Embedded newform 572.2.y.a.79.4

• Label: 572.2.y.a.79.4
• 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			79.4
Character	\(\chi\)	\(=\)	572.79
Dual form			572.2.y.a.391.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.38715 + 0.275341i) q^{2} +(-0.322447 - 0.443810i) q^{3} +(1.84837 - 0.763879i) q^{4} +(1.31077 + 4.03414i) q^{5} +(0.569482 + 0.526849i) q^{6} +(-2.94501 - 2.13968i) q^{7} +(-2.35365 + 1.56855i) q^{8} +(0.834056 - 2.56696i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 4 q^{4} + 80 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.38715	+	0.275341i	−0.980864	+	0.194695i
							\(3\)	−0.322447	−	0.443810i	−0.186165	−	0.256234i	0.705726	−	0.708485i	\(-0.250621\pi\)
−0.891891	+	0.452251i	\(0.850621\pi\)
\(5\)	1.31077	+	4.03414i	0.586196	+	1.80412i	0.594415	+	0.804158i	\(0.297384\pi\)
							−0.00821977	+	0.999966i	\(0.502616\pi\)
\(7\)	−2.94501	−	2.13968i	−1.11311	−	0.808721i	−0.129959	−	0.991519i	\(-0.541485\pi\)
							−0.983150	+	0.182798i	\(0.941485\pi\)
\(11\)	2.21047	−	2.47262i	0.666481	−	0.745522i
							\(13\)	−0.951057	−	0.309017i	−0.263776	−	0.0857059i
\(17\)	5.40322	−	1.75561i	1.31047	−	0.425799i								0.431260	−	0.902228i	\(-0.358069\pi\)
							0.879213	+	0.476429i	\(0.158069\pi\)
\(19\)	2.70816	−	1.96759i	0.621294	−	0.451397i	−0.232079	−	0.972697i	\(-0.574553\pi\)
							0.853373	+	0.521300i	\(0.174553\pi\)
\(23\)			1.22734i			0.255918i	0.991779	+	0.127959i	\(0.0408425\pi\)
							−0.991779	+	0.127959i	\(0.959157\pi\)
\(29\)	1.69654	−	2.33509i	0.315040	−	0.433616i	−0.621905	−	0.783093i	\(-0.713641\pi\)
							0.936945	+	0.349477i	\(0.113641\pi\)
\(31\)	7.41294	+	2.40861i	1.33140	+	0.432599i	0.886396	−	0.462928i	\(-0.153201\pi\)
							0.445007	+	0.895527i	\(0.353201\pi\)
\(37\)	3.86367	+	2.80712i	0.635183	+	0.461488i	0.858192	−	0.513329i	\(-0.171588\pi\)
							−0.223009	+	0.974816i	\(0.571588\pi\)
\(41\)	2.34835	+	3.23222i	0.366750	+	0.504788i	0.952014	−	0.306055i	\(-0.0990090\pi\)
							−0.585264	+	0.810843i	\(0.699009\pi\)
\(43\)	−2.09501			−0.319486			−0.159743	−	0.987159i	\(-0.551067\pi\)
							−0.159743	+	0.987159i	\(0.551067\pi\)
\(47\)	6.90126	+	9.49877i	1.00665	+	1.38554i	0.921153	+	0.389200i	\(0.127249\pi\)
							0.0854997	+	0.996338i	\(0.472751\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	572.2.y.a.79.4	✓	288
4.3	odd	2	inner	572.2.y.a.79.19	yes	288
11.6	odd	10	inner	572.2.y.a.391.19	yes	288
44.39	even	10	inner	572.2.y.a.391.4	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.y.a.79.4	✓	288	1.1	even	1	trivial
572.2.y.a.79.19	yes	288	4.3	odd	2	inner
572.2.y.a.391.4	yes	288	44.39	even	10	inner
572.2.y.a.391.19	yes	288	11.6	odd	10	inner