Embedded newform 882.2.bk.a.131.53

• Label: 882.2.bk.a.131.53
• 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			131.53
Character	\(\chi\)	\(=\)	882.131
Dual form			882.2.bk.a.101.53

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.781831 + 0.623490i) q^{2} +(1.55649 + 0.759824i) q^{3} +(0.222521 + 0.974928i) q^{4} +(-0.205620 - 2.74381i) q^{5} +(0.743172 + 1.56451i) q^{6} +(-2.41710 - 1.07593i) q^{7} +(-0.433884 + 0.900969i) q^{8} +(1.84533 + 2.36532i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 672 q + 112 q^{4} + 14 q^{6} - 2 q^{7} + 10 q^{9}+O(q^{10}) \)

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
							\(3\)	1.55649	+	0.759824i	0.898641	+	0.438685i
\(5\)	−0.205620	−	2.74381i	−0.0919562	−	1.22707i								−0.832102	−	0.554623i	\(-0.812862\pi\)
							0.740145	−	0.672447i	\(-0.234757\pi\)
\(7\)	−2.41710	−	1.07593i	−0.913578	−	0.406663i
							\(11\)	4.21690	−	1.65501i	1.27144	−	0.499005i	0.368847	−	0.929490i	\(-0.379753\pi\)
0.902598	+	0.430485i	\(0.141657\pi\)
\(13\)	4.33907	−	1.70296i	1.20344	−	0.472316i	0.322967	−	0.946410i	\(-0.395320\pi\)
							0.880474	+	0.474094i	\(0.157224\pi\)
\(17\)	−0.785427	+	0.242272i	−0.190494	+	0.0587596i	−0.388534	−	0.921434i	\(-0.627018\pi\)
							0.198040	+	0.980194i	\(0.436542\pi\)
\(19\)	−1.96478	−	1.13437i	−0.450752	−	0.260242i	0.257396	−	0.966306i	\(-0.417136\pi\)
							−0.708148	+	0.706064i	\(0.750469\pi\)
\(23\)	5.24232	+	5.64988i	1.09310	+	1.17808i	0.982894	+	0.184172i	\(0.0589604\pi\)
							0.110206	+	0.993909i	\(0.464849\pi\)
\(29\)	−1.03221	−	3.34636i	−0.191677	−	0.621403i	−0.999489	−	0.0319621i	\(-0.989824\pi\)
							0.807812	−	0.589441i	\(-0.200652\pi\)
\(31\)		−	4.41601i		−	0.793139i	−0.918005	−	0.396570i	\(-0.870201\pi\)
							0.918005	−	0.396570i	\(-0.129799\pi\)
\(37\)	4.13935	+	3.84076i	0.680505	+	0.631417i	0.942717	−	0.333594i	\(-0.108261\pi\)
							−0.262212	+	0.965010i	\(0.584452\pi\)
\(41\)	−9.52306	+	6.49271i	−1.48725	+	1.01399i	−0.498096	+	0.867122i	\(0.665967\pi\)
							−0.989156	+	0.146869i	\(0.953080\pi\)
\(43\)	−2.29271	−	1.56314i	−0.349635	−	0.238377i	0.375749	−	0.926721i	\(-0.377385\pi\)
							−0.725384	+	0.688344i	\(0.758338\pi\)
\(47\)	4.99198	−	6.25975i	0.728155	−	0.913078i	−0.270613	−	0.962688i	\(-0.587226\pi\)
							0.998769	+	0.0496101i	\(0.0157979\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.2.bk.a.131.53	yes	672
9.2	odd	6		882.2.bc.a.425.24	✓	672
49.3	odd	42		882.2.bc.a.689.24	yes	672
441.101	even	42	inner	882.2.bk.a.101.53	yes	672

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.2.bc.a.425.24	✓	672	9.2	odd	6
882.2.bc.a.689.24	yes	672	49.3	odd	42
882.2.bk.a.101.53	yes	672	441.101	even	42	inner
882.2.bk.a.131.53	yes	672	1.1	even	1	trivial