Embedded newform 572.2.y.a.79.20

• Label: 572.2.y.a.79.20
• Level: $572$
• Weight: $2$
• Character: 572.79
• Analytic conductor: $4.567$
• Analytic rank: $0$
• Dimension: $288$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	572.y (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.56744299562\)
Analytic rank:	\(0\)
Dimension:	\(288\)
Relative dimension:	\(72\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			79.20
Character	\(\chi\)	\(=\)	572.79
Dual form			572.2.y.a.391.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.937016 - 1.05925i) q^{2} +(-1.45995 - 2.00944i) q^{3} +(-0.244001 + 1.98506i) q^{4} +(0.521225 + 1.60416i) q^{5} +(-0.760501 + 3.42932i) q^{6} +(-0.136261 - 0.0989992i) q^{7} +(2.33130 - 1.60158i) q^{8} +(-0.979372 + 3.01420i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 4 q^{4} + 80 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/572\mathbb{Z}\right)^\times\).

\(n\)	\(287\)	\(353\)	\(365\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{1}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.937016	−	1.05925i	−0.662571	−	0.749000i
							\(3\)	−1.45995	−	2.00944i	−0.842901	−	1.16015i	−0.985383	−	0.170356i	\(-0.945508\pi\)
0.142482	−	0.989797i	\(-0.454492\pi\)
\(5\)	0.521225	+	1.60416i	0.233099	+	0.717404i	0.997368	+	0.0725068i	\(0.0230999\pi\)
							−0.764269	+	0.644897i	\(0.776900\pi\)
\(7\)	−0.136261	−	0.0989992i	−0.0515017	−	0.0374182i	0.561737	−	0.827316i	\(-0.310133\pi\)
							−0.613238	+	0.789898i	\(0.710133\pi\)
\(11\)	3.14292	−	1.05927i	0.947626	−	0.319381i
							\(13\)	−0.951057	−	0.309017i	−0.263776	−	0.0857059i
\(17\)	−2.68876	+	0.873631i	−0.652120	+	0.211887i								−0.616349	−	0.787473i	\(-0.711389\pi\)
							−0.0357711	+	0.999360i	\(0.511389\pi\)
\(19\)	4.32965	−	3.14567i	0.993289	−	0.721667i	0.0326501	−	0.999467i	\(-0.489605\pi\)
							0.960639	+	0.277800i	\(0.0896053\pi\)
\(23\)		−	7.05167i		−	1.47037i	−0.677864	−	0.735187i	\(-0.737094\pi\)
							0.677864	−	0.735187i	\(-0.262906\pi\)
\(29\)	2.33881	−	3.21910i	0.434307	−	0.597772i	−0.534628	−	0.845087i	\(-0.679548\pi\)
							0.968935	+	0.247315i	\(0.0795483\pi\)
\(31\)	−3.85851	−	1.25371i	−0.693009	−	0.225172i	−0.0587268	−	0.998274i	\(-0.518704\pi\)
							−0.634282	+	0.773102i	\(0.718704\pi\)
\(37\)	−0.290825	−	0.211297i	−0.0478114	−	0.0347370i	0.563623	−	0.826032i	\(-0.309407\pi\)
							−0.611434	+	0.791295i	\(0.709407\pi\)
\(41\)	2.97223	+	4.09092i	0.464184	+	0.638895i	0.975370	−	0.220575i	\(-0.0707935\pi\)
							−0.511185	+	0.859470i	\(0.670793\pi\)
\(43\)	−6.03970			−0.921045			−0.460523	−	0.887648i	\(-0.652338\pi\)
							−0.460523	+	0.887648i	\(0.652338\pi\)
\(47\)	−0.964887	−	1.32805i	−0.140743	−	0.193716i	0.732826	−	0.680416i	\(-0.238201\pi\)
							−0.873570	+	0.486699i	\(0.838201\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	572.2.y.a.79.20	yes	288
4.3	odd	2	inner	572.2.y.a.79.5	✓	288
11.6	odd	10	inner	572.2.y.a.391.5	yes	288
44.39	even	10	inner	572.2.y.a.391.20	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.y.a.79.5	✓	288	4.3	odd	2	inner
572.2.y.a.79.20	yes	288	1.1	even	1	trivial
572.2.y.a.391.5	yes	288	11.6	odd	10	inner
572.2.y.a.391.20	yes	288	44.39	even	10	inner