Embedded newform 882.2.y.a.709.16

• Label: 882.2.y.a.709.16
• Level: $882$
• Weight: $2$
• Character: 882.709
• 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			709.16
Character	\(\chi\)	\(=\)	882.709
Dual form			882.2.y.a.571.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.826239 - 0.563320i) q^{2} +(0.382088 - 1.68938i) q^{3} +(0.365341 + 0.930874i) q^{4} +(-0.888346 + 3.89210i) q^{5} +(-1.26736 + 1.18059i) q^{6} +(-0.839750 + 2.50895i) q^{7} +(0.222521 - 0.974928i) q^{8} +(-2.70802 - 1.29098i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 28 q^{2} - 13 q^{3} + 28 q^{4} - 4 q^{5} - 5 q^{6} - q^{7} + 56 q^{8} + 29 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(785\)
\(\chi(n)\)	\(e\left(\frac{19}{21}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.826239	−	0.563320i	−0.584239	−	0.398327i
							\(3\)	0.382088	−	1.68938i	0.220599	−	0.975365i
\(5\)	−0.888346	+	3.89210i	−0.397280	+	1.74060i								0.240750	+	0.970587i	\(0.422607\pi\)
							−0.638030	+	0.770011i	\(0.720251\pi\)
\(7\)	−0.839750	+	2.50895i	−0.317396	+	0.948293i
							\(11\)	2.05428	−	0.989291i	0.619390	−	0.298282i	−0.0977508	−	0.995211i	\(-0.531165\pi\)
0.717141	+	0.696929i	\(0.245451\pi\)
\(13\)	−3.47680	−	2.37044i	−0.964292	−	0.657443i	−0.0246836	−	0.999695i	\(-0.507858\pi\)
							−0.939608	+	0.342252i	\(0.888810\pi\)
\(17\)	1.93671	−	4.93466i	0.469722	−	1.19683i	−0.478178	−	0.878263i	\(-0.658703\pi\)
							0.947900	−	0.318569i	\(-0.103202\pi\)
\(19\)	−2.48033	+	4.29607i	−0.569028	+	0.985585i	0.427635	+	0.903952i	\(0.359347\pi\)
							−0.996662	+	0.0816334i	\(0.973986\pi\)
\(23\)	0.0537973	−	0.0674597i	0.0112175	−	0.0140663i	−0.776191	−	0.630498i	\(-0.782851\pi\)
							0.787409	+	0.616431i	\(0.211422\pi\)
\(29\)	−6.91184	+	1.04179i	−1.28350	+	0.193456i	−0.755138	−	0.655566i	\(-0.772430\pi\)
							−0.528358	+	0.849022i	\(0.677192\pi\)
\(31\)	1.11649	−	1.93381i	0.200527	−	0.347322i	−0.748172	−	0.663505i	\(-0.769068\pi\)
							0.948698	+	0.316183i	\(0.102401\pi\)
\(37\)	−8.42858	+	1.27040i	−1.38565	+	0.208853i	−0.799167	−	0.601108i	\(-0.794726\pi\)
							−0.586483	+	0.809962i	\(0.699488\pi\)
\(41\)	7.29061	−	2.24886i	1.13860	−	0.351212i	0.332567	−	0.943080i	\(-0.392085\pi\)
							0.806036	+	0.591867i	\(0.201609\pi\)
\(43\)	−8.26988	−	2.55092i	−1.26115	−	0.389012i	−0.409105	−	0.912487i	\(-0.634159\pi\)
							−0.852041	+	0.523475i	\(0.824635\pi\)
\(47\)	−1.41474	−	0.964556i	−0.206362	−	0.140695i	0.455730	−	0.890118i	\(-0.349378\pi\)
							−0.662091	+	0.749423i	\(0.730331\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.2.y.a.709.16	yes	336
9.4	even	3		882.2.bb.b.121.21	yes	336
49.32	even	21		882.2.bb.b.277.21	yes	336
441.130	even	21	inner	882.2.y.a.571.16	✓	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.2.y.a.571.16	✓	336	441.130	even	21	inner
882.2.y.a.709.16	yes	336	1.1	even	1	trivial
882.2.bb.b.121.21	yes	336	9.4	even	3
882.2.bb.b.277.21	yes	336	49.32	even	21