Embedded newform 882.2.bk.a.131.13

• Label: 882.2.bk.a.131.13
• 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			131.13
Character	\(\chi\)	\(=\)	882.131
Dual form			882.2.bk.a.101.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.781831 - 0.623490i) q^{2} +(-0.473838 - 1.66598i) q^{3} +(0.222521 + 0.974928i) q^{4} +(-0.256905 - 3.42815i) q^{5} +(-0.668258 + 1.59795i) q^{6} +(-2.20525 - 1.46180i) q^{7} +(0.433884 - 0.900969i) q^{8} +(-2.55095 + 1.57881i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 672 q + 112 q^{4} + 14 q^{6} - 2 q^{7} + 10 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.473838	−	1.66598i	−0.273571	−	0.961852i
\(5\)	−0.256905	−	3.42815i	−0.114891	−	1.53312i								−0.695257	−	0.718761i	\(-0.744710\pi\)
							0.580366	−	0.814356i	\(-0.302910\pi\)
\(7\)	−2.20525	−	1.46180i	−0.833508	−	0.552508i
							\(11\)	0.161627	−	0.0634337i	0.0487322	−	0.0191260i	−0.340850	−	0.940118i	\(-0.610715\pi\)
0.389582	+	0.920992i	\(0.372619\pi\)
\(13\)	−4.51890	+	1.77354i	−1.25332	+	0.491891i	−0.896816	−	0.442403i	\(-0.854126\pi\)
							−0.356502	+	0.934294i	\(0.616031\pi\)
\(17\)	−0.839010	+	0.258800i	−0.203490	+	0.0627683i	−0.394824	−	0.918757i	\(-0.629195\pi\)
							0.191335	+	0.981525i	\(0.438718\pi\)
\(19\)	1.69103	+	0.976316i	0.387949	+	0.223982i	0.681271	−	0.732031i	\(-0.261428\pi\)
							−0.293322	+	0.956014i	\(0.594761\pi\)
\(23\)	4.64961	+	5.01109i	0.969510	+	1.04488i	0.998973	+	0.0453177i	\(0.0144300\pi\)
							−0.0294623	+	0.999566i	\(0.509380\pi\)
\(29\)	−1.96015	−	6.35466i	−0.363991	−	1.18003i	−0.933661	−	0.358157i	\(-0.883405\pi\)
							0.569670	−	0.821873i	\(-0.307071\pi\)
\(31\)		−	8.07638i		−	1.45056i	−0.688453	−	0.725281i	\(-0.741710\pi\)
							0.688453	−	0.725281i	\(-0.258290\pi\)
\(37\)	2.26172	+	2.09857i	0.371825	+	0.345003i	0.843847	−	0.536583i	\(-0.180285\pi\)
							−0.472022	+	0.881587i	\(0.656476\pi\)
\(41\)	5.83496	−	3.97821i	0.911267	−	0.621292i	−0.0142904	−	0.999898i	\(-0.504549\pi\)
							0.925558	+	0.378606i	\(0.123597\pi\)
\(43\)	−9.50084	−	6.47756i	−1.44886	−	0.987819i	−0.995325	−	0.0965835i	\(-0.969209\pi\)
							−0.453540	−	0.891236i	\(-0.649839\pi\)
\(47\)	−3.61766	+	4.53640i	−0.527690	+	0.661702i	−0.972222	−	0.234061i	\(-0.924799\pi\)
							0.444532	+	0.895763i	\(0.353370\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.2.bk.a.131.13	yes	672
9.2	odd	6		882.2.bc.a.425.42	✓	672
49.3	odd	42		882.2.bc.a.689.42	yes	672
441.101	even	42	inner	882.2.bk.a.101.13	yes	672

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.2.bc.a.425.42	✓	672	9.2	odd	6
882.2.bc.a.689.42	yes	672	49.3	odd	42
882.2.bk.a.101.13	yes	672	441.101	even	42	inner
882.2.bk.a.131.13	yes	672	1.1	even	1	trivial