Embedded newform 882.2.y.b.571.18

• Label: 882.2.y.b.571.18
• Level: $882$
• Weight: $2$
• Character: 882.571
• 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			571.18
Character	\(\chi\)	\(=\)	882.571
Dual form			882.2.y.b.709.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.826239 - 0.563320i) q^{2} +(0.614400 - 1.61942i) q^{3} +(0.365341 - 0.930874i) q^{4} +(-0.185229 - 0.811542i) q^{5} +(-0.404609 - 1.68413i) q^{6} +(-2.13018 + 1.56918i) q^{7} +(-0.222521 - 0.974928i) q^{8} +(-2.24503 - 1.98994i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 28 q^{2} + 13 q^{3} + 28 q^{4} - 4 q^{5} - 5 q^{6} + 3 q^{7} - 56 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(785\)
\(\chi(n)\)	\(e\left(\frac{2}{21}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.826239	−	0.563320i	0.584239	−	0.398327i
							\(3\)	0.614400	−	1.61942i	0.354724	−	0.934971i
\(5\)	−0.185229	−	0.811542i	−0.0828370	−	0.362932i								0.916472	−	0.400099i	\(-0.131024\pi\)
							−0.999309	+	0.0371660i	\(0.988167\pi\)
\(7\)	−2.13018	+	1.56918i	−0.805132	+	0.593095i
							\(11\)	−4.51434	−	2.17399i	−1.36113	−	0.655484i	−0.396238	−	0.918148i	\(-0.629685\pi\)
−0.964887	+	0.262664i	\(0.915399\pi\)
\(13\)	0.656305	−	0.447461i	0.182026	−	0.124103i	−0.468879	−	0.883262i	\(-0.655342\pi\)
							0.650905	+	0.759159i	\(0.274390\pi\)
\(17\)	1.82049	+	4.63854i	0.441534	+	1.12501i	0.962464	+	0.271411i	\(0.0874902\pi\)
							−0.520929	+	0.853600i	\(0.674415\pi\)
\(19\)	−2.94343	−	5.09817i	−0.675269	−	1.16960i	−0.976390	−	0.216014i	\(-0.930694\pi\)
							0.301122	−	0.953586i	\(-0.402639\pi\)
\(23\)	−0.910439	−	1.14165i	−0.189840	−	0.238051i	0.677798	−	0.735248i	\(-0.262934\pi\)
							−0.867638	+	0.497197i	\(0.834363\pi\)
\(29\)	−6.11085	−	0.921063i	−1.13476	−	0.171037i	−0.445312	−	0.895375i	\(-0.646907\pi\)
							−0.689445	+	0.724338i	\(0.742145\pi\)
\(31\)	−3.39782	−	5.88520i	−0.610266	−	1.05701i	−0.991195	−	0.132408i	\(-0.957729\pi\)
							0.380929	−	0.924604i	\(-0.375604\pi\)
\(37\)	6.13711	+	0.925021i	1.00893	+	0.152072i	0.632655	−	0.774434i	\(-0.281965\pi\)
							0.376280	+	0.926506i	\(0.377203\pi\)
\(41\)	5.75577	+	1.77542i	0.898900	+	0.277274i	0.709566	−	0.704639i	\(-0.248891\pi\)
							0.189334	+	0.981913i	\(0.439367\pi\)
\(43\)	6.91019	−	2.13151i	1.05379	−	0.325052i	0.280999	−	0.959708i	\(-0.409334\pi\)
							0.772795	+	0.634656i	\(0.218858\pi\)
\(47\)	5.47819	−	3.73497i	0.799076	−	0.544801i	−0.0935414	−	0.995615i	\(-0.529819\pi\)
							0.892618	+	0.450814i	\(0.148866\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.2.y.b.571.18	✓	336
9.7	even	3		882.2.bb.a.277.12	yes	336
49.23	even	21		882.2.bb.a.121.12	yes	336
441.268	even	21	inner	882.2.y.b.709.18	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.2.y.b.571.18	✓	336	1.1	even	1	trivial
882.2.y.b.709.18	yes	336	441.268	even	21	inner
882.2.bb.a.121.12	yes	336	49.23	even	21
882.2.bb.a.277.12	yes	336	9.7	even	3