Embedded newform 882.2.bk.a.131.7

• Label: 882.2.bk.a.131.7
• Level: $882$
• Weight: $2$
• Character: 882.131
• Analytic conductor: $7.043$
• Analytic rank: $0$
• Dimension: $672$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	882.bk (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.04280545828\)
Analytic rank:	\(0\)
Dimension:	\(672\)
Relative dimension:	\(56\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			131.7
Character	\(\chi\)	\(=\)	882.131
Dual form			882.2.bk.a.101.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.781831 - 0.623490i) q^{2} +(-1.29187 - 1.15372i) q^{3} +(0.222521 + 0.974928i) q^{4} +(-0.0660114 - 0.880861i) q^{5} +(0.290689 + 1.70748i) q^{6} +(2.29848 - 1.31034i) q^{7} +(0.433884 - 0.900969i) q^{8} +(0.337848 + 2.98092i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 672 q + 112 q^{4} + 14 q^{6} - 2 q^{7} + 10 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(785\)
\(\chi(n)\)	\(e\left(\frac{41}{42}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.781831	−	0.623490i	−0.552838	−	0.440874i
							\(3\)	−1.29187	−	1.15372i	−0.745861	−	0.666102i
\(5\)	−0.0660114	−	0.880861i	−0.0295212	−	0.393933i								−0.992319	−	0.123709i	\(-0.960521\pi\)
							0.962797	−	0.270224i	\(-0.0870979\pi\)
\(7\)	2.29848	−	1.31034i	0.868743	−	0.495263i
							\(11\)	0.355253	−	0.139427i	0.107113	−	0.0420387i	−0.311179	−	0.950351i	\(-0.600724\pi\)
0.418292	+	0.908312i	\(0.362629\pi\)
\(13\)	2.95152	−	1.15839i	0.818605	−	0.321279i	0.0811727	−	0.996700i	\(-0.474133\pi\)
							0.737432	+	0.675421i	\(0.236038\pi\)
\(17\)	−1.20381	+	0.371325i	−0.291966	+	0.0900596i	−0.437278	−	0.899326i	\(-0.644058\pi\)
							0.145312	+	0.989386i	\(0.453581\pi\)
\(19\)	4.37325	+	2.52490i	1.00329	+	0.579251i	0.909221	−	0.416314i	\(-0.136678\pi\)
							0.0940716	+	0.995565i	\(0.470012\pi\)
\(23\)	3.11276	+	3.35476i	0.649056	+	0.699516i	0.968760	−	0.248002i	\(-0.0797739\pi\)
							−0.319704	+	0.947518i	\(0.603583\pi\)
\(29\)	1.29888	+	4.21087i	0.241196	+	0.781938i	0.992528	+	0.122014i	\(0.0389353\pi\)
							−0.751332	+	0.659924i	\(0.770588\pi\)
\(31\)		−	2.86404i		−	0.514396i	−0.966359	−	0.257198i	\(-0.917201\pi\)
							0.966359	−	0.257198i	\(-0.0827993\pi\)
\(37\)	2.27767	+	2.11337i	0.374446	+	0.347435i	0.844838	−	0.535023i	\(-0.179697\pi\)
							−0.470392	+	0.882458i	\(0.655887\pi\)
\(41\)	−3.02416	+	2.06184i	−0.472294	+	0.322005i	−0.775969	−	0.630771i	\(-0.782739\pi\)
							0.303675	+	0.952776i	\(0.401786\pi\)
\(43\)	−2.66536	−	1.81721i	−0.406464	−	0.277123i	0.342781	−	0.939415i	\(-0.388631\pi\)
							−0.749245	+	0.662293i	\(0.769584\pi\)
\(47\)	6.75672	−	8.47266i	0.985569	−	1.23586i	0.0138064	−	0.999905i	\(-0.495605\pi\)
							0.971763	−	0.235960i	\(-0.0758234\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.2.bk.a.131.7	yes	672
9.2	odd	6		882.2.bc.a.425.36	✓	672
49.3	odd	42		882.2.bc.a.689.36	yes	672
441.101	even	42	inner	882.2.bk.a.101.7	yes	672

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.2.bc.a.425.36	✓	672	9.2	odd	6
882.2.bc.a.689.36	yes	672	49.3	odd	42
882.2.bk.a.101.7	yes	672	441.101	even	42	inner
882.2.bk.a.131.7	yes	672	1.1	even	1	trivial