Embedded newform 432.2.bj.a.131.4

• Label: 432.2.bj.a.131.4
• Level: $432$
• Weight: $2$
• Character: 432.131
• 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.bj (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			131.4
Character	\(\chi\)	\(=\)	432.131
Dual form			432.2.bj.a.155.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39279 + 0.245235i) q^{2} +(1.19939 - 1.24959i) q^{3} +(1.87972 - 0.683120i) q^{4} +(1.61043 - 1.12764i) q^{5} +(-1.36405 + 2.03454i) q^{6} +(-0.0809125 + 0.0678936i) q^{7} +(-2.45053 + 1.41241i) q^{8} +(-0.122946 - 2.99748i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 24 q^{7} - 18 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(271\)	\(325\)	\(353\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{11}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.39279	+	0.245235i	−0.984850	+	0.173407i
							\(3\)	1.19939	−	1.24959i	0.692466	−	0.721451i
\(5\)	1.61043	−	1.12764i	0.720208	−	0.504295i								−0.155077	−	0.987902i	\(-0.549563\pi\)
							0.875285	+	0.483607i	\(0.160674\pi\)
\(7\)	−0.0809125	+	0.0678936i	−0.0305820	+	0.0256614i	−0.657951	−	0.753061i	\(-0.728576\pi\)
							0.627369	+	0.778723i	\(0.284132\pi\)
\(11\)	3.90992	+	2.73776i	1.17889	+	0.825465i	0.987742	−	0.156097i	\(-0.0498913\pi\)
							0.191144	+	0.981562i	\(0.438780\pi\)
\(13\)	5.45719	+	2.54473i	1.51355	+	0.705780i	0.989315	−	0.145793i	\(-0.0465733\pi\)
							0.524236	+	0.851573i	\(0.324351\pi\)
\(17\)	−7.03471	−	4.06149i	−1.70617	−	0.985056i	−0.939200	−	0.343371i	\(-0.888431\pi\)
							−0.766968	−	0.641685i	\(-0.778235\pi\)
\(19\)	2.37687	+	0.636879i	0.545290	+	0.146110i	0.520940	−	0.853593i	\(-0.325582\pi\)
							0.0243504	+	0.999703i	\(0.492248\pi\)
\(23\)	−0.0549500	+	0.0654869i	−0.0114579	+	0.0136550i	−0.771743	−	0.635935i	\(-0.780615\pi\)
							0.760285	+	0.649590i	\(0.225059\pi\)
\(29\)	−5.88948	+	2.74631i	−1.09365	+	0.509977i	−0.883851	−	0.467768i	\(-0.845058\pi\)
							−0.209798	+	0.977745i	\(0.567281\pi\)
\(31\)	1.20335	−	1.43410i	0.216128	−	0.257571i	−0.647077	−	0.762424i	\(-0.724009\pi\)
							0.863205	+	0.504853i	\(0.168453\pi\)
\(37\)	−2.22143	−	8.29048i	−0.365200	−	1.36295i	−0.867149	−	0.498049i	\(-0.834050\pi\)
							0.501949	−	0.864898i	\(-0.332617\pi\)
\(41\)	−4.12925	+	1.50292i	−0.644880	+	0.234717i	−0.643695	−	0.765282i	\(-0.722599\pi\)
							−0.00118519	+	0.999999i	\(0.500377\pi\)
\(43\)	3.53213	−	5.04440i	0.538644	−	0.769263i	−0.453913	−	0.891046i	\(-0.649972\pi\)
							0.992557	+	0.121783i	\(0.0388611\pi\)
\(47\)	6.29165	−	5.27932i	0.917732	−	0.770068i	−0.0558426	−	0.998440i	\(-0.517784\pi\)
							0.973574	+	0.228371i	\(0.0733401\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.2.bj.a.131.4	✓	840
16.11	odd	4	inner	432.2.bj.a.347.51	yes	840
27.20	odd	18	inner	432.2.bj.a.371.51	yes	840
432.155	even	36	inner	432.2.bj.a.155.4	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.2.bj.a.131.4	✓	840	1.1	even	1	trivial
432.2.bj.a.155.4	yes	840	432.155	even	36	inner
432.2.bj.a.347.51	yes	840	16.11	odd	4	inner
432.2.bj.a.371.51	yes	840	27.20	odd	18	inner