Embedded newform 882.2.y.b.583.4

• Label: 882.2.y.b.583.4
• 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			583.4
Character	\(\chi\)	\(=\)	882.583
Dual form			882.2.y.b.823.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.365341 - 0.930874i) q^{2} +(-1.60341 - 0.655034i) q^{3} +(-0.733052 - 0.680173i) q^{4} +(-0.804410 + 0.387384i) q^{5} +(-1.19555 + 1.25326i) q^{6} +(-0.673052 + 2.55871i) q^{7} +(-0.900969 + 0.433884i) q^{8} +(2.14186 + 2.10058i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 28 q^{2} + 13 q^{3} + 28 q^{4} - 4 q^{5} - 5 q^{6} + 3 q^{7} - 56 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

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

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

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
							\(3\)	−1.60341	−	0.655034i	−0.925730	−	0.378184i
\(5\)	−0.804410	+	0.387384i	−0.359743	+	0.173243i								−0.605021	−	0.796210i	\(-0.706835\pi\)
							0.245278	+	0.969453i	\(0.421121\pi\)
\(7\)	−0.673052	+	2.55871i	−0.254390	+	0.967102i
							\(11\)	0.0570263	+	0.0715087i	0.0171941	+	0.0215607i	0.790354	−	0.612650i	\(-0.209897\pi\)
−0.773160	+	0.634211i	\(0.781325\pi\)
\(13\)	0.0452609	−	0.115323i	0.0125531	−	0.0319848i	−0.924460	−	0.381280i	\(-0.875484\pi\)
							0.937013	+	0.349295i	\(0.113579\pi\)
\(17\)	4.72244	−	4.38178i	1.14536	−	1.06274i	0.148084	−	0.988975i	\(-0.452689\pi\)
							0.997276	−	0.0737637i	\(-0.0235011\pi\)
\(19\)	2.76963	−	4.79713i	0.635396	−	1.10054i	−0.351036	−	0.936362i	\(-0.614170\pi\)
							0.986431	−	0.164175i	\(-0.0524963\pi\)
\(23\)	0.969317	−	4.24685i	0.202117	−	0.885530i	−0.767529	−	0.641015i	\(-0.778514\pi\)
							0.969645	−	0.244516i	\(-0.0786290\pi\)
\(29\)	−3.69304	−	1.13915i	−0.685780	−	0.211535i	−0.0677648	−	0.997701i	\(-0.521587\pi\)
							−0.618015	+	0.786166i	\(0.712063\pi\)
\(31\)	2.68683	−	4.65373i	0.482569	−	0.835834i	−0.517231	−	0.855846i	\(-0.673037\pi\)
							0.999800	+	0.0200121i	\(0.00637049\pi\)
\(37\)	1.24798	+	0.384952i	0.205167	+	0.0632857i	0.395635	−	0.918408i	\(-0.370524\pi\)
							−0.190468	+	0.981693i	\(0.561001\pi\)
\(41\)	6.90403	+	4.70709i	1.07823	+	0.735124i	0.965996	−	0.258558i	\(-0.0832473\pi\)
							0.112233	+	0.993682i	\(0.464200\pi\)
\(43\)	−7.49090	+	5.10721i	−1.14235	+	0.778843i	−0.978215	−	0.207595i	\(-0.933436\pi\)
							−0.164137	+	0.986437i	\(0.552484\pi\)
\(47\)	−0.0117065	+	0.0298277i	−0.00170757	+	0.00435082i	−0.931725	−	0.363165i	\(-0.881696\pi\)
							0.930017	+	0.367515i	\(0.119791\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.2.y.b.583.4	✓	336
9.4	even	3		882.2.bb.a.877.7	yes	336
49.39	even	21		882.2.bb.a.529.7	yes	336
441.382	even	21	inner	882.2.y.b.823.4	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.2.y.b.583.4	✓	336	1.1	even	1	trivial
882.2.y.b.823.4	yes	336	441.382	even	21	inner
882.2.bb.a.529.7	yes	336	49.39	even	21
882.2.bb.a.877.7	yes	336	9.4	even	3