Embedded newform 547.2.h.a.13.1

• Label: 547.2.h.a.13.1
• Level: $547$
• Weight: $2$
• Character: 547.13
• Analytic conductor: $4.368$
• Analytic rank: $0$
• Dimension: $540$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 547 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	547.h (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.36781699056\)
Analytic rank:	\(0\)
Dimension:	\(540\)
Relative dimension:	\(45\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			13.1
Character	\(\chi\)	\(=\)	547.13
Dual form			547.2.h.a.505.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.208129 + 2.77729i) q^{2} +(1.56284 - 1.95974i) q^{3} +(-5.69234 - 0.857982i) q^{4} +(-1.65592 + 1.53647i) q^{5} +(5.11748 + 4.74833i) q^{6} +(4.14244 - 0.624372i) q^{7} +(2.32812 - 10.2002i) q^{8} +(-0.730547 - 3.20074i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 540 q - 13 q^{2} - 10 q^{3} + 33 q^{4} - 15 q^{5} + 10 q^{6} - 2 q^{7} - 26 q^{8} - 96 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{11}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.208129	+	2.77729i	−0.147169	+	1.96384i	0.100344	+	0.994953i	\(0.468006\pi\)
							−0.247513	+	0.968885i	\(0.579613\pi\)
\(3\)	1.56284	−	1.95974i	0.902306	−	1.13146i	−0.0884876	−	0.996077i	\(-0.528203\pi\)
							0.990794	−	0.135379i	\(-0.0432252\pi\)
\(5\)	−1.65592	+	1.53647i	−0.740551	+	0.687131i	−0.957300	−	0.289096i	\(-0.906645\pi\)
							0.216749	+	0.976227i	\(0.430455\pi\)
\(7\)	4.14244	−	0.624372i	1.56569	−	0.235990i	0.691793	−	0.722095i	\(-0.256821\pi\)
							0.873900	+	0.486105i	\(0.161583\pi\)
\(11\)	2.54895	−	4.41491i	0.768538	−	1.33115i	−0.169818	−	0.985475i	\(-0.554318\pi\)
							0.938356	−	0.345671i	\(-0.112349\pi\)
\(13\)	−0.512562	+	0.349459i	−0.142159	+	0.0969224i	−0.632311	−	0.774714i	\(-0.717894\pi\)
							0.490152	+	0.871637i	\(0.336941\pi\)
\(17\)	0.675132	+	0.208251i	0.163744	+	0.0505082i	0.375543	−	0.926805i	\(-0.377456\pi\)
							−0.211800	+	0.977313i	\(0.567932\pi\)
\(19\)	−0.192601	−	2.57008i	−0.0441858	−	0.589618i	−0.974722	−	0.223419i	\(-0.928278\pi\)
							0.930537	−	0.366199i	\(-0.119341\pi\)
\(23\)	0.580052	+	7.74026i	0.120949	+	1.61395i	0.646310	+	0.763075i	\(0.276311\pi\)
							−0.525361	+	0.850880i	\(0.676070\pi\)
\(29\)	0.822729	−	3.60461i	0.152777	−	0.669359i	−0.839294	−	0.543678i	\(-0.817031\pi\)
							0.992071	−	0.125681i	\(-0.0401116\pi\)
\(31\)	3.73296	−	4.68098i	0.670459	−	0.840729i	−0.323977	−	0.946065i	\(-0.605020\pi\)
							0.994437	+	0.105335i	\(0.0335916\pi\)
\(37\)	4.40818	+	1.35974i	0.724700	+	0.223540i	0.635092	−	0.772437i	\(-0.280962\pi\)
							0.0896085	+	0.995977i	\(0.471438\pi\)
\(41\)	−1.46154	+	2.53147i	−0.228255	+	0.395349i	−0.957291	−	0.289126i	\(-0.906635\pi\)
							0.729036	+	0.684475i	\(0.239969\pi\)
\(43\)	1.12929	−	0.170213i	0.172215	−	0.0259573i	−0.0623685	−	0.998053i	\(-0.519865\pi\)
							0.234584	+	0.972096i	\(0.424627\pi\)
\(47\)	−1.91378	−	3.31476i	−0.279153	−	0.483508i	0.692021	−	0.721877i	\(-0.256720\pi\)
							−0.971175	+	0.238369i	\(0.923387\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.2.h.a.13.1	✓	540
547.505	even	21	inner	547.2.h.a.505.1	yes	540

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.h.a.13.1	✓	540	1.1	even	1	trivial
547.2.h.a.505.1	yes	540	547.505	even	21	inner