Embedded newform 552.2.t.b.19.9

• Label: 552.2.t.b.19.9
• Level: $552$
• Weight: $2$
• Character: 552.19
• 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			19.9
Character	\(\chi\)	\(=\)	552.19
Dual form			552.2.t.b.523.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.562315 + 1.29761i) q^{2} +(-0.959493 + 0.281733i) q^{3} +(-1.36760 - 1.45934i) q^{4} +(-1.62830 - 1.04644i) q^{5} +(0.173957 - 1.40347i) q^{6} +(0.707273 + 4.91919i) q^{7} +(2.66268 - 0.954016i) q^{8} +(0.841254 - 0.540641i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 4 q^{2} - 24 q^{3} - 4 q^{4} - 7 q^{6} + 4 q^{8} - 24 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(97\)	\(185\)	\(277\)	\(415\)
\(\chi(n)\)	\(e\left(\frac{15}{22}\right)\)	\(1\)	\(-1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.562315	+	1.29761i	−0.397617	+	0.917552i
							\(3\)	−0.959493	+	0.281733i	−0.553964	+	0.162658i
\(5\)	−1.62830	−	1.04644i	−0.728197	−	0.467984i								0.123283	−	0.992372i	\(-0.460658\pi\)
							−0.851480	+	0.524388i	\(0.824294\pi\)
\(7\)	0.707273	+	4.91919i	0.267324	+	1.85928i	0.473539	+	0.880773i	\(0.342976\pi\)
							−0.206215	+	0.978507i	\(0.566115\pi\)
\(11\)	0.738167	+	0.337110i	0.222566	+	0.101642i	0.523577	−	0.851978i	\(-0.324597\pi\)
							−0.301012	+	0.953620i	\(0.597324\pi\)
\(13\)	−6.04330	−	0.868895i	−1.67611	−	0.240988i	−0.762327	−	0.647192i	\(-0.775943\pi\)
							−0.913783	+	0.406204i	\(0.866852\pi\)
\(17\)	−0.419003	+	0.363069i	−0.101623	+	0.0880571i	−0.704191	−	0.710011i	\(-0.748690\pi\)
							0.602567	+	0.798068i	\(0.294144\pi\)
\(19\)	−2.00375	−	1.73626i	−0.459692	−	0.398326i	0.393994	−	0.919113i	\(-0.371093\pi\)
							−0.853687	+	0.520787i	\(0.825639\pi\)
\(23\)	0.178890	−	4.79249i	0.0373011	−	0.999304i
							\(29\)	6.77419	−	5.86987i	1.25794	−	1.09001i	0.265907	−	0.963999i	\(-0.414329\pi\)
0.992029	−	0.126009i	\(-0.0402168\pi\)
\(31\)	1.26344	−	4.30290i	0.226921	−	0.772823i	−0.764783	−	0.644287i	\(-0.777154\pi\)
							0.991705	−	0.128536i	\(-0.0410277\pi\)
\(37\)	−6.72361	+	4.32100i	−1.10536	+	0.710369i	−0.960276	−	0.279051i	\(-0.909980\pi\)
							−0.145079	+	0.989420i	\(0.546344\pi\)
\(41\)	−2.21438	−	1.42310i	−0.345828	−	0.222250i	0.356183	−	0.934416i	\(-0.384078\pi\)
							−0.702011	+	0.712166i	\(0.747714\pi\)
\(43\)	−1.47492	−	5.02311i	−0.224923	−	0.766017i	−0.992195	−	0.124695i	\(-0.960205\pi\)
							0.767272	−	0.641322i	\(-0.221614\pi\)
\(47\)			0.376410i			0.0549051i	0.999623	+	0.0274525i	\(0.00873951\pi\)
							−0.999623	+	0.0274525i	\(0.991260\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.2.t.b.19.9	✓	240
8.3	odd	2	inner	552.2.t.b.19.21	yes	240
23.17	odd	22	inner	552.2.t.b.523.21	yes	240
184.155	even	22	inner	552.2.t.b.523.9	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.t.b.19.9	✓	240	1.1	even	1	trivial
552.2.t.b.19.21	yes	240	8.3	odd	2	inner
552.2.t.b.523.9	yes	240	184.155	even	22	inner
552.2.t.b.523.21	yes	240	23.17	odd	22	inner