Embedded newform 536.2.bf.a.21.7

• Label: 536.2.bf.a.21.7
• 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			21.7
Character	\(\chi\)	\(=\)	536.21
Dual form			536.2.bf.a.485.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.36003 - 0.387694i) q^{2} +(0.794198 + 0.362698i) q^{3} +(1.69939 + 1.05456i) q^{4} +(-0.649579 + 2.21226i) q^{5} +(-0.939521 - 0.801188i) q^{6} +(1.59465 - 4.60745i) q^{7} +(-1.90238 - 2.09307i) q^{8} +(-1.46538 - 1.69114i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1320 q - 19 q^{2} - 23 q^{4} - 23 q^{6} - 38 q^{7} + 5 q^{8} + 88 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.36003	−	0.387694i	−0.961689	−	0.274141i
							\(3\)	0.794198	+	0.362698i	0.458531	+	0.209404i	0.631269	−	0.775564i	\(-0.282534\pi\)
−0.172739	+	0.984968i	\(0.555262\pi\)
\(5\)	−0.649579	+	2.21226i	−0.290500	+	0.989353i	0.676894	+	0.736080i	\(0.263325\pi\)
							−0.967395	+	0.253273i	\(0.918493\pi\)
\(7\)	1.59465	−	4.60745i	0.602722	−	1.74145i	−0.0658775	−	0.997828i	\(-0.520985\pi\)
							0.668599	−	0.743623i	\(-0.266894\pi\)
\(11\)	−1.51742	+	0.368123i	−0.457520	+	0.110993i	−0.457894	−	0.889007i	\(-0.651396\pi\)
							0.000374103		1.00000i	\(0.499881\pi\)
\(13\)	−1.76077	−	3.41542i	−0.488349	−	0.947266i	−0.996305	−	0.0858826i	\(-0.972629\pi\)
							0.507956	−	0.861383i	\(-0.330401\pi\)
\(17\)	−6.54467	−	2.62009i	−1.58732	−	0.635465i	−0.600839	−	0.799370i	\(-0.705167\pi\)
							−0.986476	+	0.163904i	\(0.947591\pi\)
\(19\)	3.13370	−	1.08458i	0.718920	−	0.248821i	0.0569777	−	0.998375i	\(-0.481854\pi\)
							0.661942	+	0.749555i	\(0.269732\pi\)
\(23\)	−1.62909	−	2.28773i	−0.339688	−	0.477025i	0.609031	−	0.793147i	\(-0.291559\pi\)
							−0.948719	+	0.316122i	\(0.897619\pi\)
\(29\)	4.51548	−	2.60701i	0.838503	−	0.484110i	−0.0182523	−	0.999833i	\(-0.505810\pi\)
							0.856755	+	0.515724i	\(0.172477\pi\)
\(31\)	6.32147	+	3.25894i	1.13537	+	0.585323i	0.920276	−	0.391270i	\(-0.127964\pi\)
							0.215093	+	0.976594i	\(0.430995\pi\)
\(37\)	−0.358791	−	0.207148i	−0.0589849	−	0.0340549i	0.470218	−	0.882551i	\(-0.344175\pi\)
							−0.529202	+	0.848496i	\(0.677509\pi\)
\(41\)	1.38721	−	1.09091i	0.216646	−	0.170372i	−0.503938	−	0.863740i	\(-0.668116\pi\)
							0.720584	+	0.693367i	\(0.243874\pi\)
\(43\)	−11.2662	+	1.61983i	−1.71808	+	0.247022i	−0.929738	−	0.368222i	\(-0.879967\pi\)
							−0.788337	+	0.615243i	\(0.789058\pi\)
\(47\)	−0.299693	+	0.0286172i	−0.0437146	+	0.00417424i	−0.116890	−	0.993145i	\(-0.537293\pi\)
							0.0731758	+	0.997319i	\(0.476687\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	536.2.bf.a.21.7	✓	1320
8.5	even	2	inner	536.2.bf.a.21.14	yes	1320
67.16	even	33	inner	536.2.bf.a.485.14	yes	1320
536.485	even	66	inner	536.2.bf.a.485.7	yes	1320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
536.2.bf.a.21.7	✓	1320	1.1	even	1	trivial
536.2.bf.a.21.14	yes	1320	8.5	even	2	inner
536.2.bf.a.485.7	yes	1320	536.485	even	66	inner
536.2.bf.a.485.14	yes	1320	67.16	even	33	inner