Embedded newform 552.2.bb.a.13.1

• Label: 552.2.bb.a.13.1
• 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			13.1
Character	\(\chi\)	\(=\)	552.13
Dual form			552.2.bb.a.85.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.40928 - 0.118034i) q^{2} +(0.540641 - 0.841254i) q^{3} +(1.97214 + 0.332685i) q^{4} +(-3.96495 - 1.81073i) q^{5} +(-0.861210 + 1.12175i) q^{6} +(-1.61967 - 0.475578i) q^{7} +(-2.74002 - 0.701625i) q^{8} +(-0.415415 - 0.909632i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q + 4 q^{2} + 4 q^{6} + 8 q^{7} + 4 q^{8} + 48 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.40928	−	0.118034i	−0.996511	−	0.0834626i
							\(3\)	0.540641	−	0.841254i	0.312139	−	0.485698i
\(5\)	−3.96495	−	1.81073i	−1.77318	−	0.809784i								−0.979499	−	0.201450i	\(-0.935434\pi\)
							−0.793681	−	0.608334i	\(-0.791838\pi\)
\(7\)	−1.61967	−	0.475578i	−0.612178	−	0.179752i	−0.0390769	−	0.999236i	\(-0.512442\pi\)
							−0.573101	+	0.819484i	\(0.694260\pi\)
\(11\)	−0.819826	−	0.710383i	−0.247187	−	0.214189i	0.522451	−	0.852669i	\(-0.325018\pi\)
							−0.769638	+	0.638481i	\(0.779563\pi\)
\(13\)	0.113504	+	0.386558i	0.0314803	+	0.107212i	0.973734	−	0.227687i	\(-0.0731161\pi\)
							−0.942254	+	0.334899i	\(0.891298\pi\)
\(17\)	0.783629	+	5.45026i	0.190058	+	1.32188i	0.831849	+	0.555002i	\(0.187283\pi\)
							−0.641791	+	0.766880i	\(0.721808\pi\)
\(19\)	0.673908	+	0.0968933i	0.154605	+	0.0222289i	0.219183	−	0.975684i	\(-0.429661\pi\)
							−0.0645776	+	0.997913i	\(0.520570\pi\)
\(23\)	3.43668	+	3.34503i	0.716597	+	0.697487i
							\(29\)	−2.81279	+	0.404418i	−0.522322	+	0.0750985i	−0.398433	−	0.917198i	\(-0.630446\pi\)
−0.123889	+	0.992296i	\(0.539537\pi\)
\(31\)	−2.24367	+	1.44192i	−0.402976	+	0.258977i	−0.726393	−	0.687280i	\(-0.758805\pi\)
							0.323417	+	0.946257i	\(0.395168\pi\)
\(37\)	−3.01332	+	1.37614i	−0.495386	+	0.226235i	−0.647411	−	0.762141i	\(-0.724148\pi\)
							0.152024	+	0.988377i	\(0.451421\pi\)
\(41\)	3.05516	−	6.68988i	0.477137	−	1.04478i	−0.506104	−	0.862472i	\(-0.668915\pi\)
							0.983241	−	0.182311i	\(-0.0583578\pi\)
\(43\)	0.367868	−	0.572414i	0.0560993	−	0.0872922i	−0.812071	−	0.583559i	\(-0.801660\pi\)
							0.868170	+	0.496267i	\(0.165296\pi\)
\(47\)	−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.13.1	✓	480
8.5	even	2	inner	552.2.bb.a.13.27	yes	480
23.16	even	11	inner	552.2.bb.a.85.27	yes	480
184.85	even	22	inner	552.2.bb.a.85.1	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.bb.a.13.1	✓	480	1.1	even	1	trivial
552.2.bb.a.13.27	yes	480	8.5	even	2	inner
552.2.bb.a.85.1	yes	480	184.85	even	22	inner
552.2.bb.a.85.27	yes	480	23.16	even	11	inner