Embedded newform 572.2.y.a.79.1

• Label: 572.2.y.a.79.1
• 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			79.1
Character	\(\chi\)	\(=\)	572.79
Dual form			572.2.y.a.391.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41285 + 0.0620317i) q^{2} +(1.52448 + 2.09827i) q^{3} +(1.99230 - 0.175283i) q^{4} +(-0.870694 - 2.67972i) q^{5} +(-2.28403 - 2.86998i) q^{6} +(-2.76319 - 2.00757i) q^{7} +(-2.80396 + 0.371235i) q^{8} +(-1.15164 + 3.54438i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 4 q^{4} + 80 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41285	+	0.0620317i	−0.999038	+	0.0438630i
							\(3\)	1.52448	+	2.09827i	0.880160	+	1.21144i	0.976377	+	0.216076i	\(0.0693259\pi\)
−0.0962165	+	0.995360i	\(0.530674\pi\)
\(5\)	−0.870694	−	2.67972i	−0.389386	−	1.19841i	−0.933248	−	0.359233i	\(-0.883038\pi\)
							0.543862	−	0.839175i	\(-0.316962\pi\)
\(7\)	−2.76319	−	2.00757i	−1.04439	−	0.758791i	−0.0732492	−	0.997314i	\(-0.523337\pi\)
							−0.971137	+	0.238523i	\(0.923337\pi\)
\(11\)	3.15304	−	1.02876i	0.950677	−	0.310182i
							\(13\)	−0.951057	−	0.309017i	−0.263776	−	0.0857059i
\(17\)	−4.75309	+	1.54437i	−1.15279	+	0.374565i								−0.822195	−	0.569206i	\(-0.807251\pi\)
							−0.330598	+	0.943772i	\(0.607251\pi\)
\(19\)	4.96578	−	3.60785i	1.13923	−	0.827697i	0.152216	−	0.988347i	\(-0.451359\pi\)
							0.987011	+	0.160650i	\(0.0513590\pi\)
\(23\)		−	7.52137i		−	1.56831i	−0.620563	−	0.784157i	\(-0.713096\pi\)
							0.620563	−	0.784157i	\(-0.286904\pi\)
\(29\)	3.50705	−	4.82704i	0.651243	−	0.896359i	−0.347909	−	0.937528i	\(-0.613108\pi\)
							0.999152	+	0.0411692i	\(0.0131083\pi\)
\(31\)	−3.64973	−	1.18587i	−0.655510	−	0.212988i	−0.0376682	−	0.999290i	\(-0.511993\pi\)
							−0.617842	+	0.786302i	\(0.711993\pi\)
\(37\)	7.28414	+	5.29224i	1.19751	+	0.870039i	0.994037	−	0.109043i	\(-0.0347786\pi\)
							0.203468	+	0.979081i	\(0.434779\pi\)
\(41\)	−6.03143	−	8.30155i	−0.941951	−	1.29648i	−0.955011	−	0.296569i	\(-0.904157\pi\)
							0.0130605	−	0.999915i	\(-0.495843\pi\)
\(43\)	5.76598			0.879303			0.439651	−	0.898168i	\(-0.355102\pi\)
							0.439651	+	0.898168i	\(0.355102\pi\)
\(47\)	−0.697316	−	0.959773i	−0.101714	−	0.139997i	0.755126	−	0.655580i	\(-0.227576\pi\)
							−0.856840	+	0.515582i	\(0.827576\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	572.2.y.a.79.1	✓	288
4.3	odd	2	inner	572.2.y.a.79.15	yes	288
11.6	odd	10	inner	572.2.y.a.391.15	yes	288
44.39	even	10	inner	572.2.y.a.391.1	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.y.a.79.1	✓	288	1.1	even	1	trivial
572.2.y.a.79.15	yes	288	4.3	odd	2	inner
572.2.y.a.391.1	yes	288	44.39	even	10	inner
572.2.y.a.391.15	yes	288	11.6	odd	10	inner