Embedded newform 882.2.y.b.583.9

• Label: 882.2.y.b.583.9
• Level: $882$
• Weight: $2$
• Character: 882.583
• Analytic conductor: $7.043$
• Analytic rank: $0$
• Dimension: $336$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	882.y (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.04280545828\)
Analytic rank:	\(0\)
Dimension:	\(336\)
Relative dimension:	\(28\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			583.9
Character	\(\chi\)	\(=\)	882.583
Dual form			882.2.y.b.823.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.365341 - 0.930874i) q^{2} +(-0.987241 + 1.42315i) q^{3} +(-0.733052 - 0.680173i) q^{4} +(-3.29145 + 1.58508i) q^{5} +(0.964093 + 1.43893i) q^{6} +(-2.61026 + 0.431934i) q^{7} +(-0.900969 + 0.433884i) q^{8} +(-1.05071 - 2.80998i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 28 q^{2} + 13 q^{3} + 28 q^{4} - 4 q^{5} - 5 q^{6} + 3 q^{7} - 56 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\)	\(199\)	\(785\)
\(\chi(n)\)	\(e\left(\frac{4}{21}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.365341	−	0.930874i	0.258335	−	0.658227i
							\(3\)	−0.987241	+	1.42315i	−0.569984	+	0.821656i
\(5\)	−3.29145	+	1.58508i	−1.47198	+	0.708869i								−0.986253	−	0.165241i	\(-0.947160\pi\)
							−0.485728	+	0.874110i	\(0.661445\pi\)
\(7\)	−2.61026	+	0.431934i	−0.986584	+	0.163256i
							\(11\)	0.844326	+	1.05875i	0.254574	+	0.319225i	0.892652	−	0.450746i	\(-0.148842\pi\)
−0.638079	+	0.769971i	\(0.720271\pi\)
\(13\)	−0.995399	+	2.53624i	−0.276074	+	0.703425i	0.723835	+	0.689973i	\(0.242377\pi\)
							−0.999910	+	0.0134527i	\(0.995718\pi\)
\(17\)	2.70139	−	2.50652i	0.655182	−	0.607920i	−0.280885	−	0.959741i	\(-0.590628\pi\)
							0.936067	+	0.351821i	\(0.114437\pi\)
\(19\)	1.45053	−	2.51239i	0.332774	−	0.576381i	−0.650281	−	0.759694i	\(-0.725349\pi\)
							0.983055	+	0.183313i	\(0.0586821\pi\)
\(23\)	1.85471	−	8.12602i	0.386734	−	1.69439i	−0.289067	−	0.957309i	\(-0.593345\pi\)
							0.675801	−	0.737084i	\(-0.263798\pi\)
\(29\)	5.84547	+	1.80309i	1.08548	+	0.334825i	0.785336	−	0.619069i	\(-0.212490\pi\)
							0.300141	+	0.953895i	\(0.402966\pi\)
\(31\)	−5.31815	+	9.21131i	−0.955169	+	1.65440i	−0.221187	+	0.975231i	\(0.570993\pi\)
							−0.733982	+	0.679169i	\(0.762340\pi\)
\(37\)	−3.59226	−	1.10806i	−0.590564	−	0.182165i	−0.0149523	−	0.999888i	\(-0.504760\pi\)
							−0.575611	+	0.817723i	\(0.695236\pi\)
\(41\)	−7.13179	−	4.86237i	−1.11380	−	0.759375i	−0.140766	−	0.990043i	\(-0.544957\pi\)
							−0.973033	+	0.230668i	\(0.925909\pi\)
\(43\)	5.63084	−	3.83904i	0.858696	−	0.585449i	−0.0519441	−	0.998650i	\(-0.516542\pi\)
							0.910640	+	0.413201i	\(0.135589\pi\)
\(47\)	0.213779	−	0.544701i	0.0311829	−	0.0794528i	−0.914434	−	0.404735i	\(-0.867364\pi\)
							0.945617	+	0.325282i	\(0.105459\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.2.y.b.583.9	✓	336
9.4	even	3		882.2.bb.a.877.20	yes	336
49.39	even	21		882.2.bb.a.529.20	yes	336
441.382	even	21	inner	882.2.y.b.823.9	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.2.y.b.583.9	✓	336	1.1	even	1	trivial
882.2.y.b.823.9	yes	336	441.382	even	21	inner
882.2.bb.a.529.20	yes	336	49.39	even	21
882.2.bb.a.877.20	yes	336	9.4	even	3