Embedded newform 547.2.f.a.46.1

• Label: 547.2.f.a.46.1
• 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			46.1
Character	\(\chi\)	\(=\)	547.46
Dual form			547.2.f.a.440.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.335888 + 2.76629i) q^{2} +1.13158 q^{3} +(-5.59764 - 1.37970i) q^{4} +(-0.153172 - 0.403882i) q^{5} +(-0.380085 + 3.13028i) q^{6} +(-0.393363 - 3.23964i) q^{7} +(3.72053 - 9.81022i) q^{8} -1.71952 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 528 q - 8 q^{2} - 46 q^{3} - 50 q^{4} - 9 q^{5} - 21 q^{6} - 3 q^{7} + 14 q^{8} + 466 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.335888	+	2.76629i	−0.237509	+	1.95606i	0.0513265	+	0.998682i	\(0.483655\pi\)
							−0.288835	+	0.957379i	\(0.593268\pi\)
\(3\)	1.13158			0.653319			0.326660	−	0.945142i	\(-0.394077\pi\)
							0.326660	+	0.945142i	\(0.394077\pi\)
\(5\)	−0.153172	−	0.403882i	−0.0685008	−	0.180622i	0.896394	−	0.443258i	\(-0.146177\pi\)
							−0.964895	+	0.262636i	\(0.915408\pi\)
\(7\)	−0.393363	−	3.23964i	−0.148677	−	1.22447i	−0.855773	−	0.517351i	\(-0.826918\pi\)
							0.707096	−	0.707118i	\(-0.250005\pi\)
\(11\)	−1.03652	+	1.50166i	−0.312523	+	0.452767i	−0.947496	−	0.319766i	\(-0.896396\pi\)
							0.634974	+	0.772534i	\(0.281011\pi\)
\(13\)	−2.97441			−0.824953			−0.412476	−	0.910968i	\(-0.635336\pi\)
							−0.412476	+	0.910968i	\(0.635336\pi\)
\(17\)	−3.47290	−	5.03136i	−0.842301	−	1.22028i	−0.973445	−	0.228921i	\(-0.926480\pi\)
							0.131144	−	0.991363i	\(-0.458135\pi\)
\(19\)	7.15502	−	3.75524i	1.64147	−	0.861512i	0.646333	−	0.763056i	\(-0.276302\pi\)
							0.995141	−	0.0984562i	\(-0.0313904\pi\)
\(23\)	0.718306	−	0.636363i	0.149777	−	0.132691i	−0.584907	−	0.811100i	\(-0.698869\pi\)
							0.734684	+	0.678410i	\(0.237330\pi\)
\(29\)	−0.687698	−	0.609247i	−0.127702	−	0.113134i	0.596854	−	0.802350i	\(-0.296417\pi\)
							−0.724557	+	0.689215i	\(0.757955\pi\)
\(31\)	7.11428	+	3.73386i	1.27776	+	0.670622i	0.960649	−	0.277766i	\(-0.0895939\pi\)
							0.317114	+	0.948387i	\(0.397286\pi\)
\(37\)	0.641860	+	0.568638i	0.105521	+	0.0934836i	0.714234	−	0.699907i	\(-0.246775\pi\)
							−0.608713	+	0.793391i	\(0.708314\pi\)
\(41\)	−11.1341			−1.73885			−0.869427	−	0.494062i	\(-0.835512\pi\)
							−0.869427	+	0.494062i	\(0.835512\pi\)
\(43\)	−7.82547	−	1.92880i	−1.19337	−	0.294140i	−0.407894	−	0.913029i	\(-0.633737\pi\)
							−0.785478	+	0.618889i	\(0.787583\pi\)
\(47\)	−5.04451	−	4.46904i	−0.735817	−	0.651877i	0.209583	−	0.977791i	\(-0.432789\pi\)
							−0.945399	+	0.325914i	\(0.894328\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.2.f.a.46.1	✓	528
547.440	even	13	inner	547.2.f.a.440.1	yes	528

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.f.a.46.1	✓	528	1.1	even	1	trivial
547.2.f.a.440.1	yes	528	547.440	even	13	inner