Embedded newform 550.2.be.a.17.13

• Label: 550.2.be.a.17.13
• Level: $550$
• Weight: $2$
• Character: 550.17
• Analytic conductor: $4.392$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	550.be (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.39177211117\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(30\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			17.13
Character	\(\chi\)	\(=\)	550.17
Dual form			550.2.be.a.453.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.156434 + 0.987688i) q^{2} +(1.46950 - 1.46950i) q^{3} +(-0.951057 - 0.309017i) q^{4} +(0.972993 - 2.01328i) q^{5} +(1.22153 + 1.68129i) q^{6} +(0.161998 - 1.02281i) q^{7} +(0.453990 - 0.891007i) q^{8} -1.31888i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 4 q^{3} + 12 q^{5}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/550\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(177\)
\(\chi(n)\)	\(e\left(\frac{9}{10}\right)\)	\(e\left(\frac{13}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.156434	+	0.987688i	−0.110616	+	0.698401i
							\(3\)	1.46950	−	1.46950i	0.848419	−	0.848419i	−0.141517	−	0.989936i	\(-0.545198\pi\)
0.989936	+	0.141517i	\(0.0451981\pi\)
\(5\)	0.972993	−	2.01328i	0.435136	−	0.900365i
							\(7\)	0.161998	−	1.02281i	0.0612294	−	0.386587i	−0.937975	−	0.346703i	\(-0.887301\pi\)
0.999204	−	0.0398842i	\(-0.0126989\pi\)
\(11\)	3.23773	+	0.719084i	0.976213	+	0.216812i
							\(13\)	−4.91366	−	4.91366i	−1.36280	−	1.36280i	−0.870322	−	0.492482i	\(-0.836090\pi\)
−0.492482	−	0.870322i	\(-0.663910\pi\)
\(17\)	−0.246034	+	1.55340i	−0.0596720	+	0.376754i	0.939722	+	0.341940i	\(0.111084\pi\)
							−0.999394	+	0.0348143i	\(0.988916\pi\)
\(19\)	0.735428	+	2.26341i	0.168719	+	0.519263i	0.999291	−	0.0376479i	\(-0.0119865\pi\)
							−0.830572	+	0.556911i	\(0.811987\pi\)
\(23\)	−1.06743	−	0.543884i	−0.222575	−	0.113408i	0.339148	−	0.940733i	\(-0.389861\pi\)
							−0.561723	+	0.827325i	\(0.689861\pi\)
\(29\)	0.273553	−	0.841909i	0.0507975	−	0.156339i	−0.922440	−	0.386141i	\(-0.873808\pi\)
							0.973237	+	0.229802i	\(0.0738079\pi\)
\(31\)	2.08433	−	6.41492i	0.374357	−	1.15215i	−0.569554	−	0.821954i	\(-0.692884\pi\)
							0.943911	−	0.330199i	\(-0.107116\pi\)
\(37\)	−0.487121	+	0.956029i	−0.0800822	+	0.157170i	−0.927575	−	0.373636i	\(-0.878111\pi\)
							0.847493	+	0.530806i	\(0.178111\pi\)
\(41\)	2.11773	+	2.91480i	0.330734	+	0.455216i	0.941706	−	0.336436i	\(-0.109222\pi\)
							−0.610973	+	0.791652i	\(0.709222\pi\)
\(43\)	4.21993	+	4.21993i	0.643534	+	0.643534i	0.951422	−	0.307889i	\(-0.0996225\pi\)
							−0.307889	+	0.951422i	\(0.599622\pi\)
\(47\)	−4.53728	−	4.53728i	−0.661831	−	0.661831i	0.293980	−	0.955811i	\(-0.405020\pi\)
							−0.955811	+	0.293980i	\(0.905020\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	550.2.be.a.17.13	✓	240
11.2	odd	10		550.2.bp.a.167.3	yes	240
25.3	odd	20		550.2.bp.a.303.3	yes	240
275.178	even	20	inner	550.2.be.a.453.13	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
550.2.be.a.17.13	✓	240	1.1	even	1	trivial
550.2.be.a.453.13	yes	240	275.178	even	20	inner
550.2.bp.a.167.3	yes	240	11.2	odd	10
550.2.bp.a.303.3	yes	240	25.3	odd	20