Embedded newform 572.2.j.a.463.16

• Label: 572.2.j.a.463.16
• 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			463.16
Character	\(\chi\)	\(=\)	572.463
Dual form			572.2.j.a.551.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.09876 + 0.890354i) q^{2} +3.11188i q^{3} +(0.414539 - 1.95657i) q^{4} +(-0.115988 - 0.115988i) q^{5} +(-2.77067 - 3.41920i) q^{6} +(-2.35663 - 2.35663i) q^{7} +(1.28656 + 2.51888i) q^{8} -6.68380 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q + 4 q^{5} - 12 q^{6} + 12 q^{8} - 140 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.09876	+	0.890354i	−0.776939	+	0.629575i
							\(3\)			3.11188i			1.79664i	0.439337	+	0.898322i	\(0.355213\pi\)
−0.439337	+	0.898322i	\(0.644787\pi\)
\(5\)	−0.115988	−	0.115988i	−0.0518716	−	0.0518716i	0.680695	−	0.732567i	\(-0.261678\pi\)
							−0.732567	+	0.680695i	\(0.761678\pi\)
\(7\)	−2.35663	−	2.35663i	−0.890724	−	0.890724i	0.103868	−	0.994591i	\(-0.466878\pi\)
							−0.994591	+	0.103868i	\(0.966878\pi\)
\(11\)	0.707107	+	0.707107i	0.213201	+	0.213201i
							\(13\)	−1.86603	−	3.08512i	−0.517543	−	0.855657i
\(17\)		−	1.38819i		−	0.336685i								−0.985729	−	0.168342i	\(-0.946159\pi\)
							0.985729	−	0.168342i	\(-0.0538414\pi\)
\(19\)	3.96735	−	3.96735i	0.910173	−	0.910173i	−0.0861123	−	0.996285i	\(-0.527444\pi\)
							0.996285	+	0.0861123i	\(0.0274444\pi\)
\(23\)	−7.41909			−1.54699			−0.773493	−	0.633804i	\(-0.781492\pi\)
							−0.773493	+	0.633804i	\(0.781492\pi\)
\(29\)	−0.897280			−0.166621			−0.0833103	−	0.996524i	\(-0.526549\pi\)
							−0.0833103	+	0.996524i	\(0.526549\pi\)
\(31\)	−6.48848	+	6.48848i	−1.16537	+	1.16537i	−0.182082	+	0.983283i	\(0.558284\pi\)
							−0.983283	+	0.182082i	\(0.941716\pi\)
\(37\)	4.47417	−	4.47417i	0.735548	−	0.735548i	−0.236165	−	0.971713i	\(-0.575890\pi\)
							0.971713	+	0.236165i	\(0.0758905\pi\)
\(41\)	5.69272	+	5.69272i	0.889054	+	0.889054i	0.994432	−	0.105378i	\(-0.0336053\pi\)
							−0.105378	+	0.994432i	\(0.533605\pi\)
\(43\)	−1.15908			−0.176758			−0.0883789	−	0.996087i	\(-0.528169\pi\)
							−0.0883789	+	0.996087i	\(0.528169\pi\)
\(47\)	−5.22153	−	5.22153i	−0.761638	−	0.761638i	0.214980	−	0.976618i	\(-0.431031\pi\)
							−0.976618	+	0.214980i	\(0.931031\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	572.2.j.a.463.16	✓	140
4.3	odd	2	inner	572.2.j.a.463.52	yes	140
13.5	odd	4	inner	572.2.j.a.551.52	yes	140
52.31	even	4	inner	572.2.j.a.551.16	yes	140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.j.a.463.16	✓	140	1.1	even	1	trivial
572.2.j.a.463.52	yes	140	4.3	odd	2	inner
572.2.j.a.551.16	yes	140	52.31	even	4	inner
572.2.j.a.551.52	yes	140	13.5	odd	4	inner