Embedded newform 432.2.bj.a.203.59

• Label: 432.2.bj.a.203.59
• 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			203.59
Character	\(\chi\)	\(=\)	432.203
Dual form			432.2.bj.a.83.59

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.21338 - 0.726440i) q^{2} +(0.00235717 - 1.73205i) q^{3} +(0.944570 - 1.76289i) q^{4} +(0.335014 - 0.0293099i) q^{5} +(-1.25537 - 2.10334i) q^{6} +(0.985784 + 0.358796i) q^{7} +(-0.134515 - 2.82523i) q^{8} +(-2.99999 - 0.00816548i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 24 q^{7} - 18 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.21338	−	0.726440i	0.857987	−	0.513671i
							\(3\)	0.00235717	−	1.73205i	0.00136091	−	0.999999i
\(5\)	0.335014	−	0.0293099i	0.149823	−	0.0131078i								−0.0119976	−	0.999928i	\(-0.503819\pi\)
							0.161820	+	0.986820i	\(0.448263\pi\)
\(7\)	0.985784	+	0.358796i	0.372591	+	0.135612i	0.521527	−	0.853235i	\(-0.325363\pi\)
							−0.148936	+	0.988847i	\(0.547585\pi\)
\(11\)	1.57076	+	0.137424i	0.473602	+	0.0414348i	0.321458	−	0.946924i	\(-0.395827\pi\)
							0.152143	+	0.988358i	\(0.451383\pi\)
\(13\)	−1.01861	−	1.45472i	−0.282511	−	0.403467i	0.652667	−	0.757645i	\(-0.273650\pi\)
							−0.935178	+	0.354177i	\(0.884761\pi\)
\(17\)	1.59110	+	0.918623i	0.385899	+	0.222799i	0.680382	−	0.732858i	\(-0.261814\pi\)
							−0.294483	+	0.955657i	\(0.595147\pi\)
\(19\)	0.0467046	−	0.174304i	0.0107148	−	0.0399881i	−0.960361	−	0.278758i	\(-0.910077\pi\)
							0.971076	+	0.238770i	\(0.0767441\pi\)
\(23\)	1.77712	+	4.88258i	0.370554	+	1.01809i	0.975148	+	0.221555i	\(0.0711132\pi\)
							−0.604594	+	0.796534i	\(0.706665\pi\)
\(29\)	0.00856336	−	0.0122297i	0.00159018	−	0.00227101i	−0.818356	−	0.574711i	\(-0.805114\pi\)
							0.819946	+	0.572440i	\(0.194003\pi\)
\(31\)	0.766291	+	2.10537i	0.137630	+	0.378135i	0.989291	−	0.145958i	\(-0.0466265\pi\)
							−0.851661	+	0.524093i	\(0.824404\pi\)
\(37\)	4.49702	−	1.20497i	0.739306	−	0.198097i	0.130537	−	0.991443i	\(-0.458330\pi\)
							0.608770	+	0.793347i	\(0.291663\pi\)
\(41\)	0.0884751	−	0.501767i	0.0138175	−	0.0783629i	−0.977119	−	0.212692i	\(-0.931777\pi\)
							0.990937	+	0.134329i	\(0.0428880\pi\)
\(43\)	−0.198417	+	2.26792i	−0.0302583	+	0.345854i	0.965942	+	0.258760i	\(0.0833138\pi\)
							−0.996200	+	0.0870946i	\(0.972242\pi\)
\(47\)	7.94170	+	2.89054i	1.15842	+	0.421629i	0.848531	−	0.529145i	\(-0.177487\pi\)
							0.309885	+	0.950774i	\(0.399709\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.2.bj.a.203.59	yes	840
16.3	odd	4	inner	432.2.bj.a.419.67	yes	840
27.2	odd	18	inner	432.2.bj.a.299.67	yes	840
432.83	even	36	inner	432.2.bj.a.83.59	✓	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.2.bj.a.83.59	✓	840	432.83	even	36	inner
432.2.bj.a.203.59	yes	840	1.1	even	1	trivial
432.2.bj.a.299.67	yes	840	27.2	odd	18	inner
432.2.bj.a.419.67	yes	840	16.3	odd	4	inner