Embedded newform 882.2.y.a.457.24

• Label: 882.2.y.a.457.24
• 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			457.24
Character	\(\chi\)	\(=\)	882.457
Dual form			882.2.y.a.193.24

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.988831 + 0.149042i) q^{2} +(1.52036 - 0.829767i) q^{3} +(0.955573 + 0.294755i) q^{4} +(1.93094 + 0.929890i) q^{5} +(1.62705 - 0.593902i) q^{6} +(1.37818 + 2.25845i) q^{7} +(0.900969 + 0.433884i) q^{8} +(1.62297 - 2.52309i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 28 q^{2} - 13 q^{3} + 28 q^{4} - 4 q^{5} - 5 q^{6} - q^{7} + 56 q^{8} + 29 q^{9}+O(q^{10}) \)

Character values

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

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

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
							\(3\)	1.52036	−	0.829767i	0.877779	−	0.479066i
\(5\)	1.93094	+	0.929890i	0.863541	+	0.415859i								0.812586	−	0.582842i	\(-0.198059\pi\)
							0.0509551	+	0.998701i	\(0.483773\pi\)
\(7\)	1.37818	+	2.25845i	0.520904	+	0.853615i
							\(11\)	1.91946	−	2.40693i	0.578740	−	0.725716i	−0.403158	−	0.915130i	\(-0.632087\pi\)
0.981897	+	0.189414i	\(0.0606588\pi\)
\(13\)	−3.96983	−	0.598355i	−1.10103	−	0.165954i	−0.426708	−	0.904390i	\(-0.640327\pi\)
							−0.674324	+	0.738436i	\(0.735565\pi\)
\(17\)	−6.85949	+	2.11587i	−1.66367	+	0.513174i	−0.976899	−	0.213700i	\(-0.931449\pi\)
							−0.686771	+	0.726874i	\(0.740972\pi\)
\(19\)	0.586921	−	1.01658i	0.134649	−	0.233219i	−0.790814	−	0.612056i	\(-0.790343\pi\)
							0.925463	+	0.378837i	\(0.123676\pi\)
\(23\)	1.49974	+	6.57078i	0.312717	+	1.37010i	0.850037	+	0.526724i	\(0.176580\pi\)
							−0.537320	+	0.843379i	\(0.680563\pi\)
\(29\)	−6.37414	−	5.91434i	−1.18365	−	1.09826i	−0.993182	−	0.116571i	\(-0.962810\pi\)
							−0.190465	−	0.981694i	\(-0.561000\pi\)
\(31\)	0.810990	−	1.40468i	0.145658	−	0.252287i	−0.783960	−	0.620811i	\(-0.786803\pi\)
							0.929618	+	0.368524i	\(0.120137\pi\)
\(37\)	−0.692678	−	0.642711i	−0.113876	−	0.105661i	0.621180	−	0.783668i	\(-0.286654\pi\)
							−0.735056	+	0.678007i	\(0.762844\pi\)
\(41\)	0.595055	+	7.94046i	0.0929320	+	1.24009i	0.827464	+	0.561518i	\(0.189783\pi\)
							−0.734532	+	0.678574i	\(0.762598\pi\)
\(43\)	−0.381057	+	5.08485i	−0.0581107	+	0.775433i	0.889689	+	0.456567i	\(0.150921\pi\)
							−0.947800	+	0.318866i	\(0.896698\pi\)
\(47\)	−4.27016	−	0.643624i	−0.622867	−	0.0938821i	−0.169976	−	0.985448i	\(-0.554369\pi\)
							−0.452891	+	0.891566i	\(0.649607\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.2.y.a.457.24	yes	336
9.4	even	3		882.2.bb.b.751.3	yes	336
49.46	even	21		882.2.bb.b.781.3	yes	336
441.193	even	21	inner	882.2.y.a.193.24	✓	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.2.y.a.193.24	✓	336	441.193	even	21	inner
882.2.y.a.457.24	yes	336	1.1	even	1	trivial
882.2.bb.b.751.3	yes	336	9.4	even	3
882.2.bb.b.781.3	yes	336	49.46	even	21