Embedded newform 432.2.bj.a.11.3

• Label: 432.2.bj.a.11.3
• Level: $432$
• Weight: $2$
• Character: 432.11
• 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			11.3
Character	\(\chi\)	\(=\)	432.11
Dual form			432.2.bj.a.275.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41099 - 0.0954482i) q^{2} +(1.71137 - 0.266838i) q^{3} +(1.98178 + 0.269353i) q^{4} +(-2.91453 + 1.35907i) q^{5} +(-2.44020 + 0.213158i) q^{6} +(-0.433865 + 2.46057i) q^{7} +(-2.77056 - 0.569211i) q^{8} +(2.85760 - 0.913318i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 24 q^{7} - 18 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(271\)	\(325\)	\(353\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{13}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41099	−	0.0954482i	−0.997720	−	0.0674920i
							\(3\)	1.71137	−	0.266838i	0.988062	−	0.154059i
\(5\)	−2.91453	+	1.35907i	−1.30342	+	0.607794i								−0.945397	−	0.325920i	\(-0.894326\pi\)
							−0.358020	+	0.933714i	\(0.616548\pi\)
\(7\)	−0.433865	+	2.46057i	−0.163985	+	0.930008i	0.786119	+	0.618075i	\(0.212087\pi\)
							−0.950104	+	0.311932i	\(0.899024\pi\)
\(11\)	−1.40954	−	0.657278i	−0.424992	−	0.198177i	0.198342	−	0.980133i	\(-0.436444\pi\)
							−0.623334	+	0.781956i	\(0.714222\pi\)
\(13\)	−0.212049	+	2.42373i	−0.0588119	+	0.672223i	0.908375	+	0.418156i	\(0.137324\pi\)
							−0.967187	+	0.254066i	\(0.918232\pi\)
\(17\)	−4.56592	+	2.63614i	−1.10740	+	0.639357i	−0.938154	−	0.346218i	\(-0.887466\pi\)
							−0.169244	+	0.985574i	\(0.554133\pi\)
\(19\)	−7.24585	+	1.94152i	−1.66231	+	0.445415i	−0.963022	−	0.269423i	\(-0.913167\pi\)
							−0.699290	+	0.714838i	\(0.746501\pi\)
\(23\)	4.21822	−	0.743785i	0.879559	−	0.155090i	0.284407	−	0.958704i	\(-0.408203\pi\)
							0.595151	+	0.803614i	\(0.297092\pi\)
\(29\)	0.470260	+	5.37510i	0.0873251	+	0.998130i	0.905833	+	0.423635i	\(0.139246\pi\)
							−0.818508	+	0.574495i	\(0.805198\pi\)
\(31\)	−1.76621	+	0.311430i	−0.317220	+	0.0559345i	−0.329991	−	0.943984i	\(-0.607046\pi\)
							0.0127711	+	0.999918i	\(0.495935\pi\)
\(37\)	−0.774072	+	2.88887i	−0.127257	+	0.474928i	−0.999910	−	0.0134119i	\(-0.995731\pi\)
							0.872653	+	0.488340i	\(0.162397\pi\)
\(41\)	6.74425	−	5.65909i	1.05327	−	0.883802i	0.0598401	−	0.998208i	\(-0.480941\pi\)
							0.993434	+	0.114406i	\(0.0364965\pi\)
\(43\)	1.33444	−	2.86171i	0.203500	−	0.436407i	−0.777898	−	0.628390i	\(-0.783714\pi\)
							0.981398	+	0.191983i	\(0.0614919\pi\)
\(47\)	−2.05060	+	11.6296i	−0.299111	+	1.69635i	0.350894	+	0.936415i	\(0.385878\pi\)
							−0.650005	+	0.759930i	\(0.725233\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.2.bj.a.11.3	✓	840
16.3	odd	4	inner	432.2.bj.a.227.40	yes	840
27.5	odd	18	inner	432.2.bj.a.59.40	yes	840
432.275	even	36	inner	432.2.bj.a.275.3	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.2.bj.a.11.3	✓	840	1.1	even	1	trivial
432.2.bj.a.59.40	yes	840	27.5	odd	18	inner
432.2.bj.a.227.40	yes	840	16.3	odd	4	inner
432.2.bj.a.275.3	yes	840	432.275	even	36	inner