Embedded newform 572.2.j.a.463.19

• Label: 572.2.j.a.463.19
• Level: $572$
• Weight: $2$
• Character: 572.463
• Analytic conductor: $4.567$
• Analytic rank: $0$
• Dimension: $140$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(4.56744299562\)
Analytic rank:	\(0\)
Dimension:	\(140\)
Relative dimension:	\(70\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			463.19
Character	\(\chi\)	\(=\)	572.463
Dual form			572.2.j.a.551.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.981374 - 1.01828i) q^{2} -1.60363i q^{3} +(-0.0738084 + 1.99864i) q^{4} +(0.228079 + 0.228079i) q^{5} +(-1.63295 + 1.57376i) q^{6} +(-0.445735 - 0.445735i) q^{7} +(2.10762 - 1.88625i) q^{8} +0.428375 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q + 4 q^{5} - 12 q^{6} + 12 q^{8} - 140 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(287\)	\(353\)	\(365\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.981374	−	1.01828i	−0.693937	−	0.720036i
							\(3\)		−	1.60363i		−	0.925856i	−0.886396	−	0.462928i	\(-0.846799\pi\)
0.886396	−	0.462928i	\(-0.153201\pi\)
\(5\)	0.228079	+	0.228079i	0.102000	+	0.102000i	0.756265	−	0.654265i	\(-0.227022\pi\)
							−0.654265	+	0.756265i	\(0.727022\pi\)
\(7\)	−0.445735	−	0.445735i	−0.168472	−	0.168472i	0.617835	−	0.786307i	\(-0.288010\pi\)
							−0.786307	+	0.617835i	\(0.788010\pi\)
\(11\)	−0.707107	−	0.707107i	−0.213201	−	0.213201i
							\(13\)	3.46692	−	0.990191i	0.961550	−	0.274630i
\(17\)		−	0.467488i		−	0.113382i								−0.998392	−	0.0566912i	\(-0.981945\pi\)
							0.998392	−	0.0566912i	\(-0.0180550\pi\)
\(19\)	2.45194	−	2.45194i	0.562514	−	0.562514i	−0.367507	−	0.930021i	\(-0.619788\pi\)
							0.930021	+	0.367507i	\(0.119788\pi\)
\(23\)	−6.17368			−1.28730			−0.643650	−	0.765320i	\(-0.722581\pi\)
							−0.643650	+	0.765320i	\(0.722581\pi\)
\(29\)	1.01520			0.188517			0.0942585	−	0.995548i	\(-0.469952\pi\)
							0.0942585	+	0.995548i	\(0.469952\pi\)
\(31\)	7.81438	−	7.81438i	1.40350	−	1.40350i	0.614894	−	0.788610i	\(-0.289199\pi\)
							0.788610	−	0.614894i	\(-0.210801\pi\)
\(37\)	−6.62006	+	6.62006i	−1.08833	+	1.08833i	−0.0926306	+	0.995701i	\(0.529528\pi\)
							−0.995701	+	0.0926306i	\(0.970472\pi\)
\(41\)	6.24022	+	6.24022i	0.974558	+	0.974558i	0.999684	−	0.0251260i	\(-0.00799869\pi\)
							−0.0251260	+	0.999684i	\(0.507999\pi\)
\(43\)	−9.25190			−1.41090			−0.705451	−	0.708759i	\(-0.749256\pi\)
							−0.705451	+	0.708759i	\(0.749256\pi\)
\(47\)	−3.82133	−	3.82133i	−0.557399	−	0.557399i	0.371167	−	0.928566i	\(-0.378958\pi\)
							−0.928566	+	0.371167i	\(0.878958\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	572.2.j.a.463.19	yes	140
4.3	odd	2	inner	572.2.j.a.463.18	✓	140
13.5	odd	4	inner	572.2.j.a.551.18	yes	140
52.31	even	4	inner	572.2.j.a.551.19	yes	140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.j.a.463.18	✓	140	4.3	odd	2	inner
572.2.j.a.463.19	yes	140	1.1	even	1	trivial
572.2.j.a.551.18	yes	140	13.5	odd	4	inner
572.2.j.a.551.19	yes	140	52.31	even	4	inner