Embedded newform 552.2.t.b.19.11

• Label: 552.2.t.b.19.11
• 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			19.11
Character	\(\chi\)	\(=\)	552.19
Dual form			552.2.t.b.523.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.218084 + 1.39730i) q^{2} +(-0.959493 + 0.281733i) q^{3} +(-1.90488 - 0.609457i) q^{4} +(-0.659581 - 0.423887i) q^{5} +(-0.184414 - 1.40214i) q^{6} +(-0.167612 - 1.16577i) q^{7} +(1.26702 - 2.52877i) q^{8} +(0.841254 - 0.540641i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 4 q^{2} - 24 q^{3} - 4 q^{4} - 7 q^{6} + 4 q^{8} - 24 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.218084	+	1.39730i	−0.154209	+	0.988038i
							\(3\)	−0.959493	+	0.281733i	−0.553964	+	0.162658i
\(5\)	−0.659581	−	0.423887i	−0.294973	−	0.189568i								0.384781	−	0.923008i	\(-0.374277\pi\)
							−0.679754	+	0.733440i	\(0.737913\pi\)
\(7\)	−0.167612	−	1.16577i	−0.0633515	−	0.440620i	−0.996668	−	0.0815649i	\(-0.974008\pi\)
							0.933316	−	0.359055i	\(-0.116901\pi\)
\(11\)	1.56705	+	0.715646i	0.472482	+	0.215775i	0.637402	−	0.770532i	\(-0.280009\pi\)
							−0.164920	+	0.986307i	\(0.552736\pi\)
\(13\)	3.17756	+	0.456864i	0.881296	+	0.126711i	0.568082	−	0.822972i	\(-0.307685\pi\)
							0.313213	+	0.949683i	\(0.398595\pi\)
\(17\)	−3.13544	+	2.71687i	−0.760455	+	0.658938i	−0.946172	−	0.323665i	\(-0.895085\pi\)
							0.185717	+	0.982603i	\(0.440539\pi\)
\(19\)	6.17872	+	5.35389i	1.41750	+	1.22827i	0.936086	+	0.351770i	\(0.114420\pi\)
							0.481409	+	0.876496i	\(0.340125\pi\)
\(23\)	−3.84566	+	2.86546i	−0.801877	+	0.597490i
							\(29\)	4.90056	−	4.24636i	0.910010	−	0.788528i	−0.0678699	−	0.997694i	\(-0.521620\pi\)
0.977880	+	0.209166i	\(0.0670748\pi\)
\(31\)	−1.95081	+	6.64385i	−0.350376	+	1.19327i	0.576246	+	0.817276i	\(0.304517\pi\)
							−0.926622	+	0.375994i	\(0.877301\pi\)
\(37\)	6.79820	−	4.36894i	1.11762	−	0.718249i	0.154678	−	0.987965i	\(-0.450566\pi\)
							0.962940	+	0.269716i	\(0.0869297\pi\)
\(41\)	7.75417	+	4.98330i	1.21100	+	0.778261i	0.980825	−	0.194889i	\(-0.0624346\pi\)
							0.230172	+	0.973150i	\(0.426071\pi\)
\(43\)	−0.0435530	−	0.148328i	−0.00664178	−	0.0226198i	0.956105	−	0.293023i	\(-0.0946612\pi\)
							−0.962747	+	0.270403i	\(0.912843\pi\)
\(47\)		−	0.275978i		−	0.0402555i	−0.999797	−	0.0201277i	\(-0.993593\pi\)
							0.999797	−	0.0201277i	\(-0.00640729\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.2.t.b.19.11	✓	240
8.3	odd	2	inner	552.2.t.b.19.20	yes	240
23.17	odd	22	inner	552.2.t.b.523.20	yes	240
184.155	even	22	inner	552.2.t.b.523.11	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.t.b.19.11	✓	240	1.1	even	1	trivial
552.2.t.b.19.20	yes	240	8.3	odd	2	inner
552.2.t.b.523.11	yes	240	184.155	even	22	inner
552.2.t.b.523.20	yes	240	23.17	odd	22	inner