Embedded newform 552.2.bb.a.85.27

• Label: 552.2.bb.a.85.27
• 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			85.27
Character	\(\chi\)	\(=\)	552.85
Dual form			552.2.bb.a.13.27

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0837289 + 1.41173i) q^{2} +(-0.540641 - 0.841254i) q^{3} +(-1.98598 + 0.236406i) q^{4} +(3.96495 - 1.81073i) q^{5} +(1.14236 - 0.833678i) q^{6} +(-1.61967 + 0.475578i) q^{7} +(-0.500025 - 2.78388i) q^{8} +(-0.415415 + 0.909632i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q + 4 q^{2} + 4 q^{6} + 8 q^{7} + 4 q^{8} + 48 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0837289	+	1.41173i	0.0592053	+	0.998246i
							\(3\)	−0.540641	−	0.841254i	−0.312139	−	0.485698i
\(5\)	3.96495	−	1.81073i	1.77318	−	0.809784i								0.793681	−	0.608334i	\(-0.208162\pi\)
							0.979499	−	0.201450i	\(-0.0645655\pi\)
\(7\)	−1.61967	+	0.475578i	−0.612178	+	0.179752i	−0.573101	−	0.819484i	\(-0.694260\pi\)
							−0.0390769	+	0.999236i	\(0.512442\pi\)
\(11\)	0.819826	−	0.710383i	0.247187	−	0.214189i	−0.522451	−	0.852669i	\(-0.674982\pi\)
							0.769638	+	0.638481i	\(0.220437\pi\)
\(13\)	−0.113504	+	0.386558i	−0.0314803	+	0.107212i	−0.973734	−	0.227687i	\(-0.926884\pi\)
							0.942254	+	0.334899i	\(0.108702\pi\)
\(17\)	0.783629	−	5.45026i	0.190058	−	1.32188i	−0.641791	−	0.766880i	\(-0.721808\pi\)
							0.831849	−	0.555002i	\(-0.187283\pi\)
\(19\)	−0.673908	+	0.0968933i	−0.154605	+	0.0222289i	−0.219183	−	0.975684i	\(-0.570339\pi\)
							0.0645776	+	0.997913i	\(0.479430\pi\)
\(23\)	3.43668	−	3.34503i	0.716597	−	0.697487i
							\(29\)	2.81279	+	0.404418i	0.522322	+	0.0750985i	0.398433	−	0.917198i	\(-0.369554\pi\)
0.123889	+	0.992296i	\(0.460463\pi\)
\(31\)	−2.24367	−	1.44192i	−0.402976	−	0.258977i	0.323417	−	0.946257i	\(-0.395168\pi\)
							−0.726393	+	0.687280i	\(0.758805\pi\)
\(37\)	3.01332	+	1.37614i	0.495386	+	0.226235i	0.647411	−	0.762141i	\(-0.275852\pi\)
							−0.152024	+	0.988377i	\(0.548579\pi\)
\(41\)	3.05516	+	6.68988i	0.477137	+	1.04478i	0.983241	+	0.182311i	\(0.0583578\pi\)
							−0.506104	+	0.862472i	\(0.668915\pi\)
\(43\)	−0.367868	−	0.572414i	−0.0560993	−	0.0872922i	0.812071	−	0.583559i	\(-0.198340\pi\)
							−0.868170	+	0.496267i	\(0.834704\pi\)
\(47\)	−2.72025			−0.396790			−0.198395	−	0.980122i	\(-0.563573\pi\)
							−0.198395	+	0.980122i	\(0.563573\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.2.bb.a.85.27	yes	480
8.5	even	2	inner	552.2.bb.a.85.1	yes	480
23.13	even	11	inner	552.2.bb.a.13.1	✓	480
184.13	even	22	inner	552.2.bb.a.13.27	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.bb.a.13.1	✓	480	23.13	even	11	inner
552.2.bb.a.13.27	yes	480	184.13	even	22	inner
552.2.bb.a.85.1	yes	480	8.5	even	2	inner
552.2.bb.a.85.27	yes	480	1.1	even	1	trivial