Embedded newform 547.2.h.a.13.18

• Label: 547.2.h.a.13.18
• 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			13.18
Character	\(\chi\)	\(=\)	547.13
Dual form			547.2.h.a.505.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0388648 + 0.518615i) q^{2} +(-0.378079 + 0.474096i) q^{3} +(1.71021 + 0.257773i) q^{4} +(-2.53370 + 2.35093i) q^{5} +(-0.231179 - 0.214503i) q^{6} +(-3.39754 + 0.512096i) q^{7} +(-0.431604 + 1.89098i) q^{8} +(0.585740 + 2.56629i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 540 q - 13 q^{2} - 10 q^{3} + 33 q^{4} - 15 q^{5} + 10 q^{6} - 2 q^{7} - 26 q^{8} - 96 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0388648	+	0.518615i	−0.0274816	+	0.366716i	0.966461	+	0.256814i	\(0.0826727\pi\)
							−0.993942	+	0.109902i	\(0.964946\pi\)
\(3\)	−0.378079	+	0.474096i	−0.218284	+	0.273719i	−0.878902	−	0.477003i	\(-0.841723\pi\)
							0.660618	+	0.750723i	\(0.270294\pi\)
\(5\)	−2.53370	+	2.35093i	−1.13311	+	1.05137i	−0.134939	+	0.990854i	\(0.543084\pi\)
							−0.998167	+	0.0605150i	\(0.980726\pi\)
\(7\)	−3.39754	+	0.512096i	−1.28415	+	0.193554i	−0.755422	−	0.655238i	\(-0.772568\pi\)
							−0.528726	+	0.848792i	\(0.677330\pi\)
\(11\)	2.52179	−	4.36787i	0.760348	−	1.31696i	−0.182323	−	0.983239i	\(-0.558362\pi\)
							0.942671	−	0.333723i	\(-0.108305\pi\)
\(13\)	−1.75126	+	1.19399i	−0.485711	+	0.331152i	−0.781292	−	0.624165i	\(-0.785439\pi\)
							0.295581	+	0.955318i	\(0.404487\pi\)
\(17\)	−6.03477	−	1.86148i	−1.46365	−	0.451476i	−0.542290	−	0.840191i	\(-0.682443\pi\)
							−0.921358	+	0.388716i	\(0.872919\pi\)
\(19\)	−0.455004	−	6.07161i	−0.104385	−	1.39292i	−0.765885	−	0.642978i	\(-0.777699\pi\)
							0.661500	−	0.749945i	\(-0.269920\pi\)
\(23\)	0.0234974	+	0.313551i	0.00489955	+	0.0653800i	0.999118	−	0.0419999i	\(-0.0133729\pi\)
							−0.994218	+	0.107380i	\(0.965754\pi\)
\(29\)	−1.18049	+	5.17206i	−0.219211	+	0.960427i	0.738851	+	0.673868i	\(0.235369\pi\)
							−0.958063	+	0.286559i	\(0.907489\pi\)
\(31\)	−1.32667	+	1.66359i	−0.238276	+	0.298789i	−0.886563	−	0.462607i	\(-0.846914\pi\)
							0.648287	+	0.761396i	\(0.275486\pi\)
\(37\)	5.58838	+	1.72379i	0.918723	+	0.283389i	0.717825	−	0.696223i	\(-0.245138\pi\)
							0.200898	+	0.979612i	\(0.435614\pi\)
\(41\)	−5.56698	+	9.64229i	−0.869416	+	1.50587i	−0.00682177	+	0.999977i	\(0.502171\pi\)
							−0.862594	+	0.505896i	\(0.831162\pi\)
\(43\)	−4.02698	+	0.606969i	−0.614108	+	0.0925620i	−0.448729	−	0.893668i	\(-0.648123\pi\)
							−0.165380	+	0.986230i	\(0.552885\pi\)
\(47\)	6.46193	+	11.1924i	0.942569	+	1.63258i	0.760546	+	0.649284i	\(0.224931\pi\)
							0.182023	+	0.983294i	\(0.441735\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.2.h.a.13.18	✓	540
547.505	even	21	inner	547.2.h.a.505.18	yes	540

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.h.a.13.18	✓	540	1.1	even	1	trivial
547.2.h.a.505.18	yes	540	547.505	even	21	inner