Embedded newform 552.2.bb.a.13.42

• Label: 552.2.bb.a.13.42
• 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			13.42
Character	\(\chi\)	\(=\)	552.13
Dual form			552.2.bb.a.85.42

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.30213 - 0.551786i) q^{2} +(-0.540641 + 0.841254i) q^{3} +(1.39107 - 1.43699i) q^{4} +(-3.52460 - 1.60963i) q^{5} +(-0.239791 + 1.39374i) q^{6} +(4.53885 + 1.33273i) q^{7} +(1.01843 - 2.63871i) q^{8} +(-0.415415 - 0.909632i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q + 4 q^{2} + 4 q^{6} + 8 q^{7} + 4 q^{8} + 48 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.30213	−	0.551786i	0.920742	−	0.390171i
							\(3\)	−0.540641	+	0.841254i	−0.312139	+	0.485698i
\(5\)	−3.52460	−	1.60963i	−1.57625	−	0.719849i								−0.580702	−	0.814116i	\(-0.697222\pi\)
							−0.995547	+	0.0942670i	\(0.969949\pi\)
\(7\)	4.53885	+	1.33273i	1.71552	+	0.503723i	0.984012	−	0.178105i	\(-0.0569966\pi\)
							0.731512	+	0.681828i	\(0.238815\pi\)
\(11\)	2.16140	+	1.87287i	0.651688	+	0.564691i	0.916711	−	0.399551i	\(-0.130834\pi\)
							−0.265023	+	0.964242i	\(0.585380\pi\)
\(13\)	−1.08690	−	3.70164i	−0.301452	−	1.02665i	−0.961358	−	0.275302i	\(-0.911222\pi\)
							0.659906	−	0.751348i	\(-0.270596\pi\)
\(17\)	−0.821395	−	5.71293i	−0.199218	−	1.38559i	−0.806562	−	0.591150i	\(-0.798674\pi\)
							0.607344	−	0.794439i	\(-0.292235\pi\)
\(19\)	4.11880	+	0.592194i	0.944917	+	0.135859i	0.597515	−	0.801858i	\(-0.296155\pi\)
							0.347402	+	0.937716i	\(0.387064\pi\)
\(23\)	0.624683	−	4.75497i	0.130255	−	0.991480i
							\(29\)	2.26772	−	0.326049i	0.421105	−	0.0605458i	0.0714949	−	0.997441i	\(-0.477223\pi\)
0.349610	+	0.936895i	\(0.386314\pi\)
\(31\)	−4.43896	+	2.85275i	−0.797261	+	0.512368i	−0.874721	−	0.484627i	\(-0.838955\pi\)
							0.0774600	+	0.996995i	\(0.475319\pi\)
\(37\)	3.15297	−	1.43991i	0.518344	−	0.236720i	−0.139024	−	0.990289i	\(-0.544397\pi\)
							0.657369	+	0.753569i	\(0.271669\pi\)
\(41\)	−3.72589	+	8.15855i	−0.581886	+	1.27415i	0.358337	+	0.933592i	\(0.383344\pi\)
							−0.940223	+	0.340560i	\(0.889383\pi\)
\(43\)	−2.93821	+	4.57195i	−0.448073	+	0.697216i	−0.989661	−	0.143429i	\(-0.954187\pi\)
							0.541587	+	0.840645i	\(0.317824\pi\)
\(47\)	−1.81148			−0.264232			−0.132116	−	0.991234i	\(-0.542177\pi\)
							−0.132116	+	0.991234i	\(0.542177\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.2.bb.a.13.42	yes	480
8.5	even	2	inner	552.2.bb.a.13.16	✓	480
23.16	even	11	inner	552.2.bb.a.85.16	yes	480
184.85	even	22	inner	552.2.bb.a.85.42	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.bb.a.13.16	✓	480	8.5	even	2	inner
552.2.bb.a.13.42	yes	480	1.1	even	1	trivial
552.2.bb.a.85.16	yes	480	23.16	even	11	inner
552.2.bb.a.85.42	yes	480	184.85	even	22	inner