Embedded newform 882.2.y.a.583.9

• Label: 882.2.y.a.583.9
• Level: $882$
• Weight: $2$
• Character: 882.583
• 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			583.9
Character	\(\chi\)	\(=\)	882.583
Dual form			882.2.y.a.823.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.365341 + 0.930874i) q^{2} +(-1.08885 + 1.34700i) q^{3} +(-0.733052 - 0.680173i) q^{4} +(-0.0755207 + 0.0363689i) q^{5} +(-0.856083 - 1.50570i) q^{6} +(1.62664 - 2.08663i) q^{7} +(0.900969 - 0.433884i) q^{8} +(-0.628806 - 2.93336i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 28 q^{2} - 13 q^{3} + 28 q^{4} - 4 q^{5} - 5 q^{6} - q^{7} + 56 q^{8} + 29 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(785\)
\(\chi(n)\)	\(e\left(\frac{4}{21}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.365341	+	0.930874i	−0.258335	+	0.658227i
							\(3\)	−1.08885	+	1.34700i	−0.628649	+	0.777690i
\(5\)	−0.0755207	+	0.0363689i	−0.0337739	+	0.0162647i								−0.450694	−	0.892678i	\(-0.648824\pi\)
							0.416921	+	0.908943i	\(0.363109\pi\)
\(7\)	1.62664	−	2.08663i	0.614813	−	0.788673i
							\(11\)	0.316805	+	0.397261i	0.0955203	+	0.119779i	0.827294	−	0.561769i	\(-0.189879\pi\)
−0.731774	+	0.681547i	\(0.761307\pi\)
\(13\)	−1.94456	+	4.95466i	−0.539324	+	1.37418i	0.358380	+	0.933576i	\(0.383329\pi\)
							−0.897704	+	0.440599i	\(0.854766\pi\)
\(17\)	1.92860	−	1.78948i	0.467753	−	0.434012i	−0.410750	−	0.911748i	\(-0.634733\pi\)
							0.878503	+	0.477736i	\(0.158543\pi\)
\(19\)	−2.58479	+	4.47699i	−0.592992	+	1.02709i	0.400835	+	0.916150i	\(0.368720\pi\)
							−0.993827	+	0.110942i	\(0.964613\pi\)
\(23\)	−0.209428	+	0.917563i	−0.0436687	+	0.191325i	−0.992058	−	0.125785i	\(-0.959855\pi\)
							0.948389	+	0.317110i	\(0.102712\pi\)
\(29\)	−8.47854	−	2.61528i	−1.57443	−	0.485646i	−0.620127	−	0.784501i	\(-0.712919\pi\)
							−0.954298	+	0.298856i	\(0.903395\pi\)
\(31\)	−4.62989	+	8.01921i	−0.831553	+	1.44029i	0.0652525	+	0.997869i	\(0.479215\pi\)
							−0.896806	+	0.442424i	\(0.854119\pi\)
\(37\)	5.66043	+	1.74601i	0.930568	+	0.287042i	0.722738	−	0.691123i	\(-0.242884\pi\)
							0.207831	+	0.978165i	\(0.433360\pi\)
\(41\)	6.57752	+	4.48448i	1.02724	+	0.700358i	0.954871	−	0.297021i	\(-0.0959930\pi\)
							0.0723648	+	0.997378i	\(0.476945\pi\)
\(43\)	−0.535205	+	0.364897i	−0.0816180	+	0.0556462i	−0.603442	−	0.797407i	\(-0.706204\pi\)
							0.521824	+	0.853053i	\(0.325252\pi\)
\(47\)	1.50359	−	3.83109i	0.219321	−	0.558821i	−0.778290	−	0.627905i	\(-0.783912\pi\)
							0.997611	+	0.0690841i	\(0.0220077\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.2.y.a.583.9	✓	336
9.4	even	3		882.2.bb.b.877.20	yes	336
49.39	even	21		882.2.bb.b.529.20	yes	336
441.382	even	21	inner	882.2.y.a.823.9	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.2.y.a.583.9	✓	336	1.1	even	1	trivial
882.2.y.a.823.9	yes	336	441.382	even	21	inner
882.2.bb.b.529.20	yes	336	49.39	even	21
882.2.bb.b.877.20	yes	336	9.4	even	3