Embedded newform 882.2.y.a.583.15

• Label: 882.2.y.a.583.15
• 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			583.15
Character	\(\chi\)	\(=\)	882.583
Dual form			882.2.y.a.823.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.365341 + 0.930874i) q^{2} +(-0.127611 + 1.72734i) q^{3} +(-0.733052 - 0.680173i) q^{4} +(-2.77743 + 1.33754i) q^{5} +(-1.56132 - 0.749859i) q^{6} +(-0.904156 - 2.48646i) q^{7} +(0.900969 - 0.433884i) q^{8} +(-2.96743 - 0.440855i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 28 q^{2} - 13 q^{3} + 28 q^{4} - 4 q^{5} - 5 q^{6} - q^{7} + 56 q^{8} + 29 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.127611	+	1.72734i	−0.0736760	+	0.997282i
\(5\)	−2.77743	+	1.33754i	−1.24210	+	0.598165i								−0.935384	−	0.353635i	\(-0.884946\pi\)
							−0.306719	+	0.951800i	\(0.599231\pi\)
\(7\)	−0.904156	−	2.48646i	−0.341739	−	0.939795i
							\(11\)	0.415078	+	0.520492i	0.125151	+	0.156934i	0.840459	−	0.541875i	\(-0.182285\pi\)
−0.715308	+	0.698809i	\(0.753714\pi\)
\(13\)	1.15886	−	2.95272i	0.321409	−	0.818937i	−0.675275	−	0.737566i	\(-0.735975\pi\)
							0.996684	−	0.0813707i	\(-0.0259298\pi\)
\(17\)	1.43684	−	1.33319i	0.348485	−	0.323347i	−0.486436	−	0.873716i	\(-0.661703\pi\)
							0.834921	+	0.550369i	\(0.185513\pi\)
\(19\)	0.384655	−	0.666242i	0.0882459	−	0.152846i	−0.818524	−	0.574473i	\(-0.805207\pi\)
							0.906770	+	0.421626i	\(0.138541\pi\)
\(23\)	−0.675410	+	2.95917i	−0.140833	+	0.617029i	0.854410	+	0.519600i	\(0.173919\pi\)
							−0.995242	+	0.0974289i	\(0.968938\pi\)
\(29\)	4.22410	+	1.30296i	0.784396	+	0.241954i	0.660983	−	0.750401i	\(-0.270140\pi\)
							0.123413	+	0.992355i	\(0.460616\pi\)
\(31\)	0.783299	−	1.35671i	0.140685	−	0.243673i	−0.787070	−	0.616864i	\(-0.788403\pi\)
							0.927755	+	0.373191i	\(0.121736\pi\)
\(37\)	−3.99092	−	1.23104i	−0.656103	−	0.202381i	−0.0512104	−	0.998688i	\(-0.516308\pi\)
							−0.604893	+	0.796307i	\(0.706784\pi\)
\(41\)	−2.55587	−	1.74257i	−0.399161	−	0.272143i	0.347059	−	0.937843i	\(-0.387181\pi\)
							−0.746219	+	0.665700i	\(0.768133\pi\)
\(43\)	5.72376	−	3.90239i	0.872865	−	0.595109i	−0.0418883	−	0.999122i	\(-0.513337\pi\)
							0.914753	+	0.404013i	\(0.132385\pi\)
\(47\)	0.673803	−	1.71682i	0.0982843	−	0.250424i	−0.873272	−	0.487233i	\(-0.838006\pi\)
							0.971556	+	0.236809i	\(0.0761015\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.2.y.a.583.15	✓	336
9.4	even	3		882.2.bb.b.877.25	yes	336
49.39	even	21		882.2.bb.b.529.25	yes	336
441.382	even	21	inner	882.2.y.a.823.15	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.2.y.a.583.15	✓	336	1.1	even	1	trivial
882.2.y.a.823.15	yes	336	441.382	even	21	inner
882.2.bb.b.529.25	yes	336	49.39	even	21
882.2.bb.b.877.25	yes	336	9.4	even	3