Embedded newform 432.2.bj.a.203.10

• Label: 432.2.bj.a.203.10
• Level: $432$
• Weight: $2$
• Character: 432.203
• 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			203.10
Character	\(\chi\)	\(=\)	432.203
Dual form			432.2.bj.a.83.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.29831 - 0.560707i) q^{2} +(0.335030 + 1.69934i) q^{3} +(1.37121 + 1.45594i) q^{4} +(-4.04506 + 0.353897i) q^{5} +(0.517859 - 2.39412i) q^{6} +(-0.174052 - 0.0633499i) q^{7} +(-0.963904 - 2.65911i) q^{8} +(-2.77551 + 1.13866i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 24 q^{7} - 18 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(271\)	\(325\)	\(353\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{17}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.29831	−	0.560707i	−0.918043	−	0.396480i
							\(3\)	0.335030	+	1.69934i	0.193430	+	0.981114i
\(5\)	−4.04506	+	0.353897i	−1.80901	+	0.158268i								−0.941027	−	0.338331i	\(-0.890138\pi\)
							−0.867980	+	0.496599i	\(0.834582\pi\)
\(7\)	−0.174052	−	0.0633499i	−0.0657856	−	0.0239440i	0.308918	−	0.951089i	\(-0.400033\pi\)
							−0.374703	+	0.927145i	\(0.622255\pi\)
\(11\)	3.97935	+	0.348148i	1.19982	+	0.104970i	0.669489	−	0.742822i	\(-0.266513\pi\)
							0.530329	+	0.847792i	\(0.322069\pi\)
\(13\)	−0.940926	−	1.34378i	−0.260966	−	0.372698i	0.667191	−	0.744886i	\(-0.267496\pi\)
							−0.928157	+	0.372189i	\(0.878607\pi\)
\(17\)	−6.25381	−	3.61064i	−1.51677	−	0.875708i	−0.999806	−	0.0197023i	\(-0.993728\pi\)
							−0.516966	−	0.856006i	\(-0.672939\pi\)
\(19\)	−0.550494	+	2.05447i	−0.126292	+	0.471328i	−0.999882	−	0.0153337i	\(-0.995119\pi\)
							0.873590	+	0.486662i	\(0.161786\pi\)
\(23\)	−0.510911	−	1.40372i	−0.106532	−	0.292695i	0.874960	−	0.484195i	\(-0.160887\pi\)
							−0.981493	+	0.191499i	\(0.938665\pi\)
\(29\)	3.64469	−	5.20515i	0.676802	−	0.966573i	−0.322995	−	0.946401i	\(-0.604690\pi\)
							0.999797	−	0.0201721i	\(-0.00642143\pi\)
\(31\)	−3.40417	−	9.35288i	−0.611407	−	1.67983i	−0.727091	−	0.686541i	\(-0.759128\pi\)
							0.115684	−	0.993286i	\(-0.463094\pi\)
\(37\)	2.30620	−	0.617944i	0.379137	−	0.101589i	−0.0642175	−	0.997936i	\(-0.520455\pi\)
							0.443354	+	0.896347i	\(0.353788\pi\)
\(41\)	−0.875002	+	4.96238i	−0.136652	+	0.774994i	0.837043	+	0.547138i	\(0.184283\pi\)
							−0.973695	+	0.227856i	\(0.926828\pi\)
\(43\)	0.345490	−	3.94897i	0.0526867	−	0.602212i	−0.923256	−	0.384185i	\(-0.874482\pi\)
							0.975943	−	0.218027i	\(-0.0699620\pi\)
\(47\)	−7.00088	−	2.54811i	−1.02118	−	0.371680i	−0.223466	−	0.974712i	\(-0.571737\pi\)
							−0.797718	+	0.603031i	\(0.793959\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.2.bj.a.203.10	yes	840
16.3	odd	4	inner	432.2.bj.a.419.17	yes	840
27.2	odd	18	inner	432.2.bj.a.299.17	yes	840
432.83	even	36	inner	432.2.bj.a.83.10	✓	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.2.bj.a.83.10	✓	840	432.83	even	36	inner
432.2.bj.a.203.10	yes	840	1.1	even	1	trivial
432.2.bj.a.299.17	yes	840	27.2	odd	18	inner
432.2.bj.a.419.17	yes	840	16.3	odd	4	inner