Embedded newform 552.2.t.b.19.8

• Label: 552.2.t.b.19.8
• 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			19.8
Character	\(\chi\)	\(=\)	552.19
Dual form			552.2.t.b.523.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.617779 + 1.27214i) q^{2} +(-0.959493 + 0.281733i) q^{3} +(-1.23670 - 1.57181i) q^{4} +(2.92140 + 1.87747i) q^{5} +(0.234350 - 1.39466i) q^{6} +(0.376896 + 2.62137i) q^{7} +(2.76357 - 0.602229i) q^{8} +(0.841254 - 0.540641i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 4 q^{2} - 24 q^{3} - 4 q^{4} - 7 q^{6} + 4 q^{8} - 24 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.617779	+	1.27214i	−0.436836	+	0.899541i
							\(3\)	−0.959493	+	0.281733i	−0.553964	+	0.162658i
\(5\)	2.92140	+	1.87747i	1.30649	+	0.839631i								0.993903	−	0.110258i	\(-0.0351678\pi\)
							0.312588	+	0.949889i	\(0.398804\pi\)
\(7\)	0.376896	+	2.62137i	0.142453	+	0.990784i	0.928159	+	0.372184i	\(0.121391\pi\)
							−0.785706	+	0.618600i	\(0.787700\pi\)
\(11\)	−3.76855	−	1.72104i	−1.13626	−	0.518912i	−0.243704	−	0.969850i	\(-0.578362\pi\)
							−0.892555	+	0.450938i	\(0.851090\pi\)
\(13\)	6.51950	+	0.937362i	1.80818	+	0.259978i	0.962027	−	0.272956i	\(-0.0880013\pi\)
							0.846157	+	0.532934i	\(0.178910\pi\)
\(17\)	2.64251	−	2.28975i	0.640903	−	0.555345i	−0.272626	−	0.962120i	\(-0.587892\pi\)
							0.913528	+	0.406775i	\(0.133347\pi\)
\(19\)	1.26105	+	1.09270i	0.289304	+	0.250683i	0.787405	−	0.616435i	\(-0.211424\pi\)
							−0.498102	+	0.867119i	\(0.665969\pi\)
\(23\)	2.51678	+	4.08238i	0.524785	+	0.851235i
							\(29\)	−4.71099	+	4.08210i	−0.874809	+	0.758026i	−0.971438	−	0.237294i	\(-0.923739\pi\)
0.0966288	+	0.995320i	\(0.469194\pi\)
\(31\)	1.71507	−	5.84100i	0.308036	−	1.04907i	−0.649404	−	0.760443i	\(-0.724982\pi\)
							0.957440	−	0.288631i	\(-0.0932001\pi\)
\(37\)	−8.25536	+	5.30540i	−1.35717	+	0.872202i	−0.998131	−	0.0611186i	\(-0.980533\pi\)
							−0.359043	+	0.933321i	\(0.616897\pi\)
\(41\)	−9.93891	−	6.38735i	−1.55220	−	0.997537i	−0.984719	−	0.174148i	\(-0.944283\pi\)
							−0.567478	−	0.823388i	\(-0.692081\pi\)
\(43\)	1.56829	+	5.34111i	0.239162	+	0.814511i	0.988356	+	0.152160i	\(0.0486229\pi\)
							−0.749194	+	0.662351i	\(0.769559\pi\)
\(47\)			4.03174i			0.588089i	0.955792	+	0.294045i	\(0.0950014\pi\)
							−0.955792	+	0.294045i	\(0.904999\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.2.t.b.19.8	✓	240
8.3	odd	2	inner	552.2.t.b.19.23	yes	240
23.17	odd	22	inner	552.2.t.b.523.23	yes	240
184.155	even	22	inner	552.2.t.b.523.8	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.t.b.19.8	✓	240	1.1	even	1	trivial
552.2.t.b.19.23	yes	240	8.3	odd	2	inner
552.2.t.b.523.8	yes	240	184.155	even	22	inner
552.2.t.b.523.23	yes	240	23.17	odd	22	inner