Embedded newform 882.2.bk.a.131.18

• Label: 882.2.bk.a.131.18
• 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			131.18
Character	\(\chi\)	\(=\)	882.131
Dual form			882.2.bk.a.101.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.781831 - 0.623490i) q^{2} +(0.744681 - 1.56379i) q^{3} +(0.222521 + 0.974928i) q^{4} +(-0.0165168 - 0.220401i) q^{5} +(-1.55722 + 0.758322i) q^{6} +(1.78069 - 1.95682i) q^{7} +(0.433884 - 0.900969i) q^{8} +(-1.89090 - 2.32905i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 672 q + 112 q^{4} + 14 q^{6} - 2 q^{7} + 10 q^{9}+O(q^{10}) \)

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
							\(3\)	0.744681	−	1.56379i	0.429942	−	0.902857i
\(5\)	−0.0165168	−	0.220401i	−0.00738652	−	0.0985662i								0.992292	−	0.123922i	\(-0.0395472\pi\)
							−0.999678	+	0.0253557i	\(0.991928\pi\)
\(7\)	1.78069	−	1.95682i	0.673038	−	0.739608i
							\(11\)	−4.87535	+	1.91343i	−1.46997	+	0.576921i	−0.959183	−	0.282785i	\(-0.908742\pi\)
−0.510789	+	0.859706i	\(0.670646\pi\)
\(13\)	−2.36534	+	0.928326i	−0.656026	+	0.257471i	−0.669921	−	0.742432i	\(-0.733672\pi\)
							0.0138951	+	0.999903i	\(0.495577\pi\)
\(17\)	−2.96872	+	0.915730i	−0.720021	+	0.222097i	−0.633047	−	0.774113i	\(-0.718196\pi\)
							−0.0869743	+	0.996211i	\(0.527720\pi\)
\(19\)	−3.20662	−	1.85134i	−0.735649	−	0.424727i	0.0848363	−	0.996395i	\(-0.472963\pi\)
							−0.820485	+	0.571668i	\(0.806297\pi\)
\(23\)	−2.35643	−	2.53962i	−0.491349	−	0.529548i	0.437889	−	0.899029i	\(-0.355726\pi\)
							−0.929238	+	0.369480i	\(0.879536\pi\)
\(29\)	0.973627	+	3.15642i	0.180798	+	0.586132i	0.999922	+	0.0125156i	\(0.00398396\pi\)
							−0.819124	+	0.573617i	\(0.805540\pi\)
\(31\)		−	5.27561i		−	0.947527i	−0.880652	−	0.473763i	\(-0.842895\pi\)
							0.880652	−	0.473763i	\(-0.157105\pi\)
\(37\)	−6.52263	−	6.05212i	−1.07231	−	0.994962i	−0.0723168	−	0.997382i	\(-0.523039\pi\)
							−0.999997	+	0.00241980i	\(0.999230\pi\)
\(41\)	5.70971	−	3.89282i	0.891708	−	0.607956i	−0.0284108	−	0.999596i	\(-0.509045\pi\)
							0.920118	+	0.391640i	\(0.128092\pi\)
\(43\)	1.33100	+	0.907458i	0.202975	+	0.138386i	0.660542	−	0.750789i	\(-0.270327\pi\)
							−0.457567	+	0.889175i	\(0.651279\pi\)
\(47\)	−2.23663	+	2.80464i	−0.326245	+	0.409099i	−0.917722	−	0.397223i	\(-0.869974\pi\)
							0.591477	+	0.806322i	\(0.298545\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.2.bk.a.131.18	yes	672
9.2	odd	6		882.2.bc.a.425.47	✓	672
49.3	odd	42		882.2.bc.a.689.47	yes	672
441.101	even	42	inner	882.2.bk.a.101.18	yes	672

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.2.bc.a.425.47	✓	672	9.2	odd	6
882.2.bc.a.689.47	yes	672	49.3	odd	42
882.2.bk.a.101.18	yes	672	441.101	even	42	inner
882.2.bk.a.131.18	yes	672	1.1	even	1	trivial