Embedded newform 432.2.bg.a.13.13

• Label: 432.2.bg.a.13.13
• 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			13.13
Character	\(\chi\)	\(=\)	432.13
Dual form			432.2.bg.a.133.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.19078 - 0.762913i) q^{2} +(-1.38465 + 1.04055i) q^{3} +(0.835928 + 1.81693i) q^{4} +(-2.45604 + 0.214875i) q^{5} +(2.44267 - 0.182707i) q^{6} +(-1.20765 + 3.31798i) q^{7} +(0.390748 - 2.80131i) q^{8} +(0.834499 - 2.88160i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 6 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.19078	−	0.762913i	−0.842011	−	0.539461i
							\(3\)	−1.38465	+	1.04055i	−0.799427	+	0.600764i
\(5\)	−2.45604	+	0.214875i	−1.09837	+	0.0960952i								−0.621902	−	0.783095i	\(-0.713640\pi\)
							−0.476471	+	0.879190i	\(0.658084\pi\)
\(7\)	−1.20765	+	3.31798i	−0.456447	+	1.25408i	0.471665	+	0.881778i	\(0.343653\pi\)
							−0.928112	+	0.372300i	\(0.878569\pi\)
\(11\)	−0.274422	+	3.13666i	−0.0827413	+	0.945737i	0.835441	+	0.549580i	\(0.185212\pi\)
							−0.918183	+	0.396157i	\(0.870343\pi\)
\(13\)	−1.00671	+	0.704909i	−0.279212	+	0.195507i	−0.704791	−	0.709415i	\(-0.748959\pi\)
							0.425579	+	0.904921i	\(0.360070\pi\)
\(17\)	1.68340	−	2.91574i	0.408285	−	0.707171i	−0.586413	−	0.810013i	\(-0.699460\pi\)
							0.994698	+	0.102842i	\(0.0327936\pi\)
\(19\)	−0.144120	+	0.537865i	−0.0330635	+	0.123395i	−0.980485	−	0.196593i	\(-0.937012\pi\)
							0.947422	+	0.319988i	\(0.103679\pi\)
\(23\)	−3.07183	−	8.43979i	−0.640521	−	1.75982i	−0.650083	−	0.759863i	\(-0.725266\pi\)
							0.00956233	−	0.999954i	\(-0.496956\pi\)
\(29\)	−0.674562	+	0.963374i	−0.125263	+	0.178894i	−0.876885	−	0.480700i	\(-0.840383\pi\)
							0.751622	+	0.659594i	\(0.229272\pi\)
\(31\)	6.02616	−	2.19334i	1.08233	−	0.393936i	0.261556	−	0.965188i	\(-0.415764\pi\)
							0.820775	+	0.571252i	\(0.193542\pi\)
\(37\)	−3.06597	−	11.4424i	−0.504043	−	1.88111i	−0.471942	−	0.881629i	\(-0.656447\pi\)
							−0.0321005	−	0.999485i	\(-0.510220\pi\)
\(41\)	−4.10878	−	0.724489i	−0.641684	−	0.113146i	−0.156668	−	0.987651i	\(-0.550075\pi\)
							−0.485016	+	0.874505i	\(0.661186\pi\)
\(43\)	−0.0769497	+	0.879539i	−0.0117347	+	0.134129i	−0.999802	−	0.0198938i	\(-0.993667\pi\)
							0.988067	+	0.154022i	\(0.0492228\pi\)
\(47\)	−5.32992	−	1.93993i	−0.777448	−	0.282968i	−0.0773400	−	0.997005i	\(-0.524643\pi\)
							−0.700108	+	0.714037i	\(0.746865\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.2.bg.a.13.13	✓	840
16.5	even	4	inner	432.2.bg.a.229.50	yes	840
27.25	even	9	inner	432.2.bg.a.349.50	yes	840
432.133	even	36	inner	432.2.bg.a.133.13	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.2.bg.a.13.13	✓	840	1.1	even	1	trivial
432.2.bg.a.133.13	yes	840	432.133	even	36	inner
432.2.bg.a.229.50	yes	840	16.5	even	4	inner
432.2.bg.a.349.50	yes	840	27.25	even	9	inner