Embedded newform 552.2.bb.a.85.7

• Label: 552.2.bb.a.85.7
• 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			85.7
Character	\(\chi\)	\(=\)	552.85
Dual form			552.2.bb.a.13.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.30419 + 0.546886i) q^{2} +(-0.540641 - 0.841254i) q^{3} +(1.40183 - 1.42649i) q^{4} +(-0.190010 + 0.0867748i) q^{5} +(1.16517 + 0.801487i) q^{6} +(2.41838 - 0.710100i) q^{7} +(-1.04813 + 2.62706i) q^{8} +(-0.415415 + 0.909632i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q + 4 q^{2} + 4 q^{6} + 8 q^{7} + 4 q^{8} + 48 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.30419	+	0.546886i	−0.922203	+	0.386707i
							\(3\)	−0.540641	−	0.841254i	−0.312139	−	0.485698i
\(5\)	−0.190010	+	0.0867748i	−0.0849752	+	0.0388069i								−0.457449	−	0.889236i	\(-0.651237\pi\)
							0.372474	+	0.928043i	\(0.378510\pi\)
\(7\)	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	\(0.609127\pi\)
							−0.980060	+	0.198703i	\(0.936327\pi\)
\(13\)	−1.69874	+	5.78537i	−0.471145	+	1.60457i	0.290700	+	0.956814i	\(0.406112\pi\)
							−0.761845	+	0.647759i	\(0.775706\pi\)
\(17\)	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	0.696849	−	0.717218i	\(-0.254585\pi\)
							0.870685	+	0.491840i	\(0.163676\pi\)
\(23\)	3.93147	+	2.74655i	0.819768	+	0.572696i
							\(29\)	−0.673989	−	0.0969050i	−0.125157	−	0.0179948i	0.0794518	−	0.996839i	\(-0.474683\pi\)
−0.204608	+	0.978844i	\(0.565592\pi\)
\(31\)	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	0.248748	−	0.968568i	\(-0.419981\pi\)
							−0.569100	+	0.822269i	\(0.692708\pi\)
\(41\)	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	0.993077	−	0.117466i	\(-0.0374771\pi\)
							−0.519390	+	0.854537i	\(0.673841\pi\)
\(47\)	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.85.7	yes	480
8.5	even	2	inner	552.2.bb.a.85.20	yes	480
23.13	even	11	inner	552.2.bb.a.13.20	yes	480
184.13	even	22	inner	552.2.bb.a.13.7	✓	480

    

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