Embedded newform 432.2.bg.a.13.5

• Label: 432.2.bg.a.13.5
• 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			13.5
Character	\(\chi\)	\(=\)	432.13
Dual form			432.2.bg.a.133.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.37047 - 0.349003i) q^{2} +(1.72507 + 0.155359i) q^{3} +(1.75639 + 0.956599i) q^{4} +(1.88606 - 0.165009i) q^{5} +(-2.30994 - 0.814971i) q^{6} +(-0.627115 + 1.72298i) q^{7} +(-2.07323 - 1.92398i) q^{8} +(2.95173 + 0.536012i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 6 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.37047	−	0.349003i	−0.969071	−	0.246782i
							\(3\)	1.72507	+	0.155359i	0.995969	+	0.0896968i
\(5\)	1.88606	−	0.165009i	0.843471	−	0.0737941i								0.342775	−	0.939417i	\(-0.388633\pi\)
							0.500696	+	0.865623i	\(0.333078\pi\)
\(7\)	−0.627115	+	1.72298i	−0.237027	+	0.651226i	0.762962	+	0.646444i	\(0.223745\pi\)
							−0.999989	+	0.00478252i	\(0.998478\pi\)
\(11\)	0.0485467	−	0.554891i	0.0146374	−	0.167306i	−0.985358	−	0.170501i	\(-0.945461\pi\)
							0.999995	+	0.00319491i	\(0.00101697\pi\)
\(13\)	3.25744	−	2.28088i	0.903451	−	0.632603i	−0.0267544	−	0.999642i	\(-0.508517\pi\)
							0.930206	+	0.367039i	\(0.119628\pi\)
\(17\)	0.0868965	−	0.150509i	0.0210755	−	0.0365038i	−0.855295	−	0.518141i	\(-0.826624\pi\)
							0.876371	+	0.481637i	\(0.159958\pi\)
\(19\)	−1.66131	+	6.20010i	−0.381131	+	1.42240i	0.463045	+	0.886335i	\(0.346757\pi\)
							−0.844176	+	0.536066i	\(0.819910\pi\)
\(23\)	−1.91145	−	5.25167i	−0.398565	−	1.09505i	−0.962984	−	0.269560i	\(-0.913122\pi\)
							0.564419	−	0.825489i	\(-0.309100\pi\)
\(29\)	0.600321	−	0.857347i	0.111477	−	0.159205i	−0.759537	−	0.650464i	\(-0.774574\pi\)
							0.871013	+	0.491259i	\(0.163463\pi\)
\(31\)	−1.39711	+	0.508505i	−0.250928	+	0.0913302i	−0.464422	−	0.885614i	\(-0.653738\pi\)
							0.213494	+	0.976944i	\(0.431516\pi\)
\(37\)	1.12024	+	4.18081i	0.184167	+	0.687321i	0.994807	+	0.101776i	\(0.0324526\pi\)
							−0.810640	+	0.585544i	\(0.800881\pi\)
\(41\)	−2.49300	−	0.439582i	−0.389340	−	0.0686512i	−0.0244485	−	0.999701i	\(-0.507783\pi\)
							−0.364892	+	0.931050i	\(0.618894\pi\)
\(43\)	0.0120144	−	0.137325i	0.00183218	−	0.0209419i	−0.995224	−	0.0976219i	\(-0.968876\pi\)
							0.997056	+	0.0766800i	\(0.0244320\pi\)
\(47\)	3.03824	+	1.10583i	0.443173	+	0.161302i	0.553961	−	0.832542i	\(-0.313115\pi\)
							−0.110789	+	0.993844i	\(0.535338\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.2.bg.a.13.5	✓	840
16.5	even	4	inner	432.2.bg.a.229.58	yes	840
27.25	even	9	inner	432.2.bg.a.349.58	yes	840
432.133	even	36	inner	432.2.bg.a.133.5	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.2.bg.a.13.5	✓	840	1.1	even	1	trivial
432.2.bg.a.133.5	yes	840	432.133	even	36	inner
432.2.bg.a.229.58	yes	840	16.5	even	4	inner
432.2.bg.a.349.58	yes	840	27.25	even	9	inner