Embedded newform 432.2.bg.a.13.18

• Label: 432.2.bg.a.13.18
• 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			13.18
Character	\(\chi\)	\(=\)	432.13
Dual form			432.2.bg.a.133.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.945342 - 1.05182i) q^{2} +(1.02940 + 1.39296i) q^{3} +(-0.212657 + 1.98866i) q^{4} +(1.13926 - 0.0996725i) q^{5} +(0.492003 - 2.39957i) q^{6} +(-1.44119 + 3.95964i) q^{7} +(2.29275 - 1.65629i) q^{8} +(-0.880655 + 2.86783i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 6 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.945342	−	1.05182i	−0.668458	−	0.743750i
							\(3\)	1.02940	+	1.39296i	0.594327	+	0.804224i
\(5\)	1.13926	−	0.0996725i	0.509493	−	0.0445749i								0.170489	−	0.985360i	\(-0.445465\pi\)
							0.339005	+	0.940785i	\(0.389910\pi\)
\(7\)	−1.44119	+	3.95964i	−0.544718	+	1.49660i	0.296032	+	0.955178i	\(0.404337\pi\)
							−0.840750	+	0.541423i	\(0.817886\pi\)
\(11\)	0.139825	−	1.59821i	0.0421589	−	0.481878i	−0.945540	−	0.325507i	\(-0.894465\pi\)
							0.987698	−	0.156371i	\(-0.0499795\pi\)
\(13\)	−3.57947	+	2.50637i	−0.992765	+	0.695142i	−0.952655	−	0.304053i	\(-0.901660\pi\)
							−0.0401105	+	0.999195i	\(0.512771\pi\)
\(17\)	−2.32852	+	4.03312i	−0.564749	+	0.978175i	0.432323	+	0.901719i	\(0.357694\pi\)
							−0.997073	+	0.0764562i	\(0.975639\pi\)
\(19\)	2.13211	−	7.95716i	0.489140	−	1.82550i	−0.0715044	−	0.997440i	\(-0.522780\pi\)
							0.560645	−	0.828056i	\(-0.310553\pi\)
\(23\)	0.618113	+	1.69825i	0.128886	+	0.354110i	0.987304	−	0.158839i	\(-0.0507751\pi\)
							−0.858419	+	0.512949i	\(0.828553\pi\)
\(29\)	−0.427503	+	0.610538i	−0.0793853	+	0.113374i	−0.856881	−	0.515514i	\(-0.827601\pi\)
							0.777496	+	0.628888i	\(0.216490\pi\)
\(31\)	5.93229	−	2.15918i	1.06547	−	0.387800i	0.250990	−	0.967990i	\(-0.419244\pi\)
							0.814481	+	0.580190i	\(0.197022\pi\)
\(37\)	2.26427	+	8.45037i	0.372244	+	1.38923i	0.857330	+	0.514766i	\(0.172121\pi\)
							−0.485087	+	0.874466i	\(0.661212\pi\)
\(41\)	0.412591	+	0.0727509i	0.0644359	+	0.0113618i	0.205773	−	0.978600i	\(-0.434029\pi\)
							−0.141337	+	0.989961i	\(0.545140\pi\)
\(43\)	−0.258934	+	2.95963i	−0.0394871	+	0.451340i	0.950554	+	0.310560i	\(0.100517\pi\)
							−0.990041	+	0.140780i	\(0.955039\pi\)
\(47\)	9.66711	+	3.51854i	1.41009	+	0.513232i	0.931157	−	0.364618i	\(-0.118800\pi\)
							0.478936	+	0.877850i	\(0.341023\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.2.bg.a.13.18	✓	840
16.5	even	4	inner	432.2.bg.a.229.44	yes	840
27.25	even	9	inner	432.2.bg.a.349.44	yes	840
432.133	even	36	inner	432.2.bg.a.133.18	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.2.bg.a.13.18	✓	840	1.1	even	1	trivial
432.2.bg.a.133.18	yes	840	432.133	even	36	inner
432.2.bg.a.229.44	yes	840	16.5	even	4	inner
432.2.bg.a.349.44	yes	840	27.25	even	9	inner