Embedded newform 432.2.bj.a.131.20

• Label: 432.2.bj.a.131.20
• 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			131.20
Character	\(\chi\)	\(=\)	432.131
Dual form			432.2.bj.a.155.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.845556 - 1.13359i) q^{2} +(-1.20988 + 1.23943i) q^{3} +(-0.570069 + 1.91703i) q^{4} +(-3.22870 + 2.26076i) q^{5} +(2.42803 + 0.323498i) q^{6} +(-3.12524 + 2.62238i) q^{7} +(2.65516 - 0.974735i) q^{8} +(-0.0723929 - 2.99913i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 24 q^{7} - 18 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.845556	−	1.13359i	−0.597899	−	0.801572i
							\(3\)	−1.20988	+	1.23943i	−0.698523	+	0.715588i
\(5\)	−3.22870	+	2.26076i	−1.44392	+	1.01104i								−0.450966	+	0.892541i	\(0.648920\pi\)
							−0.992954	+	0.118502i	\(0.962191\pi\)
\(7\)	−3.12524	+	2.62238i	−1.18123	+	0.991168i	−0.181258	+	0.983436i	\(0.558017\pi\)
							−0.999970	+	0.00773268i	\(0.997539\pi\)
\(11\)	−0.153662	−	0.107595i	−0.0463309	−	0.0324412i	0.550182	−	0.835045i	\(-0.314558\pi\)
							−0.596512	+	0.802604i	\(0.703447\pi\)
\(13\)	3.05063	+	1.42253i	0.846091	+	0.394539i	0.796798	−	0.604246i	\(-0.206525\pi\)
							0.0492930	+	0.998784i	\(0.484303\pi\)
\(17\)	−5.49442	−	3.17220i	−1.33259	−	0.769372i	−0.346896	−	0.937904i	\(-0.612764\pi\)
							−0.985696	+	0.168531i	\(0.946098\pi\)
\(19\)	3.06254	+	0.820604i	0.702594	+	0.188260i	0.592392	−	0.805650i	\(-0.298184\pi\)
							0.110202	+	0.993909i	\(0.464850\pi\)
\(23\)	1.26153	−	1.50343i	0.263047	−	0.313488i	−0.618313	−	0.785932i	\(-0.712184\pi\)
							0.881361	+	0.472444i	\(0.156628\pi\)
\(29\)	4.49799	−	2.09745i	0.835256	−	0.389486i	0.0425528	−	0.999094i	\(-0.486451\pi\)
							0.792703	+	0.609608i	\(0.208673\pi\)
\(31\)	−4.77543	+	5.69114i	−0.857693	+	1.02216i	0.141786	+	0.989897i	\(0.454715\pi\)
							−0.999479	+	0.0322617i	\(0.989729\pi\)
\(37\)	0.525680	+	1.96187i	0.0864213	+	0.322529i	0.995580	−	0.0939221i	\(-0.0299405\pi\)
							−0.909158	+	0.416451i	\(0.863274\pi\)
\(41\)	1.19789	−	0.435997i	0.187079	−	0.0680912i	−0.246782	−	0.969071i	\(-0.579373\pi\)
							0.433861	+	0.900980i	\(0.357151\pi\)
\(43\)	−0.0940153	+	0.134268i	−0.0143372	+	0.0204756i	−0.826257	−	0.563294i	\(-0.809534\pi\)
							0.811919	+	0.583769i	\(0.198423\pi\)
\(47\)	−5.37221	+	4.50782i	−0.783617	+	0.657533i	−0.944157	−	0.329497i	\(-0.893121\pi\)
							0.160540	+	0.987029i	\(0.448676\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.2.bj.a.131.20	✓	840
16.11	odd	4	inner	432.2.bj.a.347.66	yes	840
27.20	odd	18	inner	432.2.bj.a.371.66	yes	840
432.155	even	36	inner	432.2.bj.a.155.20	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.2.bj.a.131.20	✓	840	1.1	even	1	trivial
432.2.bj.a.155.20	yes	840	432.155	even	36	inner
432.2.bj.a.347.66	yes	840	16.11	odd	4	inner
432.2.bj.a.371.66	yes	840	27.20	odd	18	inner