Embedded newform 882.2.y.a.583.1

• Label: 882.2.y.a.583.1
• 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			583.1
Character	\(\chi\)	\(=\)	882.583
Dual form			882.2.y.a.823.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.365341 + 0.930874i) q^{2} +(-1.71522 - 0.240869i) q^{3} +(-0.733052 - 0.680173i) q^{4} +(3.67425 - 1.76943i) q^{5} +(0.850859 - 1.50865i) q^{6} +(2.48539 - 0.907112i) q^{7} +(0.900969 - 0.433884i) q^{8} +(2.88396 + 0.826286i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 28 q^{2} - 13 q^{3} + 28 q^{4} - 4 q^{5} - 5 q^{6} - q^{7} + 56 q^{8} + 29 q^{9}+O(q^{10}) \)

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
							\(3\)	−1.71522	−	0.240869i	−0.990283	−	0.139066i
\(5\)	3.67425	−	1.76943i	1.64318	−	0.791312i								0.643512	−	0.765436i	\(-0.277477\pi\)
							0.999665	−	0.0258759i	\(-0.00823747\pi\)
\(7\)	2.48539	−	0.907112i	0.939388	−	0.342856i
							\(11\)	1.23429	+	1.54776i	0.372154	+	0.466666i	0.932278	−	0.361743i	\(-0.117818\pi\)
−0.560124	+	0.828409i	\(0.689247\pi\)
\(13\)	−0.499705	+	1.27323i	−0.138593	+	0.353130i	−0.983272	−	0.182144i	\(-0.941696\pi\)
							0.844679	+	0.535274i	\(0.179792\pi\)
\(17\)	0.425727	−	0.395016i	0.103254	−	0.0958056i	−0.626857	−	0.779134i	\(-0.715659\pi\)
							0.730111	+	0.683329i	\(0.239468\pi\)
\(19\)	0.440484	−	0.762941i	0.101054	−	0.175031i	−0.811065	−	0.584956i	\(-0.801112\pi\)
							0.912119	+	0.409925i	\(0.134445\pi\)
\(23\)	0.916836	−	4.01692i	0.191173	−	0.837585i	−0.784809	−	0.619737i	\(-0.787239\pi\)
							0.975983	−	0.217848i	\(-0.0699037\pi\)
\(29\)	−0.853390	−	0.263236i	−0.158471	−	0.0488817i	0.214505	−	0.976723i	\(-0.431186\pi\)
							−0.372975	+	0.927841i	\(0.621662\pi\)
\(31\)	−1.75747	+	3.04403i	−0.315651	+	0.546724i	−0.979576	−	0.201076i	\(-0.935556\pi\)
							0.663925	+	0.747799i	\(0.268890\pi\)
\(37\)	2.21845	+	0.684303i	0.364712	+	0.112499i	0.471694	−	0.881762i	\(-0.343643\pi\)
							−0.106982	+	0.994261i	\(0.534119\pi\)
\(41\)	−1.86980	−	1.27481i	−0.292014	−	0.199092i	0.408448	−	0.912782i	\(-0.366070\pi\)
							−0.700462	+	0.713690i	\(0.747023\pi\)
\(43\)	−2.78803	+	1.90085i	−0.425171	+	0.289877i	−0.756940	−	0.653484i	\(-0.773307\pi\)
							0.331770	+	0.943360i	\(0.392354\pi\)
\(47\)	−4.37226	+	11.1403i	−0.637760	+	1.62499i	0.135412	+	0.990789i	\(0.456764\pi\)
							−0.773172	+	0.634197i	\(0.781331\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.2.y.a.583.1	✓	336
9.4	even	3		882.2.bb.b.877.10	yes	336
49.39	even	21		882.2.bb.b.529.10	yes	336
441.382	even	21	inner	882.2.y.a.823.1	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.2.y.a.583.1	✓	336	1.1	even	1	trivial
882.2.y.a.823.1	yes	336	441.382	even	21	inner
882.2.bb.b.529.10	yes	336	49.39	even	21
882.2.bb.b.877.10	yes	336	9.4	even	3