Embedded newform 552.2.bb.a.13.4

• Label: 552.2.bb.a.13.4
• 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			13.4
Character	\(\chi\)	\(=\)	552.13
Dual form			552.2.bb.a.85.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.37160 + 0.344553i) q^{2} +(0.540641 - 0.841254i) q^{3} +(1.76257 - 0.945176i) q^{4} +(2.26933 + 1.03637i) q^{5} +(-0.451686 + 1.34014i) q^{6} +(-2.49497 - 0.732589i) q^{7} +(-2.09187 + 1.90370i) q^{8} +(-0.415415 - 0.909632i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q + 4 q^{2} + 4 q^{6} + 8 q^{7} + 4 q^{8} + 48 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.37160	+	0.344553i	−0.969867	+	0.243636i
							\(3\)	0.540641	−	0.841254i	0.312139	−	0.485698i
\(5\)	2.26933	+	1.03637i	1.01488	+	0.463479i								0.852208	−	0.523202i	\(-0.175263\pi\)
							0.162668	+	0.986681i	\(0.447990\pi\)
\(7\)	−2.49497	−	0.732589i	−0.943009	−	0.276893i	−0.226136	−	0.974096i	\(-0.572609\pi\)
							−0.716874	+	0.697203i	\(0.754428\pi\)
\(11\)	−3.71536	−	3.21937i	−1.12022	−	0.970678i	−0.120465	−	0.992718i	\(-0.538439\pi\)
							−0.999757	+	0.0220398i	\(0.992984\pi\)
\(13\)	−1.81486	−	6.18084i	−0.503352	−	1.71426i	−0.682903	−	0.730509i	\(-0.739283\pi\)
							0.179551	−	0.983749i	\(-0.442535\pi\)
\(17\)	0.253473	+	1.76294i	0.0614762	+	0.427576i	0.997196	+	0.0748325i	\(0.0238422\pi\)
							−0.935720	+	0.352744i	\(0.885249\pi\)
\(19\)	−2.93640	−	0.422191i	−0.673657	−	0.0968572i	−0.203007	−	0.979177i	\(-0.565071\pi\)
							−0.470650	+	0.882320i	\(0.655981\pi\)
\(23\)	1.87931	−	4.41228i	0.391864	−	0.920023i
							\(29\)	8.21491	−	1.18113i	1.52547	−	0.219329i	0.672031	−	0.740523i	\(-0.265422\pi\)
0.853439	+	0.521193i	\(0.174513\pi\)
\(31\)	−7.54861	+	4.85119i	−1.35577	+	0.871300i	−0.998043	−	0.0625263i	\(-0.980084\pi\)
							−0.357726	+	0.933826i	\(0.616448\pi\)
\(37\)	6.44490	−	2.94329i	1.05953	−	0.483873i	0.192075	−	0.981380i	\(-0.438478\pi\)
							0.867460	+	0.497507i	\(0.165751\pi\)
\(41\)	−1.79665	+	3.93411i	−0.280589	+	0.614405i	−0.996482	−	0.0838053i	\(-0.973293\pi\)
							0.715893	+	0.698210i	\(0.246020\pi\)
\(43\)	2.68440	−	4.17701i	0.409367	−	0.636988i	−0.573949	−	0.818891i	\(-0.694589\pi\)
							0.983316	+	0.181903i	\(0.0582257\pi\)
\(47\)	8.53380			1.24478			0.622391	−	0.782706i	\(-0.286161\pi\)
							0.622391	+	0.782706i	\(0.286161\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.2.bb.a.13.4	✓	480
8.5	even	2	inner	552.2.bb.a.13.29	yes	480
23.16	even	11	inner	552.2.bb.a.85.29	yes	480
184.85	even	22	inner	552.2.bb.a.85.4	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.bb.a.13.4	✓	480	1.1	even	1	trivial
552.2.bb.a.13.29	yes	480	8.5	even	2	inner
552.2.bb.a.85.4	yes	480	184.85	even	22	inner
552.2.bb.a.85.29	yes	480	23.16	even	11	inner