Embedded newform 552.2.bb.a.85.24

• Label: 552.2.bb.a.85.24
• 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			85.24
Character	\(\chi\)	\(=\)	552.85
Dual form			552.2.bb.a.13.24

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.126837 + 1.40851i) q^{2} +(0.540641 + 0.841254i) q^{3} +(-1.96782 - 0.357302i) q^{4} +(-0.902350 + 0.412090i) q^{5} +(-1.25349 + 0.654799i) q^{6} +(-3.01070 + 0.884022i) q^{7} +(0.752858 - 2.72639i) q^{8} +(-0.415415 + 0.909632i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q + 4 q^{2} + 4 q^{6} + 8 q^{7} + 4 q^{8} + 48 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.126837	+	1.40851i	−0.0896870	+	0.995970i
							\(3\)	0.540641	+	0.841254i	0.312139	+	0.485698i
\(5\)	−0.902350	+	0.412090i	−0.403543	+	0.184292i								−0.606838	−	0.794826i	\(-0.707562\pi\)
							0.203295	+	0.979118i	\(0.434835\pi\)
\(7\)	−3.01070	+	0.884022i	−1.13794	+	0.334129i	−0.795823	−	0.605529i	\(-0.792962\pi\)
							−0.342116	+	0.939658i	\(0.611143\pi\)
\(11\)	−2.49850	+	2.16496i	−0.753325	+	0.652760i	−0.944394	−	0.328817i	\(-0.893350\pi\)
							0.191069	+	0.981577i	\(0.438805\pi\)
\(13\)	0.168470	−	0.573755i	0.0467251	−	0.159131i	−0.932826	−	0.360328i	\(-0.882665\pi\)
							0.979551	+	0.201197i	\(0.0644830\pi\)
\(17\)	1.03373	−	7.18972i	0.250715	−	1.74376i	−0.343223	−	0.939254i	\(-0.611519\pi\)
							0.593939	−	0.804510i	\(-0.297572\pi\)
\(19\)	2.99071	−	0.429999i	0.686115	−	0.0986484i	0.209564	−	0.977795i	\(-0.432796\pi\)
							0.476551	+	0.879147i	\(0.341887\pi\)
\(23\)	−4.32781	−	2.06641i	−0.902411	−	0.430876i
							\(29\)	−3.65492	−	0.525498i	−0.678701	−	0.0975825i	−0.205661	−	0.978623i	\(-0.565934\pi\)
−0.473040	+	0.881041i	\(0.656843\pi\)
\(31\)	−4.15121	−	2.66782i	−0.745579	−	0.479154i	0.111871	−	0.993723i	\(-0.464316\pi\)
							−0.857449	+	0.514569i	\(0.827952\pi\)
\(37\)	−1.10676	−	0.505438i	−0.181949	−	0.0830935i	0.322357	−	0.946618i	\(-0.395525\pi\)
							−0.504307	+	0.863524i	\(0.668252\pi\)
\(41\)	−1.87780	−	4.11180i	−0.293263	−	0.642156i	0.704450	−	0.709754i	\(-0.251194\pi\)
							−0.997713	+	0.0675980i	\(0.978466\pi\)
\(43\)	2.11759	+	3.29503i	0.322929	+	0.502487i	0.964324	−	0.264723i	\(-0.0852806\pi\)
							−0.641396	+	0.767210i	\(0.721644\pi\)
\(47\)	−10.7244			−1.56432			−0.782161	−	0.623077i	\(-0.785882\pi\)
							−0.782161	+	0.623077i	\(0.785882\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.2.bb.a.85.24	yes	480
8.5	even	2	inner	552.2.bb.a.85.3	yes	480
23.13	even	11	inner	552.2.bb.a.13.3	✓	480
184.13	even	22	inner	552.2.bb.a.13.24	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.bb.a.13.3	✓	480	23.13	even	11	inner
552.2.bb.a.13.24	yes	480	184.13	even	22	inner
552.2.bb.a.85.3	yes	480	8.5	even	2	inner
552.2.bb.a.85.24	yes	480	1.1	even	1	trivial