Embedded newform 536.2.bf.a.21.19

• Label: 536.2.bf.a.21.19
• 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			21.19
Character	\(\chi\)	\(=\)	536.21
Dual form			536.2.bf.a.485.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.931653 + 1.06397i) q^{2} +(0.275175 + 0.125668i) q^{3} +(-0.264044 - 1.98249i) q^{4} +(-0.822547 + 2.80134i) q^{5} +(-0.390075 + 0.175697i) q^{6} +(-1.37905 + 3.98451i) q^{7} +(2.35530 + 1.56606i) q^{8} +(-1.90465 - 2.19809i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1320 q - 19 q^{2} - 23 q^{4} - 23 q^{6} - 38 q^{7} + 5 q^{8} + 88 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.931653	+	1.06397i	−0.658779	+	0.752337i
							\(3\)	0.275175	+	0.125668i	0.158872	+	0.0725546i	0.493264	−	0.869880i	\(-0.335804\pi\)
−0.334392	+	0.942434i	\(0.608531\pi\)
\(5\)	−0.822547	+	2.80134i	−0.367854	+	1.25280i	0.542882	+	0.839809i	\(0.317333\pi\)
							−0.910737	+	0.412988i	\(0.864485\pi\)
\(7\)	−1.37905	+	3.98451i	−0.521233	+	1.50600i	0.310360	+	0.950619i	\(0.399550\pi\)
							−0.831593	+	0.555385i	\(0.812571\pi\)
\(11\)	0.313326	−	0.0760122i	0.0944715	−	0.0229185i	−0.188245	−	0.982122i	\(-0.560280\pi\)
							0.282716	+	0.959204i	\(0.408765\pi\)
\(13\)	0.535137	+	1.03802i	0.148420	+	0.287895i	0.951312	−	0.308230i	\(-0.0997365\pi\)
							−0.802892	+	0.596125i	\(0.796706\pi\)
\(17\)	−2.57509	−	1.03091i	−0.624550	−	0.250032i	0.0377305	−	0.999288i	\(-0.487987\pi\)
							−0.662281	+	0.749256i	\(0.730411\pi\)
\(19\)	4.03539	−	1.39666i	0.925783	−	0.320417i	0.177748	−	0.984076i	\(-0.443119\pi\)
							0.748035	+	0.663660i	\(0.230998\pi\)
\(23\)	−3.46982	−	4.87267i	−0.723507	−	1.01602i	−0.998509	−	0.0545794i	\(-0.982618\pi\)
							0.275003	−	0.961443i	\(-0.411321\pi\)
\(29\)	−6.01122	+	3.47058i	−1.11626	+	0.644471i	−0.940442	−	0.339953i	\(-0.889589\pi\)
							−0.175813	+	0.984424i	\(0.556256\pi\)
\(31\)	−1.67564	−	0.863854i	−0.300954	−	0.155153i	0.301135	−	0.953582i	\(-0.402635\pi\)
							−0.602089	+	0.798429i	\(0.705665\pi\)
\(37\)	3.40420	+	1.96542i	0.559647	+	0.323112i	0.753004	−	0.658016i	\(-0.228604\pi\)
							−0.193357	+	0.981129i	\(0.561937\pi\)
\(41\)	−9.34717	+	7.35069i	−1.45978	+	1.14799i	−0.500429	+	0.865778i	\(0.666824\pi\)
							−0.959353	+	0.282207i	\(0.908933\pi\)
\(43\)	−1.94260	+	0.279304i	−0.296244	+	0.0425935i	−0.288834	−	0.957379i	\(-0.593268\pi\)
							−0.00740967	+	0.999973i	\(0.502359\pi\)
\(47\)	−9.35448	+	0.893244i	−1.36449	+	0.130293i	−0.751467	−	0.659771i	\(-0.770653\pi\)
							−0.613024	+	0.790064i	\(0.710047\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	536.2.bf.a.21.19	yes	1320
8.5	even	2	inner	536.2.bf.a.21.11	✓	1320
67.16	even	33	inner	536.2.bf.a.485.11	yes	1320
536.485	even	66	inner	536.2.bf.a.485.19	yes	1320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
536.2.bf.a.21.11	✓	1320	8.5	even	2	inner
536.2.bf.a.21.19	yes	1320	1.1	even	1	trivial
536.2.bf.a.485.11	yes	1320	67.16	even	33	inner
536.2.bf.a.485.19	yes	1320	536.485	even	66	inner