Embedded newform 432.2.bj.a.227.1

• Label: 432.2.bj.a.227.1
• Level: $432$
• Weight: $2$
• Character: 432.227
• 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			227.1
Character	\(\chi\)	\(=\)	432.227
Dual form			432.2.bj.a.59.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41071 + 0.0994984i) q^{2} +(-1.73093 - 0.0621762i) q^{3} +(1.98020 - 0.280727i) q^{4} +(-1.58735 - 3.40408i) q^{5} +(2.44803 - 0.0845128i) q^{6} +(-0.786764 + 4.46196i) q^{7} +(-2.76555 + 0.593051i) q^{8} +(2.99227 + 0.215246i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 24 q^{7} - 18 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(271\)	\(325\)	\(353\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{13}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41071	+	0.0994984i	−0.997522	+	0.0703560i
							\(3\)	−1.73093	−	0.0621762i	−0.999355	−	0.0358974i
\(5\)	−1.58735	−	3.40408i	−0.709884	−	1.52235i								−0.846883	−	0.531779i	\(-0.821524\pi\)
							0.136999	−	0.990571i	\(-0.456254\pi\)
\(7\)	−0.786764	+	4.46196i	−0.297369	+	1.68646i	0.360046	+	0.932935i	\(0.382761\pi\)
							−0.657415	+	0.753529i	\(0.728350\pi\)
\(11\)	−1.43275	+	3.07254i	−0.431991	+	0.926407i	0.563002	+	0.826456i	\(0.309646\pi\)
							−0.994992	+	0.0999513i	\(0.968131\pi\)
\(13\)	0.373992	+	0.0327201i	0.103727	+	0.00907492i	0.138901	−	0.990306i	\(-0.455643\pi\)
							−0.0351741	+	0.999381i	\(0.511199\pi\)
\(17\)	2.48663	−	1.43566i	0.603097	−	0.348198i	−0.167162	−	0.985929i	\(-0.553460\pi\)
							0.770259	+	0.637731i	\(0.220127\pi\)
\(19\)	−1.75356	−	6.54437i	−0.402294	−	1.50138i	−0.808992	−	0.587820i	\(-0.799986\pi\)
							0.406698	−	0.913563i	\(-0.366680\pi\)
\(23\)	2.58417	−	0.455659i	0.538837	−	0.0950115i	0.102395	−	0.994744i	\(-0.467349\pi\)
							0.436442	+	0.899732i	\(0.356238\pi\)
\(29\)	4.52609	−	0.395982i	0.840474	−	0.0735320i	0.341216	−	0.939985i	\(-0.389161\pi\)
							0.499259	+	0.866453i	\(0.333606\pi\)
\(31\)	9.47680	−	1.67102i	1.70208	−	0.300123i	0.763662	−	0.645616i	\(-0.223399\pi\)
							0.938422	+	0.345492i	\(0.112288\pi\)
\(37\)	1.02951	+	0.275855i	0.169250	+	0.0453503i	0.342449	−	0.939537i	\(-0.388744\pi\)
							−0.173199	+	0.984887i	\(0.555410\pi\)
\(41\)	3.38639	−	2.84152i	0.528865	−	0.443771i	−0.338844	−	0.940843i	\(-0.610036\pi\)
							0.867709	+	0.497072i	\(0.165591\pi\)
\(43\)	4.52298	+	2.10910i	0.689748	+	0.321635i	0.735699	−	0.677309i	\(-0.236854\pi\)
							−0.0459505	+	0.998944i	\(0.514632\pi\)
\(47\)	1.03821	−	5.88800i	0.151439	−	0.858853i	−0.810531	−	0.585696i	\(-0.800821\pi\)
							0.961970	−	0.273157i	\(-0.0880677\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.2.bj.a.227.1	yes	840
16.11	odd	4	inner	432.2.bj.a.11.39	✓	840
27.5	odd	18	inner	432.2.bj.a.275.39	yes	840
432.59	even	36	inner	432.2.bj.a.59.1	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.2.bj.a.11.39	✓	840	16.11	odd	4	inner
432.2.bj.a.59.1	yes	840	432.59	even	36	inner
432.2.bj.a.227.1	yes	840	1.1	even	1	trivial
432.2.bj.a.275.39	yes	840	27.5	odd	18	inner