Embedded newform 552.2.bb.a.85.22

• Label: 552.2.bb.a.85.22
• 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			85.22
Character	\(\chi\)	\(=\)	552.85
Dual form			552.2.bb.a.13.22

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.295423 + 1.38301i) q^{2} +(-0.540641 - 0.841254i) q^{3} +(-1.82545 - 0.817147i) q^{4} +(-3.08337 + 1.40813i) q^{5} +(1.32318 - 0.499188i) q^{6} +(2.20998 - 0.648908i) q^{7} +(1.66940 - 2.28322i) q^{8} +(-0.415415 + 0.909632i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q + 4 q^{2} + 4 q^{6} + 8 q^{7} + 4 q^{8} + 48 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.295423	+	1.38301i	−0.208895	+	0.977938i
							\(3\)	−0.540641	−	0.841254i	−0.312139	−	0.485698i
\(5\)	−3.08337	+	1.40813i	−1.37893	+	0.629735i								−0.960440	−	0.278488i	\(-0.910167\pi\)
							−0.418487	+	0.908223i	\(0.637439\pi\)
\(7\)	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	−0.983076	−	0.183199i	\(-0.941355\pi\)
							0.0414276	+	0.999142i	\(0.486809\pi\)
\(13\)	1.13414	−	3.86251i	0.314553	−	1.07127i	−0.638791	−	0.769381i	\(-0.720565\pi\)
							0.953343	−	0.301888i	\(-0.0976169\pi\)
\(17\)	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	0.184777	−	0.982780i	\(-0.440844\pi\)
							0.454174	+	0.890913i	\(0.349934\pi\)
\(23\)	−0.614601	−	4.75629i	−0.128153	−	0.991754i
							\(29\)	9.62875	+	1.38441i	1.78801	+	0.257078i	0.955091	−	0.296312i	\(-0.0957568\pi\)
0.832923	+	0.553390i	\(0.186666\pi\)
\(31\)	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	0.999547	−	0.0301111i	\(-0.00958609\pi\)
							0.631807	+	0.775125i	\(0.282313\pi\)
\(41\)	−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	\(-0.849677\pi\)
							−0.0438417	−	0.999038i	\(-0.513960\pi\)
\(47\)	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.85.22	yes	480
8.5	even	2	inner	552.2.bb.a.85.6	yes	480
23.13	even	11	inner	552.2.bb.a.13.6	✓	480
184.13	even	22	inner	552.2.bb.a.13.22	yes	480

    

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