Embedded newform 882.2.y.b.457.21

• Label: 882.2.y.b.457.21
• Level: $882$
• Weight: $2$
• Character: 882.457
• 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			457.21
Character	\(\chi\)	\(=\)	882.457
Dual form			882.2.y.b.193.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.988831 - 0.149042i) q^{2} +(1.16950 + 1.27760i) q^{3} +(0.955573 + 0.294755i) q^{4} +(1.30171 + 0.626869i) q^{5} +(-0.966022 - 1.43764i) q^{6} +(1.76777 + 1.96850i) q^{7} +(-0.900969 - 0.433884i) q^{8} +(-0.264535 + 2.98831i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 28 q^{2} + 13 q^{3} + 28 q^{4} - 4 q^{5} - 5 q^{6} + 3 q^{7} - 56 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(785\)
\(\chi(n)\)	\(e\left(\frac{10}{21}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.988831	−	0.149042i	−0.699209	−	0.105389i
							\(3\)	1.16950	+	1.27760i	0.675212	+	0.737624i
\(5\)	1.30171	+	0.626869i	0.582141	+	0.280344i								0.701688	−	0.712484i	\(-0.252430\pi\)
							−0.119547	+	0.992828i	\(0.538144\pi\)
\(7\)	1.76777	+	1.96850i	0.668154	+	0.744023i
							\(11\)	3.11399	−	3.90483i	0.938905	−	1.17735i	−0.0450596	−	0.998984i	\(-0.514348\pi\)
0.983964	−	0.178365i	\(-0.0570808\pi\)
\(13\)	−1.86915	−	0.281730i	−0.518410	−	0.0781378i	−0.115375	−	0.993322i	\(-0.536807\pi\)
							−0.403036	+	0.915184i	\(0.632045\pi\)
\(17\)	7.50652	−	2.31546i	1.82060	−	0.561580i	0.820647	−	0.571435i	\(-0.193613\pi\)
							0.999951	+	0.00985444i	\(0.00313682\pi\)
\(19\)	1.48091	−	2.56500i	0.339743	−	0.588452i	−0.644641	−	0.764485i	\(-0.722993\pi\)
							0.984384	+	0.176033i	\(0.0563266\pi\)
\(23\)	0.967730	+	4.23990i	0.201786	+	0.884081i	0.969849	+	0.243707i	\(0.0783635\pi\)
							−0.768063	+	0.640374i	\(0.778779\pi\)
\(29\)	2.74085	+	2.54314i	0.508963	+	0.472248i	0.892353	−	0.451338i	\(-0.149053\pi\)
							−0.383390	+	0.923586i	\(0.625244\pi\)
\(31\)	−2.63359	+	4.56152i	−0.473007	+	0.819272i	−0.999523	−	0.0308932i	\(-0.990165\pi\)
							0.526516	+	0.850165i	\(0.323498\pi\)
\(37\)	−5.71137	−	5.29937i	−0.938943	−	0.871212i	0.0529099	−	0.998599i	\(-0.483150\pi\)
							−0.991853	+	0.127387i	\(0.959341\pi\)
\(41\)	0.265275	+	3.53985i	0.0414290	+	0.552832i	0.978843	+	0.204614i	\(0.0655939\pi\)
							−0.937414	+	0.348218i	\(0.886787\pi\)
\(43\)	0.802230	−	10.7050i	0.122339	−	1.63250i	−0.511676	−	0.859179i	\(-0.670975\pi\)
							0.634015	−	0.773321i	\(-0.281406\pi\)
\(47\)	−5.34892	−	0.806220i	−0.780220	−	0.117599i	−0.253157	−	0.967425i	\(-0.581469\pi\)
							−0.527063	+	0.849826i	\(0.676707\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.2.y.b.457.21	yes	336
9.4	even	3		882.2.bb.a.751.9	yes	336
49.46	even	21		882.2.bb.a.781.9	yes	336
441.193	even	21	inner	882.2.y.b.193.21	✓	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.2.y.b.193.21	✓	336	441.193	even	21	inner
882.2.y.b.457.21	yes	336	1.1	even	1	trivial
882.2.bb.a.751.9	yes	336	9.4	even	3
882.2.bb.a.781.9	yes	336	49.46	even	21