Embedded newform 882.2.bk.a.131.46

• Label: 882.2.bk.a.131.46
• 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			131.46
Character	\(\chi\)	\(=\)	882.131
Dual form			882.2.bk.a.101.46

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.781831 + 0.623490i) q^{2} +(0.558699 - 1.63947i) q^{3} +(0.222521 + 0.974928i) q^{4} +(0.306927 + 4.09565i) q^{5} +(1.45900 - 0.933445i) q^{6} +(0.725825 + 2.54424i) q^{7} +(-0.433884 + 0.900969i) q^{8} +(-2.37571 - 1.83194i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 672 q + 112 q^{4} + 14 q^{6} - 2 q^{7} + 10 q^{9}+O(q^{10}) \)

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
							\(3\)	0.558699	−	1.63947i	0.322565	−	0.946547i
\(5\)	0.306927	+	4.09565i	0.137262	+	1.83163i								0.463217	+	0.886245i	\(0.346695\pi\)
							−0.325955	+	0.945385i	\(0.605686\pi\)
\(7\)	0.725825	+	2.54424i	0.274336	+	0.961634i
							\(11\)	−2.96480	+	1.16360i	−0.893920	+	0.350838i	−0.767432	−	0.641131i	\(-0.778466\pi\)
−0.126488	+	0.991968i	\(0.540370\pi\)
\(13\)	2.03333	−	0.798023i	0.563944	−	0.221332i	−0.0662049	−	0.997806i	\(-0.521089\pi\)
							0.630149	+	0.776474i	\(0.282994\pi\)
\(17\)	−3.67103	+	1.13236i	−0.890357	+	0.274639i	−0.705992	−	0.708219i	\(-0.749499\pi\)
							−0.184364	+	0.982858i	\(0.559023\pi\)
\(19\)	2.94819	+	1.70214i	0.676361	+	0.390497i	0.798483	−	0.602018i	\(-0.205636\pi\)
							−0.122122	+	0.992515i	\(0.538970\pi\)
\(23\)	−6.10089	−	6.57520i	−1.27212	−	1.37102i	−0.895751	−	0.444556i	\(-0.853362\pi\)
							−0.376373	−	0.926468i	\(-0.622829\pi\)
\(29\)	1.09477	+	3.54916i	0.203294	+	0.659062i	0.998602	+	0.0528568i	\(0.0168327\pi\)
							−0.795308	+	0.606205i	\(0.792691\pi\)
\(31\)		−	1.28402i		−	0.230616i	−0.993330	−	0.115308i	\(-0.963214\pi\)
							0.993330	−	0.115308i	\(-0.0367855\pi\)
\(37\)	7.38318	+	6.85059i	1.21379	+	1.12623i	0.988386	+	0.151964i	\(0.0485598\pi\)
							0.225401	+	0.974266i	\(0.427631\pi\)
\(41\)	6.81132	−	4.64388i	1.06375	−	0.725252i	0.100792	−	0.994907i	\(-0.467862\pi\)
							0.962957	+	0.269656i	\(0.0869099\pi\)
\(43\)	9.31071	+	6.34793i	1.41987	+	0.968051i	0.998117	+	0.0613398i	\(0.0195373\pi\)
							0.421753	+	0.906711i	\(0.361415\pi\)
\(47\)	5.92038	−	7.42392i	0.863576	−	1.08289i	−0.132214	−	0.991221i	\(-0.542209\pi\)
							0.995790	−	0.0916685i	\(-0.0292200\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.2.bk.a.131.46	yes	672
9.2	odd	6		882.2.bc.a.425.19	✓	672
49.3	odd	42		882.2.bc.a.689.19	yes	672
441.101	even	42	inner	882.2.bk.a.101.46	yes	672

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.2.bc.a.425.19	✓	672	9.2	odd	6
882.2.bc.a.689.19	yes	672	49.3	odd	42
882.2.bk.a.101.46	yes	672	441.101	even	42	inner
882.2.bk.a.131.46	yes	672	1.1	even	1	trivial