Embedded newform 552.2.bb.a.85.6

• Label: 552.2.bb.a.85.6
• 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			85.6
Character	\(\chi\)	\(=\)	552.85
Dual form			552.2.bb.a.13.6

$q$-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 \(\left(\mathbb{Z}/552\mathbb{Z}\right)^\times\).

\(n\)	\(97\)	\(185\)	\(277\)	\(415\)
\(\chi(n)\)	\(e\left(\frac{4}{11}\right)\)	\(1\)	\(-1\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.32689	+	0.489239i	−0.938255	+	0.345944i
							\(3\)	0.540641	+	0.841254i	0.312139	+	0.485698i
\(5\)	3.08337	−	1.40813i	1.37893	−	0.629735i								0.418487	−	0.908223i	\(-0.362561\pi\)
							0.960440	+	0.278488i	\(0.0898333\pi\)
\(7\)	2.20998	−	0.648908i	0.835293	−	0.245264i	0.164004	−	0.986460i	\(-0.447559\pi\)
							0.671289	+	0.741195i	\(0.265741\pi\)
\(11\)	3.12309	−	2.70618i	0.941648	−	0.815943i	−0.0414276	−	0.999142i	\(-0.513191\pi\)
							0.983076	+	0.183199i	\(0.0586451\pi\)
\(13\)	−1.13414	+	3.86251i	−0.314553	+	1.07127i	0.638791	+	0.769381i	\(0.279435\pi\)
							−0.953343	+	0.301888i	\(0.902383\pi\)
\(17\)	0.524478	−	3.64782i	0.127205	−	0.884727i	−0.821870	−	0.569675i	\(-0.807069\pi\)
							0.949075	−	0.315052i	\(-0.102022\pi\)
\(19\)	−2.78512	+	0.400440i	−0.638951	+	0.0918673i	−0.454174	−	0.890913i	\(-0.650066\pi\)
							−0.184777	+	0.982780i	\(0.559156\pi\)
\(23\)	−0.614601	−	4.75629i	−0.128153	−	0.991754i
							\(29\)	−9.62875	−	1.38441i	−1.78801	−	0.257078i	−0.832923	−	0.553390i	\(-0.813334\pi\)
−0.955091	+	0.296312i	\(0.904243\pi\)
\(31\)	6.32239	+	4.06316i	1.13554	+	0.729764i	0.966708	−	0.255881i	\(-0.0823657\pi\)
							0.168827	+	0.985646i	\(0.446002\pi\)
\(37\)	−9.92314	−	4.53175i	−1.63135	−	0.745014i	−0.631807	−	0.775125i	\(-0.717687\pi\)
							−0.999547	+	0.0301111i	\(0.990414\pi\)
\(41\)	−1.35302	−	2.96271i	−0.211307	−	0.462698i	0.774067	−	0.633104i	\(-0.218219\pi\)
							−0.985374	+	0.170406i	\(0.945492\pi\)
\(43\)	6.12718	+	9.53408i	0.934387	+	1.45393i	0.890545	+	0.454895i	\(0.150323\pi\)
							0.0438417	+	0.999038i	\(0.486040\pi\)
\(47\)	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.85.6	yes	480
8.5	even	2	inner	552.2.bb.a.85.22	yes	480
23.13	even	11	inner	552.2.bb.a.13.22	yes	480
184.13	even	22	inner	552.2.bb.a.13.6	✓	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.bb.a.13.6	✓	480	184.13	even	22	inner
552.2.bb.a.13.22	yes	480	23.13	even	11	inner
552.2.bb.a.85.6	yes	480	1.1	even	1	trivial
552.2.bb.a.85.22	yes	480	8.5	even	2	inner