Embedded newform 432.2.bg.a.157.63

• Label: 432.2.bg.a.157.63
• Level: $432$
• Weight: $2$
• Character: 432.157
• 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			157.63
Character	\(\chi\)	\(=\)	432.157
Dual form			432.2.bg.a.421.63

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.31166 + 0.528719i) q^{2} +(0.315357 + 1.70310i) q^{3} +(1.44091 + 1.38700i) q^{4} +(0.0403870 - 0.0866102i) q^{5} +(-0.486819 + 2.40063i) q^{6} +(-2.54473 + 0.448705i) q^{7} +(1.15666 + 2.58111i) q^{8} +(-2.80110 + 1.07417i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 6 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(271\)	\(325\)	\(353\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{7}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.31166	+	0.528719i	0.927485	+	0.373861i
							\(3\)	0.315357	+	1.70310i	0.182071	+	0.983285i
\(5\)	0.0403870	−	0.0866102i	0.0180616	−	0.0387333i								−0.897070	−	0.441888i	\(-0.854309\pi\)
							0.915132	+	0.403155i	\(0.132086\pi\)
\(7\)	−2.54473	+	0.448705i	−0.961817	+	0.169594i	−0.632445	−	0.774606i	\(-0.717948\pi\)
							−0.329373	+	0.944200i	\(0.606837\pi\)
\(11\)	3.01229	−	1.40465i	0.908239	−	0.423519i	0.0883817	−	0.996087i	\(-0.471830\pi\)
							0.819857	+	0.572568i	\(0.194053\pi\)
\(13\)	0.220044	+	2.51512i	0.0610293	+	0.697568i	0.963656	+	0.267145i	\(0.0860802\pi\)
							−0.902627	+	0.430424i	\(0.858364\pi\)
\(17\)	−0.925008	+	1.60216i	−0.224347	+	0.388581i	−0.956123	−	0.292964i	\(-0.905358\pi\)
							0.731776	+	0.681545i	\(0.238692\pi\)
\(19\)	1.22413	−	4.56853i	0.280836	−	1.04809i	−0.670993	−	0.741464i	\(-0.734132\pi\)
							0.951829	−	0.306630i	\(-0.0992013\pi\)
\(23\)	2.27490	+	0.401126i	0.474349	+	0.0836406i	0.405712	−	0.914001i	\(-0.367024\pi\)
							0.0686376	+	0.997642i	\(0.478135\pi\)
\(29\)	−1.50275	−	0.131474i	−0.279054	−	0.0244141i	−0.0532304	−	0.998582i	\(-0.516952\pi\)
							−0.225824	+	0.974168i	\(0.572507\pi\)
\(31\)	1.87133	−	10.6128i	0.336100	−	1.90612i	−0.0799897	−	0.996796i	\(-0.525489\pi\)
							0.416090	−	0.909323i	\(-0.363400\pi\)
\(37\)	−0.866459	−	3.23367i	−0.142445	−	0.531612i	−0.999856	−	0.0169794i	\(-0.994595\pi\)
							0.857411	−	0.514633i	\(-0.172072\pi\)
\(41\)	4.62990	−	5.51770i	0.723069	−	0.861720i	−0.271856	−	0.962338i	\(-0.587638\pi\)
							0.994925	+	0.100618i	\(0.0320820\pi\)
\(43\)	8.76292	−	4.08622i	1.33633	−	0.623142i	0.382633	−	0.923901i	\(-0.375017\pi\)
							0.953700	+	0.300758i	\(0.0972397\pi\)
\(47\)	0.836072	+	4.74160i	0.121954	+	0.691633i	0.983071	+	0.183227i	\(0.0586542\pi\)
							−0.861117	+	0.508407i	\(0.830235\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.2.bg.a.157.63	✓	840
16.5	even	4	inner	432.2.bg.a.373.32	yes	840
27.16	even	9	inner	432.2.bg.a.205.32	yes	840
432.421	even	36	inner	432.2.bg.a.421.63	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.2.bg.a.157.63	✓	840	1.1	even	1	trivial
432.2.bg.a.205.32	yes	840	27.16	even	9	inner
432.2.bg.a.373.32	yes	840	16.5	even	4	inner
432.2.bg.a.421.63	yes	840	432.421	even	36	inner