Embedded newform 552.2.bb.a.13.6

• Label: 552.2.bb.a.13.6
• 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			13.6
Character	\(\chi\)	\(=\)	552.13
Dual form			552.2.bb.a.85.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.32689 - 0.489239i) q^{2} +(0.540641 - 0.841254i) q^{3} +(1.52129 + 1.29834i) q^{4} +(3.08337 + 1.40813i) q^{5} +(-1.12895 + 0.851751i) q^{6} +(2.20998 + 0.648908i) q^{7} +(-1.38339 - 2.46703i) q^{8} +(-0.415415 - 0.909632i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q + 4 q^{2} + 4 q^{6} + 8 q^{7} + 4 q^{8} + 48 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.32689	−	0.489239i	−0.938255	−	0.345944i
							\(3\)	0.540641	−	0.841254i	0.312139	−	0.485698i
\(5\)	3.08337	+	1.40813i	1.37893	+	0.629735i								0.960440	−	0.278488i	\(-0.0898333\pi\)
							0.418487	+	0.908223i	\(0.362561\pi\)
\(7\)	2.20998	+	0.648908i	0.835293	+	0.245264i	0.671289	−	0.741195i	\(-0.265741\pi\)
							0.164004	+	0.986460i	\(0.447559\pi\)
\(11\)	3.12309	+	2.70618i	0.941648	+	0.815943i	0.983076	−	0.183199i	\(-0.0586451\pi\)
							−0.0414276	+	0.999142i	\(0.513191\pi\)
\(13\)	−1.13414	−	3.86251i	−0.314553	−	1.07127i	−0.953343	−	0.301888i	\(-0.902383\pi\)
							0.638791	−	0.769381i	\(-0.279435\pi\)
\(17\)	0.524478	+	3.64782i	0.127205	+	0.884727i	0.949075	+	0.315052i	\(0.102022\pi\)
							−0.821870	+	0.569675i	\(0.807069\pi\)
\(19\)	−2.78512	−	0.400440i	−0.638951	−	0.0918673i	−0.184777	−	0.982780i	\(-0.559156\pi\)
							−0.454174	+	0.890913i	\(0.650066\pi\)
\(23\)	−0.614601	+	4.75629i	−0.128153	+	0.991754i
							\(29\)	−9.62875	+	1.38441i	−1.78801	+	0.257078i	−0.955091	−	0.296312i	\(-0.904243\pi\)
−0.832923	+	0.553390i	\(0.813334\pi\)
\(31\)	6.32239	−	4.06316i	1.13554	−	0.729764i	0.168827	−	0.985646i	\(-0.446002\pi\)
							0.966708	+	0.255881i	\(0.0823657\pi\)
\(37\)	−9.92314	+	4.53175i	−1.63135	+	0.745014i	−0.999547	−	0.0301111i	\(-0.990414\pi\)
							−0.631807	+	0.775125i	\(0.717687\pi\)
\(41\)	−1.35302	+	2.96271i	−0.211307	+	0.462698i	−0.985374	−	0.170406i	\(-0.945492\pi\)
							0.774067	+	0.633104i	\(0.218219\pi\)
\(43\)	6.12718	−	9.53408i	0.934387	−	1.45393i	0.0438417	−	0.999038i	\(-0.486040\pi\)
							0.890545	−	0.454895i	\(-0.150323\pi\)
\(47\)	6.59116			0.961419			0.480710	−	0.876880i	\(-0.340379\pi\)
							0.480710	+	0.876880i	\(0.340379\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.2.bb.a.13.6	✓	480
8.5	even	2	inner	552.2.bb.a.13.22	yes	480
23.16	even	11	inner	552.2.bb.a.85.22	yes	480
184.85	even	22	inner	552.2.bb.a.85.6	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.bb.a.13.6	✓	480	1.1	even	1	trivial
552.2.bb.a.13.22	yes	480	8.5	even	2	inner
552.2.bb.a.85.6	yes	480	184.85	even	22	inner
552.2.bb.a.85.22	yes	480	23.16	even	11	inner