Embedded newform 552.2.bb.a.13.5

• Label: 552.2.bb.a.13.5
• 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			13.5
Character	\(\chi\)	\(=\)	552.13
Dual form			552.2.bb.a.85.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.36486 + 0.370358i) q^{2} +(0.540641 - 0.841254i) q^{3} +(1.72567 - 1.01097i) q^{4} +(0.808939 + 0.369430i) q^{5} +(-0.426333 + 1.34842i) q^{6} +(0.759802 + 0.223098i) q^{7} +(-1.98087 + 2.01895i) q^{8} +(-0.415415 - 0.909632i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q + 4 q^{2} + 4 q^{6} + 8 q^{7} + 4 q^{8} + 48 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.36486	+	0.370358i	−0.965100	+	0.261882i
							\(3\)	0.540641	−	0.841254i	0.312139	−	0.485698i
\(5\)	0.808939	+	0.369430i	0.361768	+	0.165214i								0.588003	−	0.808859i	\(-0.299914\pi\)
							−0.226235	+	0.974073i	\(0.572642\pi\)
\(7\)	0.759802	+	0.223098i	0.287178	+	0.0843231i	0.422149	−	0.906526i	\(-0.361276\pi\)
							−0.134971	+	0.990850i	\(0.543094\pi\)
\(11\)	0.857262	+	0.742822i	0.258474	+	0.223969i	0.774452	−	0.632633i	\(-0.218026\pi\)
							−0.515978	+	0.856602i	\(0.672571\pi\)
\(13\)	1.80954	+	6.16272i	0.501876	+	1.70923i	0.687125	+	0.726539i	\(0.258873\pi\)
							−0.185249	+	0.982692i	\(0.559309\pi\)
\(17\)	0.262648	+	1.82676i	0.0637014	+	0.443053i	0.996564	+	0.0828217i	\(0.0263932\pi\)
							−0.932863	+	0.360231i	\(0.882698\pi\)
\(19\)	6.71287	+	0.965165i	1.54004	+	0.221424i	0.859442	−	0.511234i	\(-0.170811\pi\)
							0.680597	+	0.732658i	\(0.261720\pi\)
\(23\)	−4.76142	+	0.573480i	−0.992825	+	0.119579i
							\(29\)	−0.980596	+	0.140988i	−0.182092	+	0.0261809i	−0.232758	−	0.972535i	\(-0.574775\pi\)
0.0506654	+	0.998716i	\(0.483866\pi\)
\(31\)	3.39749	−	2.18343i	0.610207	−	0.392156i	−0.198728	−	0.980055i	\(-0.563681\pi\)
							0.808935	+	0.587899i	\(0.200045\pi\)
\(37\)	6.20415	−	2.83334i	1.01996	−	0.465798i	0.165990	−	0.986127i	\(-0.446918\pi\)
							0.853966	+	0.520329i	\(0.174191\pi\)
\(41\)	0.147591	−	0.323180i	0.0230499	−	0.0504722i	−0.897757	−	0.440491i	\(-0.854804\pi\)
							0.920807	+	0.390019i	\(0.127532\pi\)
\(43\)	2.14419	−	3.33643i	0.326986	−	0.508801i	−0.638372	−	0.769728i	\(-0.720392\pi\)
							0.965359	+	0.260927i	\(0.0840282\pi\)
\(47\)	12.7492			1.85966			0.929829	−	0.367991i	\(-0.119954\pi\)
							0.929829	+	0.367991i	\(0.119954\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.2.bb.a.13.5	✓	480
8.5	even	2	inner	552.2.bb.a.13.30	yes	480
23.16	even	11	inner	552.2.bb.a.85.30	yes	480
184.85	even	22	inner	552.2.bb.a.85.5	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.bb.a.13.5	✓	480	1.1	even	1	trivial
552.2.bb.a.13.30	yes	480	8.5	even	2	inner
552.2.bb.a.85.5	yes	480	184.85	even	22	inner
552.2.bb.a.85.30	yes	480	23.16	even	11	inner