Embedded newform 432.2.bj.a.131.57

• Label: 432.2.bj.a.131.57
• Level: $432$
• Weight: $2$
• Character: 432.131
• 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.bj (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			131.57
Character	\(\chi\)	\(=\)	432.131
Dual form			432.2.bj.a.155.57

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.10523 - 0.882313i) q^{2} +(-0.669851 + 1.59728i) q^{3} +(0.443048 - 1.95031i) q^{4} +(1.11280 - 0.779190i) q^{5} +(0.668962 + 2.35637i) q^{6} +(1.74663 - 1.46559i) q^{7} +(-1.23112 - 2.54644i) q^{8} +(-2.10260 - 2.13988i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 24 q^{7} - 18 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/432\mathbb{Z}\right)^\times\).

\(n\)	\(271\)	\(325\)	\(353\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{11}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.10523	−	0.882313i	0.781513	−	0.623889i
							\(3\)	−0.669851	+	1.59728i	−0.386739	+	0.922189i
\(5\)	1.11280	−	0.779190i	0.497659	−	0.348464i								−0.297644	−	0.954677i	\(-0.596201\pi\)
							0.795303	+	0.606213i	\(0.207312\pi\)
\(7\)	1.74663	−	1.46559i	0.660163	−	0.553942i	−0.249973	−	0.968253i	\(-0.580422\pi\)
							0.910136	+	0.414311i	\(0.135977\pi\)
\(11\)	3.44197	+	2.41009i	1.03779	+	0.726670i	0.962718	−	0.270508i	\(-0.0871917\pi\)
							0.0750744	+	0.997178i	\(0.476081\pi\)
\(13\)	−1.47465	−	0.687641i	−0.408995	−	0.190717i	0.207222	−	0.978294i	\(-0.433558\pi\)
							−0.616217	+	0.787577i	\(0.711335\pi\)
\(17\)	−0.832348	−	0.480557i	−0.201874	−	0.116552i	0.395655	−	0.918399i	\(-0.370518\pi\)
							−0.597529	+	0.801847i	\(0.703851\pi\)
\(19\)	1.34131	+	0.359404i	0.307719	+	0.0824530i	0.409374	−	0.912367i	\(-0.365747\pi\)
							−0.101655	+	0.994820i	\(0.532414\pi\)
\(23\)	3.96838	−	4.72933i	0.827465	−	0.986134i	−0.172534	−	0.985003i	\(-0.555196\pi\)
							0.999999	−	0.00113078i	\(-0.000359937\pi\)
\(29\)	4.91004	−	2.28959i	0.911772	−	0.425166i	0.0906263	−	0.995885i	\(-0.471113\pi\)
							0.821146	+	0.570719i	\(0.193335\pi\)
\(31\)	0.875552	−	1.04344i	0.157254	−	0.187408i	−0.681665	−	0.731665i	\(-0.738744\pi\)
							0.838919	+	0.544257i	\(0.183188\pi\)
\(37\)	1.99664	+	7.45158i	0.328246	+	1.22503i	0.911008	+	0.412389i	\(0.135305\pi\)
							−0.582762	+	0.812643i	\(0.698028\pi\)
\(41\)	−6.64880	+	2.41996i	−1.03837	+	0.377935i	−0.804261	−	0.594276i	\(-0.797439\pi\)
							−0.234106	+	0.972211i	\(0.575216\pi\)
\(43\)	−2.03413	+	2.90504i	−0.310202	+	0.443015i	−0.943736	−	0.330700i	\(-0.892715\pi\)
							0.633533	+	0.773715i	\(0.281604\pi\)
\(47\)	−4.49846	+	3.77465i	−0.656168	+	0.550590i	−0.908935	−	0.416937i	\(-0.863104\pi\)
							0.252768	+	0.967527i	\(0.418659\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.2.bj.a.131.57	✓	840
16.11	odd	4	inner	432.2.bj.a.347.31	yes	840
27.20	odd	18	inner	432.2.bj.a.371.31	yes	840
432.155	even	36	inner	432.2.bj.a.155.57	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.2.bj.a.131.57	✓	840	1.1	even	1	trivial
432.2.bj.a.155.57	yes	840	432.155	even	36	inner
432.2.bj.a.347.31	yes	840	16.11	odd	4	inner
432.2.bj.a.371.31	yes	840	27.20	odd	18	inner