Embedded newform 552.2.bb.a.13.23

• Label: 552.2.bb.a.13.23
• Level: $552$
• Weight: $2$
• Character: 552.13
• 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			13.23
Character	\(\chi\)	\(=\)	552.13
Dual form			552.2.bb.a.85.23

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.160789 + 1.40504i) q^{2} +(0.540641 - 0.841254i) q^{3} +(-1.94829 - 0.451832i) q^{4} +(-0.0738416 - 0.0337223i) q^{5} +(1.09507 + 0.894888i) q^{6} +(-0.400719 - 0.117662i) q^{7} +(0.948109 - 2.66479i) q^{8} +(-0.415415 - 0.909632i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q + 4 q^{2} + 4 q^{6} + 8 q^{7} + 4 q^{8} + 48 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(97\)	\(185\)	\(277\)	\(415\)
\(\chi(n)\)	\(e\left(\frac{7}{11}\right)\)	\(1\)	\(-1\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.160789	+	1.40504i	−0.113695	+	0.993516i
							\(3\)	0.540641	−	0.841254i	0.312139	−	0.485698i
\(5\)	−0.0738416	−	0.0337223i	−0.0330230	−	0.0150811i								0.398835	−	0.917023i	\(-0.369415\pi\)
							−0.431858	+	0.901942i	\(0.642142\pi\)
\(7\)	−0.400719	−	0.117662i	−0.151458	−	0.0444720i	0.205125	−	0.978736i	\(-0.434240\pi\)
							−0.356583	+	0.934264i	\(0.616058\pi\)
\(11\)	−2.76031	−	2.39183i	−0.832266	−	0.721162i	0.130515	−	0.991446i	\(-0.458337\pi\)
							−0.962781	+	0.270284i	\(0.912882\pi\)
\(13\)	−1.04525	−	3.55980i	−0.289901	−	0.987311i	−0.967709	−	0.252072i	\(-0.918888\pi\)
							0.677808	−	0.735239i	\(-0.262930\pi\)
\(17\)	−0.466666	−	3.24573i	−0.113183	−	0.787206i	−0.964789	−	0.263024i	\(-0.915280\pi\)
							0.851606	−	0.524182i	\(-0.175629\pi\)
\(19\)	−0.956420	−	0.137512i	−0.219418	−	0.0315475i	0.0317291	−	0.999497i	\(-0.489899\pi\)
							−0.251147	+	0.967949i	\(0.580808\pi\)
\(23\)	−0.872942	+	4.71572i	−0.182021	+	0.983295i
							\(29\)	−3.25048	+	0.467348i	−0.603599	+	0.0867844i	−0.437338	−	0.899297i	\(-0.644079\pi\)
−0.166261	+	0.986082i	\(0.553169\pi\)
\(31\)	4.99772	−	3.21184i	0.897616	−	0.576863i	−0.00846621	−	0.999964i	\(-0.502695\pi\)
							0.906082	+	0.423101i	\(0.139059\pi\)
\(37\)	8.45554	−	3.86152i	1.39008	−	0.634829i	0.427053	−	0.904226i	\(-0.359552\pi\)
							0.963029	+	0.269397i	\(0.0868244\pi\)
\(41\)	3.21284	−	7.03514i	0.501761	−	1.09870i	−0.474131	−	0.880454i	\(-0.657238\pi\)
							0.975893	−	0.218250i	\(-0.0700349\pi\)
\(43\)	4.77047	−	7.42299i	0.727490	−	1.13200i	−0.258633	−	0.965976i	\(-0.583272\pi\)
							0.986122	−	0.166020i	\(-0.0530917\pi\)
\(47\)	−4.94443			−0.721220			−0.360610	−	0.932717i	\(-0.617431\pi\)
							−0.360610	+	0.932717i	\(0.617431\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.2.bb.a.13.23	✓	480
8.5	even	2	inner	552.2.bb.a.13.48	yes	480
23.16	even	11	inner	552.2.bb.a.85.48	yes	480
184.85	even	22	inner	552.2.bb.a.85.23	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.bb.a.13.23	✓	480	1.1	even	1	trivial
552.2.bb.a.13.48	yes	480	8.5	even	2	inner
552.2.bb.a.85.23	yes	480	184.85	even	22	inner
552.2.bb.a.85.48	yes	480	23.16	even	11	inner