Embedded newform 552.2.bb.a.469.32

• Label: 552.2.bb.a.469.32
• Level: $552$
• Weight: $2$
• Character: 552.469
• Analytic conductor: $4.408$
• Analytic rank: $0$
• Dimension: $480$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 552 = 2^{3} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	552.bb (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.40774219157\)
Analytic rank:	\(0\)
Dimension:	\(480\)
Relative dimension:	\(48\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			469.32
Character	\(\chi\)	\(=\)	552.469
Dual form			552.2.bb.a.133.32

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.750277 + 1.19878i) q^{2} +(0.755750 - 0.654861i) q^{3} +(-0.874167 + 1.79884i) q^{4} +(4.14042 - 0.595303i) q^{5} +(1.35206 + 0.414653i) q^{6} +(-0.0911276 + 0.199542i) q^{7} +(-2.81229 + 0.301692i) q^{8} +(0.142315 - 0.989821i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q + 4 q^{2} + 4 q^{6} + 8 q^{7} + 4 q^{8} + 48 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/552\mathbb{Z}\right)^\times\).

\(n\)	\(97\)	\(185\)	\(277\)	\(415\)
\(\chi(n)\)	\(e\left(\frac{5}{11}\right)\)	\(1\)	\(-1\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.750277	+	1.19878i	0.530526	+	0.847668i
							\(3\)	0.755750	−	0.654861i	0.436332	−	0.378084i
\(5\)	4.14042	−	0.595303i	1.85165	−	0.266228i								0.875485	−	0.483245i	\(-0.160542\pi\)
							0.976168	+	0.217017i	\(0.0696327\pi\)
\(7\)	−0.0911276	+	0.199542i	−0.0344430	+	0.0754197i	−0.926072	−	0.377348i	\(-0.876836\pi\)
							0.891629	+	0.452768i	\(0.149563\pi\)
\(11\)	−1.00493	+	3.42246i	−0.302997	+	1.03191i	0.657462	+	0.753488i	\(0.271630\pi\)
							−0.960458	+	0.278424i	\(0.910188\pi\)
\(13\)	−0.385957	+	0.176261i	−0.107045	+	0.0488859i	−0.468218	−	0.883613i	\(-0.655104\pi\)
							0.361173	+	0.932499i	\(0.382377\pi\)
\(17\)	1.76125	+	1.13188i	0.427165	+	0.274522i	0.736509	−	0.676428i	\(-0.236473\pi\)
							−0.309344	+	0.950950i	\(0.600109\pi\)
\(19\)	−3.17850	−	4.94584i	−0.729198	−	1.13465i	−0.985765	−	0.168127i	\(-0.946228\pi\)
							0.256568	−	0.966526i	\(-0.417408\pi\)
\(23\)	4.45930	−	1.76485i	0.929827	−	0.367996i
							\(29\)	−0.283534	+	0.441187i	−0.0526509	+	0.0819265i	−0.866581	−	0.499036i	\(-0.833687\pi\)
0.813930	+	0.580963i	\(0.197324\pi\)
\(31\)	−3.25640	+	3.75808i	−0.584866	+	0.674971i	−0.968644	−	0.248454i	\(-0.920077\pi\)
							0.383778	+	0.923426i	\(0.374623\pi\)
\(37\)	−3.88113	−	0.558022i	−0.638054	−	0.0917382i	−0.184307	−	0.982869i	\(-0.559004\pi\)
							−0.453747	+	0.891131i	\(0.649913\pi\)
\(41\)	−0.393302	−	2.73548i	−0.0614235	−	0.427210i	−0.997210	−	0.0746434i	\(-0.976218\pi\)
							0.935787	−	0.352566i	\(-0.114691\pi\)
\(43\)	−3.77564	+	3.27161i	−0.575779	+	0.498916i	−0.893376	−	0.449310i	\(-0.851670\pi\)
							0.317596	+	0.948226i	\(0.397124\pi\)
\(47\)	−12.1022			−1.76529			−0.882643	−	0.470044i	\(-0.844238\pi\)
							−0.882643	+	0.470044i	\(0.844238\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.2.bb.a.469.32	yes	480
8.5	even	2	inner	552.2.bb.a.469.22	yes	480
23.18	even	11	inner	552.2.bb.a.133.22	✓	480
184.133	even	22	inner	552.2.bb.a.133.32	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.bb.a.133.22	✓	480	23.18	even	11	inner
552.2.bb.a.133.32	yes	480	184.133	even	22	inner
552.2.bb.a.469.22	yes	480	8.5	even	2	inner
552.2.bb.a.469.32	yes	480	1.1	even	1	trivial