Embedded newform 882.2.bk.a.131.37

• Label: 882.2.bk.a.131.37
• 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			131.37
Character	\(\chi\)	\(=\)	882.131
Dual form			882.2.bk.a.101.37

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.781831 + 0.623490i) q^{2} +(-0.861300 - 1.50272i) q^{3} +(0.222521 + 0.974928i) q^{4} +(0.100060 + 1.33520i) q^{5} +(0.263538 - 1.71188i) q^{6} +(0.473457 - 2.60304i) q^{7} +(-0.433884 + 0.900969i) q^{8} +(-1.51632 + 2.58858i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 672 q + 112 q^{4} + 14 q^{6} - 2 q^{7} + 10 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.861300	−	1.50272i	−0.497272	−	0.867595i
\(5\)	0.100060	+	1.33520i	0.0447481	+	0.597121i								0.973834	+	0.227259i	\(0.0729763\pi\)
							−0.929086	+	0.369863i	\(0.879405\pi\)
\(7\)	0.473457	−	2.60304i	0.178950	−	0.983858i
							\(11\)	−4.31266	+	1.69259i	−1.30032	+	0.510336i	−0.911622	−	0.411030i	\(-0.865169\pi\)
−0.388694	+	0.921367i	\(0.627074\pi\)
\(13\)	−5.95095	+	2.33558i	−1.65050	+	0.647772i	−0.994280	−	0.106800i	\(-0.965939\pi\)
							−0.656216	+	0.754573i	\(0.727844\pi\)
\(17\)	−3.36288	+	1.03731i	−0.815619	+	0.251585i	−0.674375	−	0.738389i	\(-0.735587\pi\)
							−0.141245	+	0.989975i	\(0.545110\pi\)
\(19\)	5.72818	+	3.30716i	1.31413	+	0.758715i	0.982778	−	0.184790i	\(-0.0591606\pi\)
							0.331356	+	0.943506i	\(0.392494\pi\)
\(23\)	−3.86844	−	4.16919i	−0.806626	−	0.869337i	0.186965	−	0.982367i	\(-0.440135\pi\)
							−0.993592	+	0.113030i	\(0.963944\pi\)
\(29\)	−1.46136	−	4.73762i	−0.271368	−	0.879755i	−0.983993	−	0.178209i	\(-0.942970\pi\)
							0.712624	−	0.701546i	\(-0.247506\pi\)
\(31\)			10.8994i			1.95759i	0.204832	+	0.978797i	\(0.434335\pi\)
							−0.204832	+	0.978797i	\(0.565665\pi\)
\(37\)	−0.694409	−	0.644317i	−0.114160	−	0.105925i	0.621028	−	0.783789i	\(-0.286715\pi\)
							−0.735188	+	0.677864i	\(0.762906\pi\)
\(41\)	−9.25270	+	6.30839i	−1.44503	+	0.985204i	−0.449252	+	0.893405i	\(0.648309\pi\)
							−0.995778	+	0.0917990i	\(0.970738\pi\)
\(43\)	−2.10273	−	1.43362i	−0.320663	−	0.218624i	0.392281	−	0.919846i	\(-0.371686\pi\)
							−0.712944	+	0.701221i	\(0.752639\pi\)
\(47\)	−3.31190	+	4.15299i	−0.483090	+	0.605775i	−0.962322	−	0.271913i	\(-0.912344\pi\)
							0.479232	+	0.877688i	\(0.340915\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.2.bk.a.131.37	yes	672
9.2	odd	6		882.2.bc.a.425.12	✓	672
49.3	odd	42		882.2.bc.a.689.12	yes	672
441.101	even	42	inner	882.2.bk.a.101.37	yes	672

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.2.bc.a.425.12	✓	672	9.2	odd	6
882.2.bc.a.689.12	yes	672	49.3	odd	42
882.2.bk.a.101.37	yes	672	441.101	even	42	inner
882.2.bk.a.131.37	yes	672	1.1	even	1	trivial