Embedded newform 432.2.bg.a.13.69

• Label: 432.2.bg.a.13.69
• 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			13.69
Character	\(\chi\)	\(=\)	432.13
Dual form			432.2.bg.a.133.69

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.41135 + 0.0898877i) q^{2} +(-0.146363 + 1.72586i) q^{3} +(1.98384 + 0.253727i) q^{4} +(-4.35765 + 0.381245i) q^{5} +(-0.361704 + 2.42264i) q^{6} +(-0.367292 + 1.00913i) q^{7} +(2.77709 + 0.536421i) q^{8} +(-2.95716 - 0.505204i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 6 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.41135	+	0.0898877i	0.997978	+	0.0635602i
							\(3\)	−0.146363	+	1.72586i	−0.0845029	+	0.996423i
\(5\)	−4.35765	+	0.381245i	−1.94880	+	0.170498i								−0.992531	−	0.121995i	\(-0.961071\pi\)
							−0.956268	+	0.292493i	\(0.905515\pi\)
\(7\)	−0.367292	+	1.00913i	−0.138823	+	0.381414i	−0.989549	−	0.144196i	\(-0.953941\pi\)
							0.850726	+	0.525610i	\(0.176163\pi\)
\(11\)	−0.291440	+	3.33118i	−0.0878726	+	1.00439i	0.816415	+	0.577465i	\(0.195958\pi\)
							−0.904288	+	0.426923i	\(0.859597\pi\)
\(13\)	−2.62109	+	1.83531i	−0.726960	+	0.509023i	−0.877497	−	0.479582i	\(-0.840788\pi\)
							0.150537	+	0.988604i	\(0.451900\pi\)
\(17\)	0.935818	−	1.62088i	0.226969	−	0.393122i	−0.729939	−	0.683512i	\(-0.760452\pi\)
							0.956908	+	0.290390i	\(0.0937850\pi\)
\(19\)	0.983165	−	3.66922i	0.225554	−	0.841777i	−0.756628	−	0.653845i	\(-0.773155\pi\)
							0.982182	−	0.187932i	\(-0.0601786\pi\)
\(23\)	2.88517	+	7.92694i	0.601599	+	1.65288i	0.748034	+	0.663660i	\(0.230998\pi\)
							−0.146435	+	0.989220i	\(0.546780\pi\)
\(29\)	−4.76495	+	6.80506i	−0.884830	+	1.26367i	0.0789256	+	0.996881i	\(0.474851\pi\)
							−0.963755	+	0.266787i	\(0.914038\pi\)
\(31\)	4.76626	−	1.73478i	0.856045	−	0.311575i	0.123543	−	0.992339i	\(-0.460574\pi\)
							0.732503	+	0.680764i	\(0.238352\pi\)
\(37\)	−0.510946	−	1.90688i	−0.0839991	−	0.313489i	0.911124	−	0.412133i	\(-0.135216\pi\)
							−0.995123	+	0.0986442i	\(0.968549\pi\)
\(41\)	2.11401	+	0.372757i	0.330153	+	0.0582148i	0.336268	−	0.941766i	\(-0.390835\pi\)
							−0.00611490	+	0.999981i	\(0.501946\pi\)
\(43\)	0.194870	−	2.22737i	0.0297174	−	0.339671i	−0.966752	−	0.255715i	\(-0.917689\pi\)
							0.996469	−	0.0839558i	\(-0.0267555\pi\)
\(47\)	−9.13687	−	3.32555i	−1.33275	−	0.485081i	−0.425228	−	0.905086i	\(-0.639806\pi\)
							−0.907522	+	0.420005i	\(0.862028\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.2.bg.a.13.69	✓	840
16.5	even	4	inner	432.2.bg.a.229.11	yes	840
27.25	even	9	inner	432.2.bg.a.349.11	yes	840
432.133	even	36	inner	432.2.bg.a.133.69	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.2.bg.a.13.69	✓	840	1.1	even	1	trivial
432.2.bg.a.133.69	yes	840	432.133	even	36	inner
432.2.bg.a.229.11	yes	840	16.5	even	4	inner
432.2.bg.a.349.11	yes	840	27.25	even	9	inner