Embedded newform 432.2.bj.a.11.6

• Label: 432.2.bj.a.11.6
• 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			11.6
Character	\(\chi\)	\(=\)	432.11
Dual form			432.2.bj.a.275.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.36790 - 0.358938i) q^{2} +(0.748873 - 1.56179i) q^{3} +(1.74233 + 0.981986i) q^{4} +(1.61065 - 0.751058i) q^{5} +(-1.58497 + 1.86758i) q^{6} +(-0.149029 + 0.845186i) q^{7} +(-2.03087 - 1.96865i) q^{8} +(-1.87838 - 2.33917i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 24 q^{7} - 18 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.36790	−	0.358938i	−0.967255	−	0.253807i
							\(3\)	0.748873	−	1.56179i	0.432362	−	0.901700i
\(5\)	1.61065	−	0.751058i	0.720304	−	0.335883i								−0.0276647	−	0.999617i	\(-0.508807\pi\)
							0.747969	+	0.663734i	\(0.231029\pi\)
\(7\)	−0.149029	+	0.845186i	−0.0563277	+	0.319450i	−0.999932	−	0.0116239i	\(-0.996300\pi\)
							0.943605	+	0.331074i	\(0.107411\pi\)
\(11\)	−4.78579	−	2.23165i	−1.44297	−	0.672868i	−0.465715	−	0.884935i	\(-0.654203\pi\)
							−0.977255	+	0.212067i	\(0.931980\pi\)
\(13\)	0.238556	−	2.72670i	0.0661634	−	0.756251i	−0.888561	−	0.458759i	\(-0.848294\pi\)
							0.954724	−	0.297492i	\(-0.0961503\pi\)
\(17\)	3.02383	−	1.74581i	0.733387	−	0.423421i	−0.0862731	−	0.996272i	\(-0.527496\pi\)
							0.819660	+	0.572850i	\(0.194162\pi\)
\(19\)	1.63007	−	0.436777i	0.373964	−	0.100203i	−0.0669413	−	0.997757i	\(-0.521324\pi\)
							0.440906	+	0.897553i	\(0.354657\pi\)
\(23\)	1.92764	−	0.339896i	0.401942	−	0.0708732i	0.0309769	−	0.999520i	\(-0.490138\pi\)
							0.370965	+	0.928647i	\(0.379027\pi\)
\(29\)	−0.647540	−	7.40142i	−0.120245	−	1.37441i	−0.781340	−	0.624106i	\(-0.785463\pi\)
							0.661095	−	0.750303i	\(-0.270092\pi\)
\(31\)	3.15788	−	0.556820i	0.567173	−	0.100008i	0.117294	−	0.993097i	\(-0.462578\pi\)
							0.449879	+	0.893089i	\(0.351467\pi\)
\(37\)	1.12719	−	4.20673i	0.185309	−	0.691581i	−0.809256	−	0.587457i	\(-0.800129\pi\)
							0.994564	−	0.104125i	\(-0.0332041\pi\)
\(41\)	5.40616	−	4.53631i	0.844300	−	0.708452i	−0.114227	−	0.993455i	\(-0.536439\pi\)
							0.958527	+	0.285003i	\(0.0919946\pi\)
\(43\)	−3.82421	+	8.20104i	−0.583187	+	1.25065i	0.364428	+	0.931232i	\(0.381265\pi\)
							−0.947615	+	0.319416i	\(0.896513\pi\)
\(47\)	−1.04824	+	5.94484i	−0.152901	+	0.867144i	0.807779	+	0.589486i	\(0.200669\pi\)
							−0.960680	+	0.277659i	\(0.910442\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.2.bj.a.11.6	✓	840
16.3	odd	4	inner	432.2.bj.a.227.46	yes	840
27.5	odd	18	inner	432.2.bj.a.59.46	yes	840
432.275	even	36	inner	432.2.bj.a.275.6	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.2.bj.a.11.6	✓	840	1.1	even	1	trivial
432.2.bj.a.59.46	yes	840	27.5	odd	18	inner
432.2.bj.a.227.46	yes	840	16.3	odd	4	inner
432.2.bj.a.275.6	yes	840	432.275	even	36	inner