Embedded newform 552.2.t.b.19.4

• Label: 552.2.t.b.19.4
• 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			19.4
Character	\(\chi\)	\(=\)	552.19
Dual form			552.2.t.b.523.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.19352 + 0.758628i) q^{2} +(-0.959493 + 0.281733i) q^{3} +(0.848969 - 1.81087i) q^{4} +(0.579145 + 0.372194i) q^{5} +(0.931442 - 1.06415i) q^{6} +(-0.117047 - 0.814077i) q^{7} +(0.360517 + 2.80536i) q^{8} +(0.841254 - 0.540641i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 4 q^{2} - 24 q^{3} - 4 q^{4} - 7 q^{6} + 4 q^{8} - 24 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.19352	+	0.758628i	−0.843944	+	0.536431i
							\(3\)	−0.959493	+	0.281733i	−0.553964	+	0.162658i
\(5\)	0.579145	+	0.372194i	0.259001	+	0.166450i								0.663697	−	0.748001i	\(-0.268986\pi\)
							−0.404696	+	0.914451i	\(0.632623\pi\)
\(7\)	−0.117047	−	0.814077i	−0.0442395	−	0.307692i	−0.999912	−	0.0132608i	\(-0.995779\pi\)
							0.955673	−	0.294431i	\(-0.0951302\pi\)
\(11\)	1.37172	+	0.626444i	0.413589	+	0.188880i	0.611332	−	0.791374i	\(-0.290634\pi\)
							−0.197743	+	0.980254i	\(0.563361\pi\)
\(13\)	−3.53680	−	0.508516i	−0.980933	−	0.141037i	−0.366852	−	0.930280i	\(-0.619564\pi\)
							−0.614081	+	0.789243i	\(0.710473\pi\)
\(17\)	5.16387	−	4.47452i	1.25242	−	1.08523i	0.259590	−	0.965719i	\(-0.416413\pi\)
							0.992833	−	0.119511i	\(-0.0381328\pi\)
\(19\)	0.514575	+	0.445882i	0.118052	+	0.102292i	0.711886	−	0.702295i	\(-0.247841\pi\)
							−0.593835	+	0.804587i	\(0.702387\pi\)
\(23\)	0.846471	+	4.72054i	0.176501	+	0.984300i
							\(29\)	3.54853	−	3.07482i	0.658946	−	0.570980i	−0.259881	−	0.965641i	\(-0.583683\pi\)
0.918827	+	0.394661i	\(0.129138\pi\)
\(31\)	1.22851	−	4.18393i	0.220647	−	0.751456i	−0.772544	−	0.634961i	\(-0.781016\pi\)
							0.993191	−	0.116495i	\(-0.0371658\pi\)
\(37\)	7.16562	−	4.60506i	1.17802	−	0.757068i	0.202999	−	0.979179i	\(-0.434931\pi\)
							0.975021	+	0.222111i	\(0.0712947\pi\)
\(41\)	5.89943	+	3.79133i	0.921336	+	0.592107i	0.913045	−	0.407859i	\(-0.133724\pi\)
							0.00829133	+	0.999966i	\(0.497361\pi\)
\(43\)	2.88406	+	9.82222i	0.439816	+	1.49787i	0.819674	+	0.572830i	\(0.194154\pi\)
							−0.379859	+	0.925045i	\(0.624027\pi\)
\(47\)		−	1.50022i		−	0.218830i	−0.993996	−	0.109415i	\(-0.965102\pi\)
							0.993996	−	0.109415i	\(-0.0348977\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.2.t.b.19.4	✓	240
8.3	odd	2	inner	552.2.t.b.19.22	yes	240
23.17	odd	22	inner	552.2.t.b.523.22	yes	240
184.155	even	22	inner	552.2.t.b.523.4	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.t.b.19.4	✓	240	1.1	even	1	trivial
552.2.t.b.19.22	yes	240	8.3	odd	2	inner
552.2.t.b.523.4	yes	240	184.155	even	22	inner
552.2.t.b.523.22	yes	240	23.17	odd	22	inner