Embedded newform 882.2.y.b.583.27

• Label: 882.2.y.b.583.27
• 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			583.27
Character	\(\chi\)	\(=\)	882.583
Dual form			882.2.y.b.823.27

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.365341 - 0.930874i) q^{2} +(1.72460 - 0.160537i) q^{3} +(-0.733052 - 0.680173i) q^{4} +(1.78847 - 0.861280i) q^{5} +(0.480626 - 1.66403i) q^{6} +(1.80785 - 1.93175i) q^{7} +(-0.900969 + 0.433884i) q^{8} +(2.94846 - 0.553721i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 28 q^{2} + 13 q^{3} + 28 q^{4} - 4 q^{5} - 5 q^{6} + 3 q^{7} - 56 q^{8} - 3 q^{9}+O(q^{10}) \)

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
							\(3\)	1.72460	−	0.160537i	0.995695	−	0.0926859i
\(5\)	1.78847	−	0.861280i	0.799827	−	0.385176i								0.0111141	−	0.999938i	\(-0.496462\pi\)
							0.788713	+	0.614762i	\(0.210748\pi\)
\(7\)	1.80785	−	1.93175i	0.683303	−	0.730135i
							\(11\)	−2.87365	−	3.60345i	−0.866439	−	1.08648i	−0.995492	−	0.0948424i	\(-0.969765\pi\)
0.129053	−	0.991638i	\(-0.458806\pi\)
\(13\)	1.16751	−	2.97478i	0.323810	−	0.825055i	−0.672569	−	0.740034i	\(-0.734809\pi\)
							0.996379	−	0.0850206i	\(-0.0270956\pi\)
\(17\)	−4.73238	+	4.39101i	−1.14777	+	1.06498i	−0.150694	+	0.988580i	\(0.548151\pi\)
							−0.997078	+	0.0763959i	\(0.975659\pi\)
\(19\)	−2.16782	+	3.75477i	−0.497331	+	0.861403i	−0.999995	−	0.00307911i	\(-0.999020\pi\)
							0.502664	+	0.864482i	\(0.332353\pi\)
\(23\)	−1.44698	+	6.33965i	−0.301717	+	1.32191i	0.565818	+	0.824530i	\(0.308561\pi\)
							−0.867534	+	0.497377i	\(0.834297\pi\)
\(29\)	5.60235	+	1.72810i	1.04033	+	0.320899i	0.767423	−	0.641141i	\(-0.221539\pi\)
							0.272907	+	0.962040i	\(0.412015\pi\)
\(31\)	1.42318	−	2.46501i	0.255610	−	0.442730i	−0.709451	−	0.704755i	\(-0.751057\pi\)
							0.965061	+	0.262025i	\(0.0843903\pi\)
\(37\)	−2.03375	−	0.627329i	−0.334346	−	0.103132i	0.123036	−	0.992402i	\(-0.460737\pi\)
							−0.457383	+	0.889270i	\(0.651213\pi\)
\(41\)	9.15267	+	6.24019i	1.42941	+	0.974553i	0.997353	+	0.0727053i	\(0.0231632\pi\)
							0.432054	+	0.901848i	\(0.357789\pi\)
\(43\)	−0.0747414	+	0.0509578i	−0.0113980	+	0.00777100i	−0.569006	−	0.822334i	\(-0.692672\pi\)
							0.557608	+	0.830105i	\(0.311719\pi\)
\(47\)	3.23868	−	8.25202i	0.472410	−	1.20368i	−0.473946	−	0.880554i	\(-0.657171\pi\)
							0.946356	−	0.323126i	\(-0.104734\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.2.y.b.583.27	✓	336
9.4	even	3		882.2.bb.a.877.17	yes	336
49.39	even	21		882.2.bb.a.529.17	yes	336
441.382	even	21	inner	882.2.y.b.823.27	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.2.y.b.583.27	✓	336	1.1	even	1	trivial
882.2.y.b.823.27	yes	336	441.382	even	21	inner
882.2.bb.a.529.17	yes	336	49.39	even	21
882.2.bb.a.877.17	yes	336	9.4	even	3