Embedded newform 552.2.bb.a.469.37

• Label: 552.2.bb.a.469.37
• Level: $552$
• Weight: $2$
• Character: 552.469
• 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			469.37
Character	\(\chi\)	\(=\)	552.469
Dual form			552.2.bb.a.133.37

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.10776 - 0.879131i) q^{2} +(-0.755750 + 0.654861i) q^{3} +(0.454259 - 1.94773i) q^{4} +(3.07552 - 0.442193i) q^{5} +(-0.261480 + 1.38983i) q^{6} +(1.27307 - 2.78762i) q^{7} +(-1.20910 - 2.55697i) q^{8} +(0.142315 - 0.989821i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q + 4 q^{2} + 4 q^{6} + 8 q^{7} + 4 q^{8} + 48 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/552\mathbb{Z}\right)^\times\).

\(n\)	\(97\)	\(185\)	\(277\)	\(415\)
\(\chi(n)\)	\(e\left(\frac{5}{11}\right)\)	\(1\)	\(-1\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.10776	−	0.879131i	0.783304	−	0.621639i
							\(3\)	−0.755750	+	0.654861i	−0.436332	+	0.378084i
\(5\)	3.07552	−	0.442193i	1.37542	−	0.197755i								0.585352	−	0.810780i	\(-0.300956\pi\)
							0.790064	+	0.613025i	\(0.210047\pi\)
\(7\)	1.27307	−	2.78762i	0.481174	−	1.05362i	−0.500966	−	0.865467i	\(-0.667022\pi\)
							0.982139	−	0.188156i	\(-0.0602510\pi\)
\(11\)	−0.100084	+	0.340854i	−0.0301764	+	0.102771i	−0.973204	−	0.229942i	\(-0.926146\pi\)
							0.943028	+	0.332714i	\(0.107964\pi\)
\(13\)	−5.09316	+	2.32597i	−1.41259	+	0.645107i	−0.968074	−	0.250665i	\(-0.919351\pi\)
							−0.444514	+	0.895772i	\(0.646623\pi\)
\(17\)	4.48672	+	2.88344i	1.08819	+	0.699337i	0.956435	−	0.291945i	\(-0.0943025\pi\)
							0.131755	+	0.991282i	\(0.457939\pi\)
\(19\)	0.803720	+	1.25061i	0.184386	+	0.286910i	0.921125	−	0.389267i	\(-0.127272\pi\)
							−0.736739	+	0.676177i	\(0.763635\pi\)
\(23\)	−3.41356	−	3.36862i	−0.711776	−	0.702407i
							\(29\)	3.99184	−	6.21143i	0.741266	−	1.15343i	−0.241826	−	0.970320i	\(-0.577746\pi\)
0.983092	−	0.183113i	\(-0.0586174\pi\)
\(31\)	−1.04581	+	1.20693i	−0.187833	+	0.216771i	−0.841854	−	0.539706i	\(-0.818535\pi\)
							0.654021	+	0.756477i	\(0.273081\pi\)
\(37\)	5.87018	+	0.844004i	0.965051	+	0.138753i	0.606785	−	0.794866i	\(-0.292459\pi\)
							0.358266	+	0.933619i	\(0.383368\pi\)
\(41\)	1.40087	+	9.74326i	0.218779	+	1.52164i	0.742555	+	0.669785i	\(0.233614\pi\)
							−0.523776	+	0.851856i	\(0.675477\pi\)
\(43\)	5.06294	−	4.38706i	0.772091	−	0.669021i	−0.176933	−	0.984223i	\(-0.556618\pi\)
							0.949025	+	0.315202i	\(0.102072\pi\)
\(47\)	−10.7149			−1.56293			−0.781465	−	0.623949i	\(-0.785527\pi\)
							−0.781465	+	0.623949i	\(0.785527\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.2.bb.a.469.37	yes	480
8.5	even	2	inner	552.2.bb.a.469.46	yes	480
23.18	even	11	inner	552.2.bb.a.133.46	yes	480
184.133	even	22	inner	552.2.bb.a.133.37	✓	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.bb.a.133.37	✓	480	184.133	even	22	inner
552.2.bb.a.133.46	yes	480	23.18	even	11	inner
552.2.bb.a.469.37	yes	480	1.1	even	1	trivial
552.2.bb.a.469.46	yes	480	8.5	even	2	inner