Embedded newform 536.2.bf.a.21.34

• Label: 536.2.bf.a.21.34
• 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			21.34
Character	\(\chi\)	\(=\)	536.21
Dual form			536.2.bf.a.485.34

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.111112 - 1.40984i) q^{2} +(0.583153 + 0.266317i) q^{3} +(-1.97531 + 0.313299i) q^{4} +(0.251516 - 0.856586i) q^{5} +(0.310670 - 0.851744i) q^{6} +(0.423817 - 1.22454i) q^{7} +(0.661182 + 2.75006i) q^{8} +(-1.69544 - 1.95664i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1320 q - 19 q^{2} - 23 q^{4} - 23 q^{6} - 38 q^{7} + 5 q^{8} + 88 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.111112	−	1.40984i	−0.0785677	−	0.996909i
							\(3\)	0.583153	+	0.266317i	0.336683	+	0.153758i	0.576581	−	0.817040i	\(-0.304387\pi\)
−0.239898	+	0.970798i	\(0.577114\pi\)
\(5\)	0.251516	−	0.856586i	0.112482	−	0.383077i	−0.883940	−	0.467600i	\(-0.845119\pi\)
							0.996421	+	0.0845234i	\(0.0269368\pi\)
\(7\)	0.423817	−	1.22454i	0.160188	−	0.462832i	−0.836134	−	0.548525i	\(-0.815189\pi\)
							0.996322	+	0.0856935i	\(0.0273106\pi\)
\(11\)	4.42426	−	1.07331i	1.33397	−	0.323617i	0.495518	−	0.868598i	\(-0.334978\pi\)
							0.838448	+	0.544981i	\(0.183463\pi\)
\(13\)	−0.306599	−	0.594718i	−0.0850352	−	0.164945i	0.842483	−	0.538722i	\(-0.181093\pi\)
							−0.927519	+	0.373777i	\(0.878062\pi\)
\(17\)	−3.86106	−	1.54574i	−0.936446	−	0.374896i	−0.147293	−	0.989093i	\(-0.547056\pi\)
							−0.789153	+	0.614197i	\(0.789480\pi\)
\(19\)	−0.115068	+	0.0398254i	−0.0263984	+	0.00913656i	−0.340235	−	0.940340i	\(-0.610507\pi\)
							0.313837	+	0.949477i	\(0.398385\pi\)
\(23\)	−3.63150	−	5.09972i	−0.757219	−	1.06337i	−0.995738	−	0.0922279i	\(-0.970601\pi\)
							0.238519	−	0.971138i	\(-0.423338\pi\)
\(29\)	1.57469	−	0.909148i	0.292413	−	0.168825i	−0.346617	−	0.938007i	\(-0.612670\pi\)
							0.639029	+	0.769182i	\(0.279336\pi\)
\(31\)	−7.23908	−	3.73200i	−1.30018	−	0.670287i	−0.337598	−	0.941290i	\(-0.609614\pi\)
							−0.962579	+	0.271003i	\(0.912645\pi\)
\(37\)	−2.12762	−	1.22838i	−0.349778	−	0.201945i	0.314809	−	0.949155i	\(-0.398059\pi\)
							−0.664588	+	0.747210i	\(0.731393\pi\)
\(41\)	5.04516	−	3.96756i	0.787921	−	0.619628i	−0.140956	−	0.990016i	\(-0.545018\pi\)
							0.928877	+	0.370388i	\(0.120775\pi\)
\(43\)	0.0424392	−	0.00610184i	0.00647192	−	0.000930522i	−0.139078	−	0.990281i	\(-0.544414\pi\)
							0.145550	+	0.989351i	\(0.453505\pi\)
\(47\)	−4.62694	+	0.441819i	−0.674908	+	0.0644459i	−0.426882	−	0.904307i	\(-0.640388\pi\)
							−0.248026	+	0.968753i	\(0.579782\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	536.2.bf.a.21.34	✓	1320
8.5	even	2	inner	536.2.bf.a.21.42	yes	1320
67.16	even	33	inner	536.2.bf.a.485.42	yes	1320
536.485	even	66	inner	536.2.bf.a.485.34	yes	1320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
536.2.bf.a.21.34	✓	1320	1.1	even	1	trivial
536.2.bf.a.21.42	yes	1320	8.5	even	2	inner
536.2.bf.a.485.34	yes	1320	536.485	even	66	inner
536.2.bf.a.485.42	yes	1320	67.16	even	33	inner