Embedded newform 882.2.y.a.583.12

• Label: 882.2.y.a.583.12
• 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			583.12
Character	\(\chi\)	\(=\)	882.583
Dual form			882.2.y.a.823.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.365341 + 0.930874i) q^{2} +(-0.323915 - 1.70149i) q^{3} +(-0.733052 - 0.680173i) q^{4} +(-1.74876 + 0.842159i) q^{5} +(1.70221 + 0.320101i) q^{6} +(-1.87592 - 1.86572i) q^{7} +(0.900969 - 0.433884i) q^{8} +(-2.79016 + 1.10228i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 28 q^{2} - 13 q^{3} + 28 q^{4} - 4 q^{5} - 5 q^{6} - q^{7} + 56 q^{8} + 29 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.323915	−	1.70149i	−0.187012	−	0.982358i
\(5\)	−1.74876	+	0.842159i	−0.782070	+	0.376625i								−0.781923	−	0.623375i	\(-0.785761\pi\)
							−0.000146949		1.00000i	\(0.500047\pi\)
\(7\)	−1.87592	−	1.86572i	−0.709031	−	0.705177i
							\(11\)	0.351619	+	0.440916i	0.106017	+	0.132941i	0.832009	−	0.554762i	\(-0.187191\pi\)
−0.725992	+	0.687703i	\(0.758619\pi\)
\(13\)	−0.244570	+	0.623155i	−0.0678316	+	0.172832i	−0.960770	−	0.277347i	\(-0.910545\pi\)
							0.892938	+	0.450179i	\(0.148640\pi\)
\(17\)	4.07982	−	3.78552i	0.989501	−	0.918123i	−0.00714325	−	0.999974i	\(-0.502274\pi\)
							0.996645	+	0.0818515i	\(0.0260833\pi\)
\(19\)	−2.59247	+	4.49029i	−0.594753	+	1.03014i	0.398828	+	0.917026i	\(0.369417\pi\)
							−0.993582	+	0.113118i	\(0.963916\pi\)
\(23\)	0.562515	−	2.46454i	0.117292	−	0.513892i	−0.881813	−	0.471600i	\(-0.843677\pi\)
							0.999105	−	0.0422923i	\(-0.0134661\pi\)
\(29\)	3.83182	+	1.18196i	0.711552	+	0.219485i	0.629340	−	0.777130i	\(-0.283325\pi\)
							0.0822119	+	0.996615i	\(0.473802\pi\)
\(31\)	−5.44679	+	9.43412i	−0.978272	+	1.69442i	−0.309588	+	0.950871i	\(0.600191\pi\)
							−0.668684	+	0.743547i	\(0.733142\pi\)
\(37\)	3.91569	+	1.20783i	0.643736	+	0.198566i	0.599399	−	0.800450i	\(-0.295406\pi\)
							0.0443368	+	0.999017i	\(0.485883\pi\)
\(41\)	3.12806	+	2.13267i	0.488520	+	0.333067i	0.782401	−	0.622775i	\(-0.213995\pi\)
							−0.293881	+	0.955842i	\(0.594947\pi\)
\(43\)	5.18583	−	3.53564i	0.790832	−	0.539180i	−0.0992060	−	0.995067i	\(-0.531630\pi\)
							0.890038	+	0.455887i	\(0.150678\pi\)
\(47\)	−3.26236	+	8.31237i	−0.475865	+	1.21248i	0.468465	+	0.883482i	\(0.344807\pi\)
							−0.944329	+	0.329001i	\(0.893288\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.2.y.a.583.12	✓	336
9.4	even	3		882.2.bb.b.877.2	yes	336
49.39	even	21		882.2.bb.b.529.2	yes	336
441.382	even	21	inner	882.2.y.a.823.12	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.2.y.a.583.12	✓	336	1.1	even	1	trivial
882.2.y.a.823.12	yes	336	441.382	even	21	inner
882.2.bb.b.529.2	yes	336	49.39	even	21
882.2.bb.b.877.2	yes	336	9.4	even	3