Embedded newform 552.2.bb.a.469.10

• Label: 552.2.bb.a.469.10
• 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			469.10
Character	\(\chi\)	\(=\)	552.469
Dual form			552.2.bb.a.133.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.11453 + 0.870525i) q^{2} +(-0.755750 + 0.654861i) q^{3} +(0.484373 - 1.94046i) q^{4} +(0.714854 - 0.102780i) q^{5} +(0.272236 - 1.38776i) q^{6} +(-0.269086 + 0.589215i) q^{7} +(1.14937 + 2.58437i) q^{8} +(0.142315 - 0.989821i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q + 4 q^{2} + 4 q^{6} + 8 q^{7} + 4 q^{8} + 48 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.11453	+	0.870525i	−0.788095	+	0.615554i
							\(3\)	−0.755750	+	0.654861i	−0.436332	+	0.378084i
\(5\)	0.714854	−	0.102780i	0.319692	−	0.0459648i								0.0193993	−	0.999812i	\(-0.493825\pi\)
							0.300293	+	0.953847i	\(0.402916\pi\)
\(7\)	−0.269086	+	0.589215i	−0.101705	+	0.222702i	−0.953643	−	0.300940i	\(-0.902700\pi\)
							0.851938	+	0.523642i	\(0.175427\pi\)
\(11\)	1.63359	−	5.56350i	0.492546	−	1.67746i	−0.219732	−	0.975560i	\(-0.570518\pi\)
							0.712278	−	0.701897i	\(-0.247663\pi\)
\(13\)	−2.71694	+	1.24079i	−0.753544	+	0.344132i	−0.754876	−	0.655867i	\(-0.772303\pi\)
							0.00133245	+	0.999999i	\(0.499576\pi\)
\(17\)	1.82767	+	1.17457i	0.443276	+	0.284876i	0.743178	−	0.669094i	\(-0.233318\pi\)
							−0.299902	+	0.953970i	\(0.596954\pi\)
\(19\)	−3.21511	−	5.00281i	−0.737597	−	1.14772i	−0.983933	−	0.178539i	\(-0.942863\pi\)
							0.246336	−	0.969185i	\(-0.420773\pi\)
\(23\)	4.59872	−	1.36082i	0.958898	−	0.283750i
							\(29\)	3.15387	−	4.90752i	0.585659	−	0.911303i	−0.414340	−	0.910122i	\(-0.635988\pi\)
0.999999	−	0.00118107i	\(-0.000375947\pi\)
\(31\)	5.19843	−	5.99931i	0.933666	−	1.07751i	−0.0631685	−	0.998003i	\(-0.520121\pi\)
							0.996834	−	0.0795050i	\(-0.0253340\pi\)
\(37\)	−3.87949	−	0.557787i	−0.637785	−	0.0916996i	−0.184165	−	0.982895i	\(-0.558958\pi\)
							−0.453619	+	0.891196i	\(0.649867\pi\)
\(41\)	−0.0678431	−	0.471859i	−0.0105953	−	0.0736920i	0.983838	−	0.179062i	\(-0.0573063\pi\)
							−0.994433	+	0.105370i	\(0.966397\pi\)
\(43\)	2.94510	−	2.55194i	0.449123	−	0.389167i	−0.400722	−	0.916200i	\(-0.631241\pi\)
							0.849845	+	0.527032i	\(0.176695\pi\)
\(47\)	8.53288			1.24465			0.622325	−	0.782759i	\(-0.286188\pi\)
							0.622325	+	0.782759i	\(0.286188\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.2.bb.a.469.10	yes	480
8.5	even	2	inner	552.2.bb.a.469.2	yes	480
23.18	even	11	inner	552.2.bb.a.133.2	✓	480
184.133	even	22	inner	552.2.bb.a.133.10	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.bb.a.133.2	✓	480	23.18	even	11	inner
552.2.bb.a.133.10	yes	480	184.133	even	22	inner
552.2.bb.a.469.2	yes	480	8.5	even	2	inner
552.2.bb.a.469.10	yes	480	1.1	even	1	trivial