Embedded newform 547.2.f.a.46.3

• Label: 547.2.f.a.46.3
• 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			46.3
Character	\(\chi\)	\(=\)	547.46
Dual form			547.2.f.a.440.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.305966 + 2.51985i) q^{2} +0.544519 q^{3} +(-4.31417 - 1.06335i) q^{4} +(0.299331 + 0.789270i) q^{5} +(-0.166604 + 1.37211i) q^{6} +(0.466827 + 3.84467i) q^{7} +(2.19924 - 5.79891i) q^{8} -2.70350 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 528 q - 8 q^{2} - 46 q^{3} - 50 q^{4} - 9 q^{5} - 21 q^{6} - 3 q^{7} + 14 q^{8} + 466 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.305966	+	2.51985i	−0.216350	+	1.78181i	0.329732	+	0.944075i	\(0.393042\pi\)
							−0.546082	+	0.837732i	\(0.683881\pi\)
\(3\)	0.544519			0.314378			0.157189	−	0.987569i	\(-0.449757\pi\)
							0.157189	+	0.987569i	\(0.449757\pi\)
\(5\)	0.299331	+	0.789270i	0.133865	+	0.352972i	0.985141	−	0.171745i	\(-0.0549404\pi\)
							−0.851277	+	0.524717i	\(0.824171\pi\)
\(7\)	0.466827	+	3.84467i	0.176444	+	1.45315i	0.764655	+	0.644440i	\(0.222909\pi\)
							−0.588211	+	0.808707i	\(0.700168\pi\)
\(11\)	−0.205384	+	0.297551i	−0.0619257	+	0.0897149i	−0.852723	−	0.522363i	\(-0.825051\pi\)
							0.790798	+	0.612078i	\(0.209666\pi\)
\(13\)	2.75284			0.763501			0.381751	−	0.924265i	\(-0.375321\pi\)
							0.381751	+	0.924265i	\(0.375321\pi\)
\(17\)	0.0100524	+	0.0145635i	0.00243807	+	0.00353216i	0.824201	−	0.566297i	\(-0.191625\pi\)
							−0.821763	+	0.569830i	\(0.807009\pi\)
\(19\)	−5.24908	+	2.75493i	−1.20422	+	0.632024i	−0.942851	−	0.333214i	\(-0.891867\pi\)
							−0.261370	+	0.965239i	\(0.584174\pi\)
\(23\)	−2.87835	+	2.54999i	−0.600177	+	0.531710i	−0.907585	−	0.419867i	\(-0.862077\pi\)
							0.307409	+	0.951578i	\(0.400538\pi\)
\(29\)	−1.95754	−	1.73423i	−0.363506	−	0.322038i	0.461524	−	0.887128i	\(-0.347303\pi\)
							−0.825029	+	0.565090i	\(0.808841\pi\)
\(31\)	0.611038	+	0.320697i	0.109746	+	0.0575989i	0.518703	−	0.854955i	\(-0.326415\pi\)
							−0.408957	+	0.912554i	\(0.634107\pi\)
\(37\)	−4.68919	−	4.15426i	−0.770899	−	0.682957i	0.182991	−	0.983115i	\(-0.441422\pi\)
							−0.953889	+	0.300158i	\(0.902961\pi\)
\(41\)	10.5623			1.64955			0.824774	−	0.565463i	\(-0.191302\pi\)
							0.824774	+	0.565463i	\(0.191302\pi\)
\(43\)	−7.91402	−	1.95063i	−1.20688	−	0.297468i	−0.415954	−	0.909386i	\(-0.636552\pi\)
							−0.790922	+	0.611917i	\(0.790399\pi\)
\(47\)	−1.98617	−	1.75959i	−0.289713	−	0.256663i	0.505730	−	0.862692i	\(-0.331223\pi\)
							−0.795443	+	0.606029i	\(0.792762\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.2.f.a.46.3	✓	528
547.440	even	13	inner	547.2.f.a.440.3	yes	528

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.f.a.46.3	✓	528	1.1	even	1	trivial
547.2.f.a.440.3	yes	528	547.440	even	13	inner