Embedded newform 552.2.bb.a.85.20

• Label: 552.2.bb.a.85.20
• 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			85.20
Character	\(\chi\)	\(=\)	552.85
Dual form			552.2.bb.a.13.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.355714 + 1.36875i) q^{2} +(0.540641 + 0.841254i) q^{3} +(-1.74694 - 0.973765i) q^{4} +(0.190010 - 0.0867748i) q^{5} +(-1.34378 + 0.440755i) q^{6} +(2.41838 - 0.710100i) q^{7} +(1.95425 - 2.04473i) q^{8} +(-0.415415 + 0.909632i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q + 4 q^{2} + 4 q^{6} + 8 q^{7} + 4 q^{8} + 48 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.355714	+	1.36875i	−0.251528	+	0.967850i
							\(3\)	0.540641	+	0.841254i	0.312139	+	0.485698i
\(5\)	0.190010	−	0.0867748i	0.0849752	−	0.0388069i								−0.372474	−	0.928043i	\(-0.621490\pi\)
							0.457449	+	0.889236i	\(0.348763\pi\)
\(7\)	2.41838	−	0.710100i	0.914061	−	0.268392i	0.209312	−	0.977849i	\(-0.432878\pi\)
							0.704749	+	0.709457i	\(0.251060\pi\)
\(11\)	4.36540	−	3.78264i	1.31622	−	1.14051i	0.336157	−	0.941806i	\(-0.390873\pi\)
							0.980060	−	0.198703i	\(-0.0636728\pi\)
\(13\)	1.69874	−	5.78537i	0.471145	−	1.60457i	−0.290700	−	0.956814i	\(-0.593888\pi\)
							0.761845	−	0.647759i	\(-0.224294\pi\)
\(17\)	0.569168	−	3.95865i	0.138043	−	0.960114i	−0.796595	−	0.604514i	\(-0.793367\pi\)
							0.934638	−	0.355600i	\(-0.115723\pi\)
\(19\)	−6.83272	+	0.982398i	−1.56753	+	0.225377i	−0.870685	−	0.491840i	\(-0.836324\pi\)
							−0.696849	+	0.717218i	\(0.745415\pi\)
\(23\)	3.93147	+	2.74655i	0.819768	+	0.572696i
							\(29\)	0.673989	+	0.0969050i	0.125157	+	0.0179948i	0.204608	−	0.978844i	\(-0.434408\pi\)
−0.0794518	+	0.996839i	\(0.525317\pi\)
\(31\)	0.246501	+	0.158417i	0.0442729	+	0.0284525i	0.562590	−	0.826736i	\(-0.309805\pi\)
							−0.518317	+	0.855188i	\(0.673441\pi\)
\(37\)	1.94862	+	0.889905i	0.320351	+	0.146300i	0.569100	−	0.822269i	\(-0.307292\pi\)
							−0.248748	+	0.968568i	\(0.580019\pi\)
\(41\)	4.71909	+	10.3334i	0.736997	+	1.61380i	0.788432	+	0.615123i	\(0.210893\pi\)
							−0.0514343	+	0.998676i	\(0.516379\pi\)
\(43\)	−3.10617	−	4.83330i	−0.473687	−	0.737071i	0.519390	−	0.854537i	\(-0.326159\pi\)
							−0.993077	+	0.117466i	\(0.962523\pi\)
\(47\)	3.14610			0.458905			0.229453	−	0.973320i	\(-0.426306\pi\)
							0.229453	+	0.973320i	\(0.426306\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.2.bb.a.85.20	yes	480
8.5	even	2	inner	552.2.bb.a.85.7	yes	480
23.13	even	11	inner	552.2.bb.a.13.7	✓	480
184.13	even	22	inner	552.2.bb.a.13.20	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.bb.a.13.7	✓	480	23.13	even	11	inner
552.2.bb.a.13.20	yes	480	184.13	even	22	inner
552.2.bb.a.85.7	yes	480	8.5	even	2	inner
552.2.bb.a.85.20	yes	480	1.1	even	1	trivial