Embedded newform 547.2.j.a.11.18

• Label: 547.2.j.a.11.18
• Level: $547$
• Weight: $2$
• Character: 547.11
• Analytic conductor: $4.368$
• Analytic rank: $0$
• Dimension: $1080$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 547 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	547.j (of order \(39\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.36781699056\)
Analytic rank:	\(0\)
Dimension:	\(1080\)
Relative dimension:	\(45\) over \(\Q(\zeta_{39})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{39}]$

Embedding invariants

Embedding label			11.18
Character	\(\chi\)	\(=\)	547.11
Dual form			547.2.j.a.199.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.849270 - 0.361840i) q^{2} +2.50274 q^{3} +(-0.795117 - 0.827805i) q^{4} +(3.63253 - 0.590344i) q^{5} +(-2.12551 - 0.905594i) q^{6} +(-1.20475 - 0.905312i) q^{7} +(1.03044 + 2.71704i) q^{8} +3.26373 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1080 q - 25 q^{2} - 74 q^{3} + 21 q^{4} - 27 q^{5} - 75 q^{6} - 28 q^{7} + 4 q^{8} + 1010 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{5}{39}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.849270	−	0.361840i	−0.600525	−	0.255860i	0.0702398	−	0.997530i	\(-0.477624\pi\)
							−0.670764	+	0.741671i	\(0.734034\pi\)
\(3\)	2.50274			1.44496			0.722480	−	0.691392i	\(-0.243002\pi\)
							0.722480	+	0.691392i	\(0.243002\pi\)
\(5\)	3.63253	−	0.590344i	1.62452	−	0.264010i	0.721120	−	0.692810i	\(-0.243628\pi\)
							0.903399	+	0.428800i	\(0.141064\pi\)
\(7\)	−1.20475	−	0.905312i	−0.455353	−	0.342176i	0.348188	−	0.937425i	\(-0.386797\pi\)
							−0.803541	+	0.595249i	\(0.797053\pi\)
\(11\)	0.575643	−	0.0464707i	0.173563	−	0.0140114i	0.00662041	−	0.999978i	\(-0.497893\pi\)
							0.166942	+	0.985967i	\(0.446611\pi\)
\(13\)	0.687511	−	1.19080i	0.190681	−	0.330269i	−0.754795	−	0.655961i	\(-0.772264\pi\)
							0.945476	+	0.325691i	\(0.105597\pi\)
\(17\)	1.82585	+	3.84790i	0.442833	+	0.933252i	0.995034	+	0.0995402i	\(0.0317372\pi\)
							−0.552200	+	0.833712i	\(0.686212\pi\)
\(19\)	−0.233546	−	5.79538i	−0.0535791	−	1.32955i	−0.772747	−	0.634715i	\(-0.781118\pi\)
							0.719168	−	0.694837i	\(-0.244523\pi\)
\(23\)	−0.697935	+	3.41871i	−0.145529	+	0.712850i	0.839897	+	0.542746i	\(0.182615\pi\)
							−0.985426	+	0.170104i	\(0.945590\pi\)
\(29\)	−3.68191	+	3.26188i	−0.683713	+	0.605717i	−0.931779	−	0.363026i	\(-0.881743\pi\)
							0.248066	+	0.968743i	\(0.420205\pi\)
\(31\)	−3.46942	+	1.82089i	−0.623127	+	0.327042i	−0.746567	−	0.665310i	\(-0.768299\pi\)
							0.123441	+	0.992352i	\(0.460607\pi\)
\(37\)	−1.79455	−	8.79029i	−0.295022	−	1.44511i	−0.809252	−	0.587462i	\(-0.800127\pi\)
							0.514229	−	0.857653i	\(-0.328078\pi\)
\(41\)	−3.18442	+	5.51558i	−0.497323	+	0.861389i	−0.999995	−	0.00308817i	\(-0.999017\pi\)
							0.502672	+	0.864477i	\(0.332350\pi\)
\(43\)	−1.54246	+	5.32520i	−0.235224	+	0.812086i	0.752862	+	0.658179i	\(0.228673\pi\)
							−0.988085	+	0.153907i	\(0.950814\pi\)
\(47\)	−0.231901	−	0.0774198i	−0.0338262	−	0.0112928i	0.299705	−	0.954032i	\(-0.403112\pi\)
							−0.333531	+	0.942739i	\(0.608240\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.2.j.a.11.18	✓	1080
547.199	even	39	inner	547.2.j.a.199.18	yes	1080

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.j.a.11.18	✓	1080	1.1	even	1	trivial
547.2.j.a.199.18	yes	1080	547.199	even	39	inner