Embedded newform 552.2.t.b.283.1

• Label: 552.2.t.b.283.1
• Level: $552$
• Weight: $2$
• Character: 552.283
• 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			283.1
Character	\(\chi\)	\(=\)	552.283
Dual form			552.2.t.b.355.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.40415 + 0.168425i) q^{2} +(0.841254 - 0.540641i) q^{3} +(1.94327 - 0.472989i) q^{4} +(0.200807 + 0.439706i) q^{5} +(-1.09019 + 0.900828i) q^{6} +(-1.89746 + 0.557146i) q^{7} +(-2.64897 + 0.991442i) q^{8} +(0.415415 - 0.909632i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 4 q^{2} - 24 q^{3} - 4 q^{4} - 7 q^{6} + 4 q^{8} - 24 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/552\mathbb{Z}\right)^\times\).

\(n\)	\(97\)	\(185\)	\(277\)	\(415\)
\(\chi(n)\)	\(e\left(\frac{19}{22}\right)\)	\(1\)	\(-1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.40415	+	0.168425i	−0.992883	+	0.119095i
							\(3\)	0.841254	−	0.540641i	0.485698	−	0.312139i
\(5\)	0.200807	+	0.439706i	0.0898035	+	0.196642i								0.949204	−	0.314660i	\(-0.101891\pi\)
							−0.859401	+	0.511302i	\(0.829163\pi\)
\(7\)	−1.89746	+	0.557146i	−0.717174	+	0.210581i	−0.619905	−	0.784677i	\(-0.712829\pi\)
							−0.0972691	+	0.995258i	\(0.531011\pi\)
\(11\)	0.250721	−	0.217251i	0.0755953	−	0.0655037i	−0.616244	−	0.787556i	\(-0.711346\pi\)
							0.691839	+	0.722052i	\(0.256801\pi\)
\(13\)	1.17528	−	4.00263i	0.325964	−	1.11013i	−0.619661	−	0.784870i	\(-0.712730\pi\)
							0.945624	−	0.325261i	\(-0.105452\pi\)
\(17\)	5.47902	+	0.787764i	1.32886	+	0.191061i	0.769916	−	0.638146i	\(-0.220298\pi\)
							0.558942	+	0.829206i	\(0.311207\pi\)
\(19\)	5.30143	−	0.762231i	1.21623	−	0.174868i	0.495827	−	0.868421i	\(-0.334865\pi\)
							0.720405	+	0.693554i	\(0.243956\pi\)
\(23\)	−2.47539	+	4.10761i	−0.516154	+	0.856496i
							\(29\)	−2.16772	−	0.311671i	−0.402535	−	0.0578758i	−0.0619260	−	0.998081i	\(-0.519724\pi\)
−0.340609	+	0.940205i	\(0.610633\pi\)
\(31\)	5.03712	−	7.83791i	0.904693	−	1.40773i	−0.00839353	−	0.999965i	\(-0.502672\pi\)
							0.913087	−	0.407765i	\(-0.133692\pi\)
\(37\)	0.319743	−	0.700140i	0.0525654	−	0.115102i	−0.881527	−	0.472134i	\(-0.843484\pi\)
							0.934092	+	0.357032i	\(0.116211\pi\)
\(41\)	−3.75955	−	8.23227i	−0.587143	−	1.28566i	−0.937153	−	0.348919i	\(-0.886549\pi\)
							0.350009	−	0.936746i	\(-0.386178\pi\)
\(43\)	5.65659	+	8.80182i	0.862621	+	1.34227i	0.938547	+	0.345152i	\(0.112173\pi\)
							−0.0759255	+	0.997113i	\(0.524191\pi\)
\(47\)			8.37778i			1.22203i	0.791621	+	0.611013i	\(0.209237\pi\)
							−0.791621	+	0.611013i	\(0.790763\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.2.t.b.283.1	✓	240
8.3	odd	2	inner	552.2.t.b.283.13	yes	240
23.10	odd	22	inner	552.2.t.b.355.13	yes	240
184.171	even	22	inner	552.2.t.b.355.1	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.t.b.283.1	✓	240	1.1	even	1	trivial
552.2.t.b.283.13	yes	240	8.3	odd	2	inner
552.2.t.b.355.1	yes	240	184.171	even	22	inner
552.2.t.b.355.13	yes	240	23.10	odd	22	inner