Embedded newform 882.2.y.b.583.7

• Label: 882.2.y.b.583.7
• 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			583.7
Character	\(\chi\)	\(=\)	882.583
Dual form			882.2.y.b.823.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.365341 - 0.930874i) q^{2} +(-1.32330 + 1.11753i) q^{3} +(-0.733052 - 0.680173i) q^{4} +(0.491504 - 0.236696i) q^{5} +(0.556818 + 1.64011i) q^{6} +(-0.0331982 + 2.64554i) q^{7} +(-0.900969 + 0.433884i) q^{8} +(0.502271 - 2.95766i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 28 q^{2} + 13 q^{3} + 28 q^{4} - 4 q^{5} - 5 q^{6} + 3 q^{7} - 56 q^{8} - 3 q^{9}+O(q^{10}) \)

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
							\(3\)	−1.32330	+	1.11753i	−0.764010	+	0.645204i
\(5\)	0.491504	−	0.236696i	0.219807	−	0.105854i								−0.320740	−	0.947167i	\(-0.603932\pi\)
							0.540547	+	0.841314i	\(0.318217\pi\)
\(7\)	−0.0331982	+	2.64554i	−0.0125477	+	0.999921i
							\(11\)	1.02283	+	1.28259i	0.308396	+	0.386717i	0.911742	−	0.410763i	\(-0.134738\pi\)
−0.603346	+	0.797480i	\(0.706166\pi\)
\(13\)	1.73672	−	4.42509i	0.481680	−	1.22730i	−0.459131	−	0.888368i	\(-0.651839\pi\)
							0.940811	−	0.338932i	\(-0.110066\pi\)
\(17\)	−1.63306	+	1.51526i	−0.396075	+	0.367504i	−0.852919	−	0.522043i	\(-0.825170\pi\)
							0.456844	+	0.889547i	\(0.348980\pi\)
\(19\)	−2.07349	+	3.59139i	−0.475691	+	0.823921i	−0.999612	−	0.0278458i	\(-0.991135\pi\)
							0.523921	+	0.851767i	\(0.324469\pi\)
\(23\)	−0.647763	+	2.83804i	−0.135068	+	0.591771i	0.861410	+	0.507911i	\(0.169582\pi\)
							−0.996478	+	0.0838604i	\(0.973275\pi\)
\(29\)	5.81009	+	1.79217i	1.07891	+	0.332799i	0.782747	−	0.622340i	\(-0.213818\pi\)
							0.296159	+	0.955139i	\(0.404294\pi\)
\(31\)	−2.31562	+	4.01078i	−0.415898	+	0.720357i	−0.995522	−	0.0945270i	\(-0.969866\pi\)
							0.579624	+	0.814884i	\(0.303199\pi\)
\(37\)	2.69276	+	0.830606i	0.442687	+	0.136551i	0.508082	−	0.861309i	\(-0.330355\pi\)
							−0.0653952	+	0.997859i	\(0.520831\pi\)
\(41\)	7.92219	+	5.40126i	1.23724	+	0.843535i	0.992038	−	0.125938i	\(-0.0401939\pi\)
							0.245200	+	0.969472i	\(0.421146\pi\)
\(43\)	2.96956	−	2.02461i	0.452853	−	0.308750i	−0.315328	−	0.948983i	\(-0.602114\pi\)
							0.768181	+	0.640233i	\(0.221162\pi\)
\(47\)	−4.62755	+	11.7908i	−0.674998	+	1.71987i	0.0189301	+	0.999821i	\(0.493974\pi\)
							−0.693928	+	0.720045i	\(0.744121\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.2.y.b.583.7	✓	336
9.4	even	3		882.2.bb.a.877.18	yes	336
49.39	even	21		882.2.bb.a.529.18	yes	336
441.382	even	21	inner	882.2.y.b.823.7	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.2.y.b.583.7	✓	336	1.1	even	1	trivial
882.2.y.b.823.7	yes	336	441.382	even	21	inner
882.2.bb.a.529.18	yes	336	49.39	even	21
882.2.bb.a.877.18	yes	336	9.4	even	3