Embedded newform 432.2.bg.a.13.6

• Label: 432.2.bg.a.13.6
• 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			13.6
Character	\(\chi\)	\(=\)	432.13
Dual form			432.2.bg.a.133.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.36057 + 0.385801i) q^{2} +(-1.47892 + 0.901550i) q^{3} +(1.70231 - 1.04982i) q^{4} +(2.53117 - 0.221449i) q^{5} +(1.66436 - 1.79719i) q^{6} +(-0.599847 + 1.64807i) q^{7} +(-1.91110 + 2.08511i) q^{8} +(1.37441 - 2.66664i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 6 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.36057	+	0.385801i	−0.962070	+	0.272803i
							\(3\)	−1.47892	+	0.901550i	−0.853855	+	0.520510i
\(5\)	2.53117	−	0.221449i	1.13197	−	0.0990349i								0.494265	−	0.869312i	\(-0.335437\pi\)
							0.637710	+	0.770277i	\(0.279882\pi\)
\(7\)	−0.599847	+	1.64807i	−0.226721	+	0.622911i	−0.999937	−	0.0112320i	\(-0.996425\pi\)
							0.773216	+	0.634143i	\(0.218647\pi\)
\(11\)	−0.139709	+	1.59688i	−0.0421239	+	0.481478i	0.945607	+	0.325312i	\(0.105469\pi\)
							−0.987731	+	0.156167i	\(0.950086\pi\)
\(13\)	4.54725	−	3.18402i	1.26118	−	0.883089i	0.264492	−	0.964388i	\(-0.414796\pi\)
							0.996690	+	0.0812992i	\(0.0259069\pi\)
\(17\)	0.0722387	−	0.125121i	0.0175205	−	0.0303463i	−0.857132	−	0.515096i	\(-0.827756\pi\)
							0.874653	+	0.484750i	\(0.161089\pi\)
\(19\)	0.875229	−	3.26640i	0.200791	−	0.749363i	−0.789900	−	0.613236i	\(-0.789868\pi\)
							0.990691	−	0.136128i	\(-0.0434658\pi\)
\(23\)	0.653321	+	1.79498i	0.136227	+	0.374280i	0.988983	−	0.148029i	\(-0.0472929\pi\)
							−0.852756	+	0.522309i	\(0.825071\pi\)
\(29\)	−3.80580	+	5.43524i	−0.706718	+	1.00930i	0.291884	+	0.956454i	\(0.405718\pi\)
							−0.998603	+	0.0528449i	\(0.983171\pi\)
\(31\)	9.01674	−	3.28182i	1.61945	−	0.589433i	0.636176	−	0.771544i	\(-0.280515\pi\)
							0.983278	+	0.182111i	\(0.0582929\pi\)
\(37\)	2.21425	+	8.26370i	0.364021	+	1.35854i	0.868743	+	0.495262i	\(0.164928\pi\)
							−0.504723	+	0.863282i	\(0.668405\pi\)
\(41\)	4.31634	+	0.761087i	0.674099	+	0.118862i	0.500211	−	0.865904i	\(-0.333256\pi\)
							0.173888	+	0.984765i	\(0.444367\pi\)
\(43\)	−0.220407	+	2.51927i	−0.0336118	+	0.384185i	0.960693	+	0.277612i	\(0.0895431\pi\)
							−0.994305	+	0.106572i	\(0.966012\pi\)
\(47\)	10.3064	+	3.75123i	1.50335	+	0.547174i	0.956925	−	0.290336i	\(-0.0937670\pi\)
							0.546423	+	0.837510i	\(0.315989\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.2.bg.a.13.6	✓	840
16.5	even	4	inner	432.2.bg.a.229.67	yes	840
27.25	even	9	inner	432.2.bg.a.349.67	yes	840
432.133	even	36	inner	432.2.bg.a.133.6	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.2.bg.a.13.6	✓	840	1.1	even	1	trivial
432.2.bg.a.133.6	yes	840	432.133	even	36	inner
432.2.bg.a.229.67	yes	840	16.5	even	4	inner
432.2.bg.a.349.67	yes	840	27.25	even	9	inner