Embedded newform 432.2.bg.a.13.46

• Label: 432.2.bg.a.13.46
• 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			13.46
Character	\(\chi\)	\(=\)	432.13
Dual form			432.2.bg.a.133.46

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.638843 - 1.26170i) q^{2} +(0.278303 - 1.70955i) q^{3} +(-1.18376 - 1.61205i) q^{4} +(2.36131 - 0.206588i) q^{5} +(-1.97914 - 1.44327i) q^{6} +(1.05303 - 2.89318i) q^{7} +(-2.79016 + 0.463699i) q^{8} +(-2.84510 - 0.951543i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 6 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.638843	−	1.26170i	0.451730	−	0.892155i
							\(3\)	0.278303	−	1.70955i	0.160678	−	0.987007i
\(5\)	2.36131	−	0.206588i	1.05601	−	0.0923890i								0.454099	−	0.890951i	\(-0.349961\pi\)
							0.601912	+	0.798562i	\(0.294406\pi\)
\(7\)	1.05303	−	2.89318i	0.398008	−	1.09352i	−0.565245	−	0.824923i	\(-0.691218\pi\)
							0.963253	−	0.268596i	\(-0.0865596\pi\)
\(11\)	−0.494555	+	5.65278i	−0.149114	+	1.70438i	0.441367	+	0.897327i	\(0.354494\pi\)
							−0.590481	+	0.807052i	\(0.701062\pi\)
\(13\)	2.53903	−	1.77785i	0.704200	−	0.493086i	−0.165787	−	0.986162i	\(-0.553016\pi\)
							0.869986	+	0.493076i	\(0.164127\pi\)
\(17\)	2.45462	−	4.25153i	0.595333	−	1.03115i	−0.398167	−	0.917313i	\(-0.630353\pi\)
							0.993500	−	0.113834i	\(-0.0363133\pi\)
\(19\)	−1.14095	+	4.25807i	−0.261751	+	0.976868i	0.702458	+	0.711725i	\(0.252086\pi\)
							−0.964209	+	0.265143i	\(0.914581\pi\)
\(23\)	1.86150	+	5.11444i	0.388150	+	1.06643i	0.967833	+	0.251592i	\(0.0809541\pi\)
							−0.579683	+	0.814842i	\(0.696824\pi\)
\(29\)	4.02805	−	5.75266i	0.747991	−	1.06824i	−0.247072	−	0.968997i	\(-0.579469\pi\)
							0.995063	−	0.0992445i	\(-0.0316426\pi\)
\(31\)	−8.66675	+	3.15444i	−1.55659	+	0.566554i	−0.969953	−	0.243293i	\(-0.921772\pi\)
							−0.586641	+	0.809847i	\(0.699550\pi\)
\(37\)	1.57988	+	5.89618i	0.259730	+	0.969325i	0.965398	+	0.260782i	\(0.0839805\pi\)
							−0.705668	+	0.708543i	\(0.749353\pi\)
\(41\)	−2.25004	−	0.396742i	−0.351397	−	0.0619607i	−0.00483607	−	0.999988i	\(-0.501539\pi\)
							−0.346561	+	0.938028i	\(0.612650\pi\)
\(43\)	0.772744	−	8.83251i	0.117842	−	1.34694i	−0.675135	−	0.737694i	\(-0.735915\pi\)
							0.792978	−	0.609251i	\(-0.208530\pi\)
\(47\)	3.18089	+	1.15775i	0.463981	+	0.168875i	0.563424	−	0.826168i	\(-0.309484\pi\)
							−0.0994428	+	0.995043i	\(0.531706\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.2.bg.a.13.46	✓	840
16.5	even	4	inner	432.2.bg.a.229.17	yes	840
27.25	even	9	inner	432.2.bg.a.349.17	yes	840
432.133	even	36	inner	432.2.bg.a.133.46	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.2.bg.a.13.46	✓	840	1.1	even	1	trivial
432.2.bg.a.133.46	yes	840	432.133	even	36	inner
432.2.bg.a.229.17	yes	840	16.5	even	4	inner
432.2.bg.a.349.17	yes	840	27.25	even	9	inner