Embedded newform 552.2.bb.a.85.28

• Label: 552.2.bb.a.85.28
• Level: $552$
• Weight: $2$
• Character: 552.85
• 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			85.28
Character	\(\chi\)	\(=\)	552.85
Dual form			552.2.bb.a.13.28

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.248651 - 1.39218i) q^{2} +(0.540641 + 0.841254i) q^{3} +(-1.87635 - 0.692334i) q^{4} +(-2.73099 + 1.24720i) q^{5} +(1.30561 - 0.543493i) q^{6} +(3.05369 - 0.896643i) q^{7} +(-1.43041 + 2.44007i) q^{8} +(-0.415415 + 0.909632i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q + 4 q^{2} + 4 q^{6} + 8 q^{7} + 4 q^{8} + 48 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(97\)	\(185\)	\(277\)	\(415\)
\(\chi(n)\)	\(e\left(\frac{4}{11}\right)\)	\(1\)	\(-1\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.248651	−	1.39218i	0.175823	−	0.984422i
							\(3\)	0.540641	+	0.841254i	0.312139	+	0.485698i
\(5\)	−2.73099	+	1.24720i	−1.22134	+	0.557766i								−0.918555	−	0.395293i	\(-0.870643\pi\)
							−0.302783	+	0.953060i	\(0.597916\pi\)
\(7\)	3.05369	−	0.896643i	1.15419	−	0.338899i	0.352015	−	0.935994i	\(-0.385497\pi\)
							0.802170	+	0.597095i	\(0.203678\pi\)
\(11\)	2.08582	−	1.80737i	0.628898	−	0.544943i	−0.281036	−	0.959697i	\(-0.590678\pi\)
							0.909933	+	0.414754i	\(0.136133\pi\)
\(13\)	1.60392	−	5.46244i	0.444846	−	1.51501i	−0.366487	−	0.930423i	\(-0.619440\pi\)
							0.811333	−	0.584584i	\(-0.198742\pi\)
\(17\)	−0.527001	+	3.66537i	−0.127817	+	0.888984i	0.820497	+	0.571650i	\(0.193697\pi\)
							−0.948314	+	0.317334i	\(0.897212\pi\)
\(19\)	7.64592	−	1.09932i	1.75409	−	0.252201i	0.811075	−	0.584942i	\(-0.198883\pi\)
							0.943018	+	0.332741i	\(0.107973\pi\)
\(23\)	4.30404	−	2.11548i	0.897454	−	0.441107i
							\(29\)	6.93992	+	0.997810i	1.28871	+	0.185289i	0.752386	−	0.658722i	\(-0.228903\pi\)
0.536326	+	0.844011i	\(0.319812\pi\)
\(31\)	−3.38529	−	2.17559i	−0.608015	−	0.390748i	0.200097	−	0.979776i	\(-0.435874\pi\)
							−0.808112	+	0.589028i	\(0.799511\pi\)
\(37\)	−6.35074	−	2.90028i	−1.04405	−	0.476804i	−0.181830	−	0.983330i	\(-0.558202\pi\)
							−0.862225	+	0.506526i	\(0.830929\pi\)
\(41\)	−0.0544825	−	0.119300i	−0.00850873	−	0.0186315i	0.905330	−	0.424709i	\(-0.139624\pi\)
							−0.913839	+	0.406077i	\(0.866896\pi\)
\(43\)	0.553019	+	0.860515i	0.0843346	+	0.131227i	0.880858	−	0.473380i	\(-0.156966\pi\)
							−0.796524	+	0.604608i	\(0.793330\pi\)
\(47\)	−2.86435			−0.417808			−0.208904	−	0.977936i	\(-0.566990\pi\)
							−0.208904	+	0.977936i	\(0.566990\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.2.bb.a.85.28	yes	480
8.5	even	2	inner	552.2.bb.a.85.44	yes	480
23.13	even	11	inner	552.2.bb.a.13.44	yes	480
184.13	even	22	inner	552.2.bb.a.13.28	✓	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.bb.a.13.28	✓	480	184.13	even	22	inner
552.2.bb.a.13.44	yes	480	23.13	even	11	inner
552.2.bb.a.85.28	yes	480	1.1	even	1	trivial
552.2.bb.a.85.44	yes	480	8.5	even	2	inner