Embedded newform 536.2.bf.a.21.52

• Label: 536.2.bf.a.21.52
• Level: $536$
• Weight: $2$
• Character: 536.21
• Analytic conductor: $4.280$
• Analytic rank: $0$
• Dimension: $1320$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 536 = 2^{3} \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	536.bf (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.27998154834\)
Analytic rank:	\(0\)
Dimension:	\(1320\)
Relative dimension:	\(66\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			21.52
Character	\(\chi\)	\(=\)	536.21
Dual form			536.2.bf.a.485.52

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.975309 - 1.02410i) q^{2} +(1.54756 + 0.706748i) q^{3} +(-0.0975454 - 1.99762i) q^{4} +(-0.760278 + 2.58927i) q^{5} +(2.23313 - 0.895555i) q^{6} +(1.06116 - 3.06602i) q^{7} +(-2.14089 - 1.84840i) q^{8} +(-0.0691271 - 0.0797770i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1320 q - 19 q^{2} - 23 q^{4} - 23 q^{6} - 38 q^{7} + 5 q^{8} + 88 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/536\mathbb{Z}\right)^\times\).

\(n\)	\(135\)	\(269\)	\(337\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{31}{33}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.975309	−	1.02410i	0.689647	−	0.724145i
							\(3\)	1.54756	+	0.706748i	0.893485	+	0.408041i	0.808606	−	0.588350i	\(-0.200222\pi\)
0.0848790	+	0.996391i	\(0.472950\pi\)
\(5\)	−0.760278	+	2.58927i	−0.340007	+	1.15796i	0.595118	+	0.803638i	\(0.297105\pi\)
							−0.935125	+	0.354318i	\(0.884713\pi\)
\(7\)	1.06116	−	3.06602i	0.401080	−	1.15885i	−0.545982	−	0.837797i	\(-0.683843\pi\)
							0.947063	−	0.321049i	\(-0.104035\pi\)
\(11\)	5.30128	−	1.28608i	1.59840	−	0.387767i	0.665066	−	0.746784i	\(-0.268403\pi\)
							0.933331	+	0.359017i	\(0.116888\pi\)
\(13\)	2.03410	+	3.94561i	0.564158	+	1.09431i	0.982541	+	0.186044i	\(0.0595667\pi\)
							−0.418383	+	0.908271i	\(0.637403\pi\)
\(17\)	1.95619	+	0.783142i	0.474447	+	0.189940i	0.596547	−	0.802578i	\(-0.296539\pi\)
							−0.122101	+	0.992518i	\(0.538963\pi\)
\(19\)	−0.522389	+	0.180801i	−0.119844	+	0.0414785i	−0.386332	−	0.922360i	\(-0.626258\pi\)
							0.266488	+	0.963838i	\(0.414137\pi\)
\(23\)	−4.15887	−	5.84031i	−0.867184	−	1.21779i	−0.974685	−	0.223584i	\(-0.928224\pi\)
							0.107501	−	0.994205i	\(-0.465715\pi\)
\(29\)	−9.08631	+	5.24598i	−1.68728	+	0.974154i	−0.730700	+	0.682698i	\(0.760806\pi\)
							−0.956584	+	0.291456i	\(0.905860\pi\)
\(31\)	8.11445	+	4.18329i	1.45740	+	0.751341i	0.991079	−	0.133276i	\(-0.0425496\pi\)
							0.466319	+	0.884617i	\(0.345580\pi\)
\(37\)	3.91204	+	2.25862i	0.643135	+	0.371314i	0.785821	−	0.618454i	\(-0.212241\pi\)
							−0.142686	+	0.989768i	\(0.545574\pi\)
\(41\)	−0.00135631	+	0.00106662i	−0.000211820	+	0.000166577i	−0.618265	−	0.785970i	\(-0.712164\pi\)
							0.618053	+	0.786136i	\(0.287922\pi\)
\(43\)	−7.24934	+	1.04230i	−1.10551	+	0.158949i	−0.670813	−	0.741626i	\(-0.734055\pi\)
							−0.434701	+	0.900575i	\(0.643146\pi\)
\(47\)	−5.67578	+	0.541971i	−0.827897	+	0.0790546i	−0.500397	−	0.865796i	\(-0.666812\pi\)
							−0.327501	+	0.944851i	\(0.606206\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	536.2.bf.a.21.52	✓	1320
8.5	even	2	inner	536.2.bf.a.21.59	yes	1320
67.16	even	33	inner	536.2.bf.a.485.59	yes	1320
536.485	even	66	inner	536.2.bf.a.485.52	yes	1320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
536.2.bf.a.21.52	✓	1320	1.1	even	1	trivial
536.2.bf.a.21.59	yes	1320	8.5	even	2	inner
536.2.bf.a.485.52	yes	1320	536.485	even	66	inner
536.2.bf.a.485.59	yes	1320	67.16	even	33	inner