Embedded newform 547.2.f.a.46.11

• Label: 547.2.f.a.46.11
• Level: $547$
• Weight: $2$
• Character: 547.46
• Analytic conductor: $4.368$
• Analytic rank: $0$
• Dimension: $528$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 547 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	547.f (of order \(13\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.36781699056\)
Analytic rank:	\(0\)
Dimension:	\(528\)
Relative dimension:	\(44\) over \(\Q(\zeta_{13})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{13}]$

Embedding invariants

Embedding label			46.11
Character	\(\chi\)	\(=\)	547.46
Dual form			547.2.f.a.440.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.188009 + 1.54840i) q^{2} -0.908881 q^{3} +(-0.420298 - 0.103594i) q^{4} +(-0.0244243 - 0.0644017i) q^{5} +(0.170878 - 1.40731i) q^{6} +(-0.367947 - 3.03032i) q^{7} +(-0.866778 + 2.28551i) q^{8} -2.17394 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 528 q - 8 q^{2} - 46 q^{3} - 50 q^{4} - 9 q^{5} - 21 q^{6} - 3 q^{7} + 14 q^{8} + 466 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/547\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{7}{13}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.188009	+	1.54840i	−0.132943	+	1.09488i	0.761647	+	0.647992i	\(0.224391\pi\)
							−0.894590	+	0.446889i	\(0.852532\pi\)
\(3\)	−0.908881			−0.524743			−0.262371	−	0.964967i	\(-0.584505\pi\)
							−0.262371	+	0.964967i	\(0.584505\pi\)
\(5\)	−0.0244243	−	0.0644017i	−0.0109229	−	0.0288013i	0.929444	−	0.368963i	\(-0.120287\pi\)
							−0.940367	+	0.340162i	\(0.889518\pi\)
\(7\)	−0.367947	−	3.03032i	−0.139071	−	1.14535i	−0.880407	−	0.474218i	\(-0.842731\pi\)
							0.741336	−	0.671134i	\(-0.234192\pi\)
\(11\)	2.43748	−	3.53129i	0.734927	−	1.06473i	−0.260145	−	0.965569i	\(-0.583770\pi\)
							0.995072	−	0.0991556i	\(-0.0316142\pi\)
\(13\)	−5.10339			−1.41543			−0.707713	−	0.706500i	\(-0.750273\pi\)
							−0.707713	+	0.706500i	\(0.750273\pi\)
\(17\)	−3.02968	−	4.38925i	−0.734805	−	1.06455i	−0.995086	−	0.0990159i	\(-0.968431\pi\)
							0.260281	−	0.965533i	\(-0.416185\pi\)
\(19\)	2.33565	−	1.22585i	0.535835	−	0.281228i	−0.175022	−	0.984565i	\(-0.556000\pi\)
							0.710857	+	0.703336i	\(0.248307\pi\)
\(23\)	−6.19747	+	5.49048i	−1.29226	+	1.14484i	−0.311370	+	0.950289i	\(0.600788\pi\)
							−0.980892	+	0.194555i	\(0.937674\pi\)
\(29\)	−1.73081	−	1.53336i	−0.321402	−	0.284738i	0.486958	−	0.873425i	\(-0.338106\pi\)
							−0.808361	+	0.588688i	\(0.799645\pi\)
\(31\)	−7.03887	−	3.69429i	−1.26422	−	0.663513i	−0.306679	−	0.951813i	\(-0.599218\pi\)
							−0.957540	+	0.288300i	\(0.906910\pi\)
\(37\)	−4.90074	−	4.34168i	−0.805677	−	0.713767i	0.156059	−	0.987748i	\(-0.450121\pi\)
							−0.961735	+	0.273981i	\(0.911660\pi\)
\(41\)	9.85025			1.53835			0.769175	−	0.639038i	\(-0.220667\pi\)
							0.769175	+	0.639038i	\(0.220667\pi\)
\(43\)	4.57063	+	1.12656i	0.697014	+	0.171799i	0.571875	−	0.820341i	\(-0.306216\pi\)
							0.125139	+	0.992139i	\(0.460062\pi\)
\(47\)	5.58274	+	4.94588i	0.814327	+	0.721431i	0.963595	−	0.267366i	\(-0.0861533\pi\)
							−0.149268	+	0.988797i	\(0.547692\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.2.f.a.46.11	✓	528
547.440	even	13	inner	547.2.f.a.440.11	yes	528

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.f.a.46.11	✓	528	1.1	even	1	trivial
547.2.f.a.440.11	yes	528	547.440	even	13	inner