Embedded newform 882.2.y.a.457.28

• Label: 882.2.y.a.457.28
• 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			457.28
Character	\(\chi\)	\(=\)	882.457
Dual form			882.2.y.a.193.28

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.988831 + 0.149042i) q^{2} +(1.71496 + 0.242685i) q^{3} +(0.955573 + 0.294755i) q^{4} +(-2.59076 - 1.24764i) q^{5} +(1.65964 + 0.495576i) q^{6} +(-1.92610 - 1.81387i) q^{7} +(0.900969 + 0.433884i) q^{8} +(2.88221 + 0.832391i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 28 q^{2} - 13 q^{3} + 28 q^{4} - 4 q^{5} - 5 q^{6} - q^{7} + 56 q^{8} + 29 q^{9}+O(q^{10}) \)

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
							\(3\)	1.71496	+	0.242685i	0.990135	+	0.140114i
\(5\)	−2.59076	−	1.24764i	−1.15862	−	0.557964i								−0.247010	−	0.969013i	\(-0.579448\pi\)
							−0.911613	+	0.411049i	\(0.865162\pi\)
\(7\)	−1.92610	−	1.81387i	−0.727998	−	0.685579i
							\(11\)	2.57078	−	3.22366i	0.775121	−	0.971970i	−0.224877	−	0.974387i	\(-0.572198\pi\)
0.999997	+	0.00241682i	\(0.000769298\pi\)
\(13\)	1.21587	+	0.183263i	0.337222	+	0.0508280i	0.315470	−	0.948935i	\(-0.397838\pi\)
							0.0217514	+	0.999763i	\(0.493076\pi\)
\(17\)	3.84733	−	1.18675i	0.933116	−	0.287828i	0.209324	−	0.977846i	\(-0.432874\pi\)
							0.723792	+	0.690018i	\(0.242398\pi\)
\(19\)	2.59185	−	4.48922i	0.594612	−	1.02990i	−0.398990	−	0.916955i	\(-0.630639\pi\)
							0.993602	−	0.112942i	\(-0.0360276\pi\)
\(23\)	1.31054	+	5.74185i	0.273266	+	1.19726i	0.906132	+	0.422996i	\(0.139022\pi\)
							−0.632865	+	0.774262i	\(0.718121\pi\)
\(29\)	−5.92973	−	5.50198i	−1.10112	−	1.02169i	−0.999639	−	0.0268551i	\(-0.991451\pi\)
							−0.101483	−	0.994837i	\(-0.532359\pi\)
\(31\)	−0.462888	+	0.801745i	−0.0831371	+	0.143998i	−0.904596	−	0.426270i	\(-0.859827\pi\)
							0.821459	+	0.570268i	\(0.193161\pi\)
\(37\)	3.81266	+	3.53763i	0.626797	+	0.581583i	0.928277	−	0.371890i	\(-0.121290\pi\)
							−0.301480	+	0.953473i	\(0.597481\pi\)
\(41\)	−0.697117	−	9.30238i	−0.108871	−	1.45279i	−0.737640	−	0.675194i	\(-0.764060\pi\)
							0.628768	−	0.777593i	\(-0.283559\pi\)
\(43\)	−0.347732	+	4.64016i	−0.0530286	+	0.707617i	0.905813	+	0.423677i	\(0.139261\pi\)
							−0.958842	+	0.283940i	\(0.908358\pi\)
\(47\)	−5.91864	−	0.892092i	−0.863323	−	0.130125i	−0.297570	−	0.954700i	\(-0.596176\pi\)
							−0.565753	+	0.824575i	\(0.691414\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.2.y.a.457.28	yes	336
9.4	even	3		882.2.bb.b.751.2	yes	336
49.46	even	21		882.2.bb.b.781.2	yes	336
441.193	even	21	inner	882.2.y.a.193.28	✓	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.2.y.a.193.28	✓	336	441.193	even	21	inner
882.2.y.a.457.28	yes	336	1.1	even	1	trivial
882.2.bb.b.751.2	yes	336	9.4	even	3
882.2.bb.b.781.2	yes	336	49.46	even	21