Embedded newform 552.2.t.b.451.10

• Label: 552.2.t.b.451.10
• Level: $552$
• Weight: $2$
• Character: 552.451
• Analytic conductor: $4.408$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 552 = 2^{3} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	552.t (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.40774219157\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(24\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			451.10
Character	\(\chi\)	\(=\)	552.451
Dual form			552.2.t.b.235.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.395410 - 1.35781i) q^{2} +(-0.654861 - 0.755750i) q^{3} +(-1.68730 + 1.07378i) q^{4} +(-0.236813 - 1.64707i) q^{5} +(-0.767227 + 1.18801i) q^{6} +(-1.60690 + 3.51862i) q^{7} +(2.12517 + 1.86645i) q^{8} +(-0.142315 + 0.989821i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 4 q^{2} - 24 q^{3} - 4 q^{4} - 7 q^{6} + 4 q^{8} - 24 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(97\)	\(185\)	\(277\)	\(415\)
\(\chi(n)\)	\(e\left(\frac{21}{22}\right)\)	\(1\)	\(-1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.395410	−	1.35781i	−0.279597	−	0.960117i
							\(3\)	−0.654861	−	0.755750i	−0.378084	−	0.436332i
\(5\)	−0.236813	−	1.64707i	−0.105906	−	0.736594i								−0.971704	−	0.236200i	\(-0.924098\pi\)
							0.865798	−	0.500393i	\(-0.166811\pi\)
\(7\)	−1.60690	+	3.51862i	−0.607350	+	1.32991i	0.317021	+	0.948418i	\(0.397317\pi\)
							−0.924372	+	0.381493i	\(0.875410\pi\)
\(11\)	1.51637	−	5.16429i	0.457203	−	1.55709i	−0.332196	−	0.943210i	\(-0.607789\pi\)
							0.789399	−	0.613881i	\(-0.210392\pi\)
\(13\)	−4.93581	+	2.25411i	−1.36895	+	0.625178i	−0.958076	−	0.286515i	\(-0.907503\pi\)
							−0.410873	+	0.911693i	\(0.634776\pi\)
\(17\)	−3.45041	+	5.36895i	−0.836848	+	1.30216i	0.114306	+	0.993446i	\(0.463536\pi\)
							−0.951154	+	0.308716i	\(0.900101\pi\)
\(19\)	4.29757	+	6.68715i	0.985931	+	1.53414i	0.838382	+	0.545084i	\(0.183502\pi\)
							0.147549	+	0.989055i	\(0.452861\pi\)
\(23\)	2.68357	+	3.97473i	0.559562	+	0.828788i
							\(29\)	−1.09187	+	1.69898i	−0.202755	+	0.315493i	−0.927709	−	0.373304i	\(-0.878225\pi\)
0.724954	+	0.688798i	\(0.241861\pi\)
\(31\)	−0.0568213	−	0.0492360i	−0.0102054	−	0.00884304i	0.649743	−	0.760154i	\(-0.274876\pi\)
							−0.659949	+	0.751311i	\(0.729422\pi\)
\(37\)	0.418616	−	2.91154i	0.0688201	−	0.478654i	−0.926042	−	0.377419i	\(-0.876812\pi\)
							0.994863	−	0.101235i	\(-0.0322794\pi\)
\(41\)	−0.237983	−	1.65521i	−0.0371667	−	0.258500i	0.962763	−	0.270346i	\(-0.0871382\pi\)
							−0.999930	+	0.0118463i	\(0.996229\pi\)
\(43\)	4.19080	−	3.63135i	0.639091	−	0.553775i	−0.273899	−	0.961759i	\(-0.588313\pi\)
							0.912989	+	0.407983i	\(0.133768\pi\)
\(47\)		−	1.86139i		−	0.271512i	−0.990742	−	0.135756i	\(-0.956654\pi\)
							0.990742	−	0.135756i	\(-0.0433464\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.2.t.b.451.10	yes	240
8.3	odd	2	inner	552.2.t.b.451.14	yes	240
23.5	odd	22	inner	552.2.t.b.235.14	yes	240
184.51	even	22	inner	552.2.t.b.235.10	✓	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.t.b.235.10	✓	240	184.51	even	22	inner
552.2.t.b.235.14	yes	240	23.5	odd	22	inner
552.2.t.b.451.10	yes	240	1.1	even	1	trivial
552.2.t.b.451.14	yes	240	8.3	odd	2	inner