + 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.667801	−	0.385555i	0.472207	−	0.272629i	−0.244956	−	0.969534i	\(-0.578774\pi\)
							0.717163	+	0.696905i	\(0.245440\pi\)
\(3\)	−1.00894	−	1.40785i	−0.582511	−	0.812823i
							\(5\)	−0.907647	+	1.57209i	−0.405912	+	0.703060i	−0.994427	−	0.105426i	\(-0.966379\pi\)
0.588515	+	0.808486i	\(0.299713\pi\)
\(7\)	1.94952	+	1.78868i	0.736848	+	0.676058i
							\(11\)			0.582432i			0.175610i	0.996138	+	0.0878049i	\(0.0279852\pi\)
−0.996138	+	0.0878049i	\(0.972015\pi\)
\(13\)	3.53127	+	0.728085i	0.979399	+	0.201935i
							\(17\)	0.479095	−	0.829817i	0.116198	−	0.201260i	−0.802060	−	0.597243i	\(-0.796263\pi\)
0.918258	+	0.395983i	\(0.129596\pi\)
\(19\)		−	1.92492i		−	0.441608i	−0.975318	−	0.220804i	\(-0.929132\pi\)
							0.975318	−	0.220804i	\(-0.0708681\pi\)
\(23\)	−7.56166	+	4.36573i	−1.57672	+	0.910317i	−0.581402	+	0.813617i	\(0.697495\pi\)
							−0.995314	+	0.0967001i	\(0.969171\pi\)
\(29\)	4.26919	+	2.46482i	0.792769	+	0.457705i	0.840936	−	0.541134i	\(-0.182005\pi\)
							−0.0481677	+	0.998839i	\(0.515338\pi\)
\(31\)	−2.83579	+	1.63724i	−0.509322	+	0.294057i	−0.732555	−	0.680708i	\(-0.761672\pi\)
							0.223233	+	0.974765i	\(0.428339\pi\)
\(37\)	0.899986	+	1.55882i	0.147957	+	0.256269i	0.930472	−	0.366363i	\(-0.119397\pi\)
							−0.782515	+	0.622631i	\(0.786064\pi\)
\(41\)	1.73088	−	2.99798i	0.270319	−	0.468205i	−0.698625	−	0.715488i	\(-0.746204\pi\)
							0.968943	+	0.247283i	\(0.0795377\pi\)
\(43\)	−0.367529	−	0.636578i	−0.0560476	−	0.0970772i	0.836640	−	0.547753i	\(-0.184517\pi\)
							−0.892688	+	0.450675i	\(0.851183\pi\)
\(47\)	3.49201	−	6.04834i	0.509362	−	0.882241i	−0.490579	−	0.871397i	\(-0.663215\pi\)
							0.999941	−	0.0108447i	\(-0.00345203\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.bf.b.152.19	yes	64
3.2	odd	2	inner	273.2.bf.b.152.14	yes	64
7.3	odd	6		273.2.r.b.269.19	yes	64
13.3	even	3		273.2.r.b.68.19	yes	64
21.17	even	6		273.2.r.b.269.14	yes	64
39.29	odd	6		273.2.r.b.68.14	✓	64
91.3	odd	6	inner	273.2.bf.b.185.14	yes	64
273.185	even	6	inner	273.2.bf.b.185.19	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.r.b.68.14	✓	64	39.29	odd	6
273.2.r.b.68.19	yes	64	13.3	even	3
273.2.r.b.269.14	yes	64	21.17	even	6
273.2.r.b.269.19	yes	64	7.3	odd	6
273.2.bf.b.152.14	yes	64	3.2	odd	2	inner
273.2.bf.b.152.19	yes	64	1.1	even	1	trivial
273.2.bf.b.185.14	yes	64	91.3	odd	6	inner
273.2.bf.b.185.19	yes	64	273.185	even	6	inner