Embedded newform 552.2.t.b.19.13

• Label: 552.2.t.b.19.13
• 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			19.13
Character	\(\chi\)	\(=\)	552.19
Dual form			552.2.t.b.523.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.175979 + 1.40322i) q^{2} +(-0.959493 + 0.281733i) q^{3} +(-1.93806 + 0.493874i) q^{4} +(1.87195 + 1.20303i) q^{5} +(-0.564184 - 1.29680i) q^{6} +(-0.493720 - 3.43390i) q^{7} +(-1.03407 - 2.63262i) q^{8} +(0.841254 - 0.540641i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 4 q^{2} - 24 q^{3} - 4 q^{4} - 7 q^{6} + 4 q^{8} - 24 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.175979	+	1.40322i	0.124436	+	0.992228i
							\(3\)	−0.959493	+	0.281733i	−0.553964	+	0.162658i
\(5\)	1.87195	+	1.20303i	0.837164	+	0.538012i								0.887547	−	0.460718i	\(-0.152408\pi\)
							−0.0503832	+	0.998730i	\(0.516044\pi\)
\(7\)	−0.493720	−	3.43390i	−0.186609	−	1.29789i	−0.840710	−	0.541486i	\(-0.817862\pi\)
							0.654101	−	0.756407i	\(-0.273047\pi\)
\(11\)	−4.20094	−	1.91850i	−1.26663	−	0.578451i	−0.335124	−	0.942174i	\(-0.608778\pi\)
							−0.931507	+	0.363723i	\(0.881505\pi\)
\(13\)	−5.08603	−	0.731260i	−1.41061	−	0.202815i	−0.605455	−	0.795880i	\(-0.707009\pi\)
							−0.805155	+	0.593065i	\(0.797918\pi\)
\(17\)	−4.30843	+	3.73328i	−1.04495	+	0.905453i	−0.995635	−	0.0933306i	\(-0.970249\pi\)
							−0.0493130	+	0.998783i	\(0.515703\pi\)
\(19\)	−5.25220	−	4.55106i	−1.20494	−	1.04408i	−0.997835	−	0.0657650i	\(-0.979051\pi\)
							−0.207102	−	0.978319i	\(-0.566403\pi\)
\(23\)	4.79338	+	0.153433i	0.999488	+	0.0319930i
							\(29\)	0.511627	−	0.443327i	0.0950067	−	0.0823238i	−0.606062	−	0.795417i	\(-0.707252\pi\)
0.701069	+	0.713093i	\(0.252706\pi\)
\(31\)	0.390868	−	1.33117i	0.0702020	−	0.239086i	−0.916916	−	0.399080i	\(-0.869329\pi\)
							0.987118	+	0.159994i	\(0.0511476\pi\)
\(37\)	−2.14108	+	1.37599i	−0.351992	+	0.226211i	−0.704673	−	0.709532i	\(-0.748906\pi\)
							0.352681	+	0.935744i	\(0.385270\pi\)
\(41\)	2.94204	+	1.89073i	0.459470	+	0.295283i	0.749826	−	0.661636i	\(-0.230137\pi\)
							−0.290356	+	0.956919i	\(0.593774\pi\)
\(43\)	3.48316	+	11.8626i	0.531177	+	1.80902i	0.585724	+	0.810511i	\(0.300810\pi\)
							−0.0545467	+	0.998511i	\(0.517371\pi\)
\(47\)		−	10.1857i		−	1.48573i	−0.669441	−	0.742865i	\(-0.733466\pi\)
							0.669441	−	0.742865i	\(-0.266534\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.2.t.b.19.13	✓	240
8.3	odd	2	inner	552.2.t.b.19.18	yes	240
23.17	odd	22	inner	552.2.t.b.523.18	yes	240
184.155	even	22	inner	552.2.t.b.523.13	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.t.b.19.13	✓	240	1.1	even	1	trivial
552.2.t.b.19.18	yes	240	8.3	odd	2	inner
552.2.t.b.523.13	yes	240	184.155	even	22	inner
552.2.t.b.523.18	yes	240	23.17	odd	22	inner