Embedded newform 432.2.bg.a.13.10

• Label: 432.2.bg.a.13.10
• Level: $432$
• Weight: $2$
• Character: 432.13
• Analytic conductor: $3.450$
• Analytic rank: $0$
• Dimension: $840$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	432.bg (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.44953736732\)
Analytic rank:	\(0\)
Dimension:	\(840\)
Relative dimension:	\(70\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label			13.10
Character	\(\chi\)	\(=\)	432.13
Dual form			432.2.bg.a.133.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.30739 + 0.539193i) q^{2} +(1.39818 + 1.02229i) q^{3} +(1.41854 - 1.40987i) q^{4} +(3.17016 - 0.277353i) q^{5} +(-2.37919 - 0.582643i) q^{6} +(0.294648 - 0.809538i) q^{7} +(-1.09439 + 2.60812i) q^{8} +(0.909840 + 2.85870i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 6 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/432\mathbb{Z}\right)^\times\).

\(n\)	\(271\)	\(325\)	\(353\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{4}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.30739	+	0.539193i	−0.924465	+	0.381267i
							\(3\)	1.39818	+	1.02229i	0.807242	+	0.590220i
\(5\)	3.17016	−	0.277353i	1.41774	−	0.124036i								0.647634	−	0.761951i	\(-0.275759\pi\)
							0.770107	+	0.637915i	\(0.220203\pi\)
\(7\)	0.294648	−	0.809538i	0.111366	−	0.305977i	−0.871472	−	0.490445i	\(-0.836834\pi\)
							0.982838	+	0.184469i	\(0.0590564\pi\)
\(11\)	−0.288468	+	3.29721i	−0.0869765	+	0.994146i	0.819832	+	0.572604i	\(0.194067\pi\)
							−0.906809	+	0.421542i	\(0.861489\pi\)
\(13\)	−4.02085	+	2.81543i	−1.11518	+	0.780860i	−0.977776	−	0.209654i	\(-0.932766\pi\)
							−0.137408	+	0.990514i	\(0.543877\pi\)
\(17\)	3.35062	−	5.80345i	0.812646	−	1.40754i	−0.0983606	−	0.995151i	\(-0.531360\pi\)
							0.911006	−	0.412393i	\(-0.135307\pi\)
\(19\)	1.40118	−	5.22927i	0.321453	−	1.19968i	−0.596377	−	0.802704i	\(-0.703394\pi\)
							0.917830	−	0.396973i	\(-0.129939\pi\)
\(23\)	2.24030	+	6.15518i	0.467135	+	1.28344i	0.920019	+	0.391874i	\(0.128173\pi\)
							−0.452883	+	0.891570i	\(0.649605\pi\)
\(29\)	2.58385	−	3.69012i	0.479809	−	0.685238i	−0.504154	−	0.863614i	\(-0.668195\pi\)
							0.983962	+	0.178376i	\(0.0570843\pi\)
\(31\)	−8.35330	+	3.04035i	−1.50030	+	0.546064i	−0.956138	−	0.292918i	\(-0.905374\pi\)
							−0.544160	+	0.838981i	\(0.683152\pi\)
\(37\)	−1.70545	−	6.36483i	−0.280374	−	1.04637i	−0.952154	−	0.305620i	\(-0.901136\pi\)
							0.671779	−	0.740751i	\(-0.265530\pi\)
\(41\)	1.01427	+	0.178843i	0.158402	+	0.0279306i	0.252287	−	0.967652i	\(-0.418817\pi\)
							−0.0938845	+	0.995583i	\(0.529928\pi\)
\(43\)	0.303089	−	3.46432i	0.0462206	−	0.528304i	−0.937425	−	0.348188i	\(-0.886797\pi\)
							0.983645	−	0.180116i	\(-0.0576473\pi\)
\(47\)	−0.388390	−	0.141363i	−0.0566526	−	0.0206198i	0.313538	−	0.949576i	\(-0.398486\pi\)
							−0.370191	+	0.928956i	\(0.620708\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.2.bg.a.13.10	✓	840
16.5	even	4	inner	432.2.bg.a.229.69	yes	840
27.25	even	9	inner	432.2.bg.a.349.69	yes	840
432.133	even	36	inner	432.2.bg.a.133.10	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.2.bg.a.13.10	✓	840	1.1	even	1	trivial
432.2.bg.a.133.10	yes	840	432.133	even	36	inner
432.2.bg.a.229.69	yes	840	16.5	even	4	inner
432.2.bg.a.349.69	yes	840	27.25	even	9	inner