Embedded newform 882.2.bk.a.131.28

• Label: 882.2.bk.a.131.28
• 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			131.28
Character	\(\chi\)	\(=\)	882.131
Dual form			882.2.bk.a.101.28

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.781831 - 0.623490i) q^{2} +(1.72464 + 0.160074i) q^{3} +(0.222521 + 0.974928i) q^{4} +(-0.0479911 - 0.640396i) q^{5} +(-1.24857 - 1.20045i) q^{6} +(-2.29764 + 1.31182i) q^{7} +(0.433884 - 0.900969i) q^{8} +(2.94875 + 0.552141i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 672 q + 112 q^{4} + 14 q^{6} - 2 q^{7} + 10 q^{9}+O(q^{10}) \)

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
							\(3\)	1.72464	+	0.160074i	0.995720	+	0.0924191i
\(5\)	−0.0479911	−	0.640396i	−0.0214623	−	0.286394i								−0.997600	−	0.0692414i	\(-0.977942\pi\)
							0.976138	−	0.217153i	\(-0.0696769\pi\)
\(7\)	−2.29764	+	1.31182i	−0.868425	+	0.495820i
							\(11\)	3.76410	−	1.47730i	1.13492	−	0.445422i	0.277849	−	0.960625i	\(-0.410378\pi\)
0.857069	+	0.515202i	\(0.172283\pi\)
\(13\)	−1.65735	+	0.650464i	−0.459668	+	0.180406i	−0.583864	−	0.811851i	\(-0.698460\pi\)
							0.124197	+	0.992258i	\(0.460365\pi\)
\(17\)	1.64184	−	0.506439i	0.398204	−	0.122830i	−0.0891842	−	0.996015i	\(-0.528426\pi\)
							0.487388	+	0.873186i	\(0.337950\pi\)
\(19\)	2.05918	+	1.18887i	0.472408	+	0.272745i	0.717247	−	0.696819i	\(-0.245402\pi\)
							−0.244839	+	0.969564i	\(0.578735\pi\)
\(23\)	1.43249	+	1.54386i	0.298695	+	0.321916i	0.864510	−	0.502616i	\(-0.167629\pi\)
							−0.565815	+	0.824532i	\(0.691438\pi\)
\(29\)	2.17850	+	7.06253i	0.404538	+	1.31148i	0.897341	+	0.441339i	\(0.145496\pi\)
							−0.492803	+	0.870141i	\(0.664028\pi\)
\(31\)		−	9.69115i		−	1.74058i	−0.492538	−	0.870291i	\(-0.663931\pi\)
							0.492538	−	0.870291i	\(-0.336069\pi\)
\(37\)	−4.09279	−	3.79755i	−0.672850	−	0.624314i	0.267887	−	0.963450i	\(-0.413675\pi\)
							−0.940737	+	0.339137i	\(0.889865\pi\)
\(41\)	3.80491	−	2.59414i	0.594227	−	0.405137i	−0.228525	−	0.973538i	\(-0.573390\pi\)
							0.822752	+	0.568401i	\(0.192438\pi\)
\(43\)	10.2911	+	7.01636i	1.56938	+	1.06999i	0.961077	+	0.276281i	\(0.0891021\pi\)
							0.608304	+	0.793704i	\(0.291850\pi\)
\(47\)	−1.97770	+	2.47995i	−0.288477	+	0.361738i	−0.904861	−	0.425707i	\(-0.860025\pi\)
							0.616384	+	0.787445i	\(0.288597\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.2.bk.a.131.28	yes	672
9.2	odd	6		882.2.bc.a.425.53	✓	672
49.3	odd	42		882.2.bc.a.689.53	yes	672
441.101	even	42	inner	882.2.bk.a.101.28	yes	672

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.2.bc.a.425.53	✓	672	9.2	odd	6
882.2.bc.a.689.53	yes	672	49.3	odd	42
882.2.bk.a.101.28	yes	672	441.101	even	42	inner
882.2.bk.a.131.28	yes	672	1.1	even	1	trivial