Embedded newform 882.2.bk.a.131.33

• Label: 882.2.bk.a.131.33
• 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			131.33
Character	\(\chi\)	\(=\)	882.131
Dual form			882.2.bk.a.101.33

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.781831 + 0.623490i) q^{2} +(-1.58878 + 0.689757i) q^{3} +(0.222521 + 0.974928i) q^{4} +(0.185877 + 2.48036i) q^{5} +(-1.67222 - 0.451317i) q^{6} +(-2.26724 + 1.36367i) q^{7} +(-0.433884 + 0.900969i) q^{8} +(2.04847 - 2.19175i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 672 q + 112 q^{4} + 14 q^{6} - 2 q^{7} + 10 q^{9}+O(q^{10}) \)

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
							\(3\)	−1.58878	+	0.689757i	−0.917285	+	0.398231i
\(5\)	0.185877	+	2.48036i	0.0831268	+	1.10925i								0.870301	+	0.492520i	\(0.163924\pi\)
							−0.787175	+	0.616730i	\(0.788457\pi\)
\(7\)	−2.26724	+	1.36367i	−0.856938	+	0.515419i
							\(11\)	−1.02586	+	0.402620i	−0.309308	+	0.121394i	−0.514918	−	0.857240i	\(-0.672178\pi\)
0.205610	+	0.978634i	\(0.434082\pi\)
\(13\)	−1.42684	+	0.559993i	−0.395734	+	0.155314i	−0.554859	−	0.831944i	\(-0.687228\pi\)
							0.159125	+	0.987258i	\(0.449133\pi\)
\(17\)	−2.55007	+	0.786594i	−0.618484	+	0.190777i	−0.588136	−	0.808762i	\(-0.700138\pi\)
							−0.0303483	+	0.999539i	\(0.509662\pi\)
\(19\)	2.02003	+	1.16626i	0.463426	+	0.267559i	0.713484	−	0.700672i	\(-0.247116\pi\)
							−0.250058	+	0.968231i	\(0.580450\pi\)
\(23\)	−0.570353	−	0.614694i	−0.118927	−	0.128173i	0.670782	−	0.741655i	\(-0.265959\pi\)
							−0.789709	+	0.613482i	\(0.789768\pi\)
\(29\)	−2.29335	−	7.43486i	−0.425864	−	1.38062i	−0.873684	−	0.486494i	\(-0.838276\pi\)
							0.447820	−	0.894124i	\(-0.352201\pi\)
\(31\)		−	0.774842i		−	0.139166i	−0.997576	−	0.0695828i	\(-0.977833\pi\)
							0.997576	−	0.0695828i	\(-0.0221668\pi\)
\(37\)	3.75095	+	3.48037i	0.616652	+	0.572169i	0.925408	−	0.378972i	\(-0.123722\pi\)
							−0.308756	+	0.951141i	\(0.599913\pi\)
\(41\)	1.87494	−	1.27831i	0.292817	−	0.199639i	−0.407997	−	0.912983i	\(-0.633773\pi\)
							0.700814	+	0.713344i	\(0.252820\pi\)
\(43\)	−6.51378	−	4.44102i	−0.993342	−	0.677249i	−0.0464898	−	0.998919i	\(-0.514803\pi\)
							−0.946852	+	0.321670i	\(0.895756\pi\)
\(47\)	−1.59388	+	1.99866i	−0.232491	+	0.291535i	−0.884368	−	0.466790i	\(-0.845410\pi\)
							0.651877	+	0.758325i	\(0.273982\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.2.bk.a.131.33	yes	672
9.2	odd	6		882.2.bc.a.425.2	✓	672
49.3	odd	42		882.2.bc.a.689.2	yes	672
441.101	even	42	inner	882.2.bk.a.101.33	yes	672

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.2.bc.a.425.2	✓	672	9.2	odd	6
882.2.bc.a.689.2	yes	672	49.3	odd	42
882.2.bk.a.101.33	yes	672	441.101	even	42	inner
882.2.bk.a.131.33	yes	672	1.1	even	1	trivial