Embedded newform 432.2.bj.a.131.70

• Label: 432.2.bj.a.131.70
• 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			131.70
Character	\(\chi\)	\(=\)	432.131
Dual form			432.2.bj.a.155.70

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.41409 + 0.0183407i) q^{2} +(1.46853 + 0.918372i) q^{3} +(1.99933 + 0.0518709i) q^{4} +(-1.31986 + 0.924175i) q^{5} +(2.05980 + 1.32560i) q^{6} +(0.654358 - 0.549071i) q^{7} +(2.82629 + 0.110019i) q^{8} +(1.31319 + 2.69732i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 24 q^{7} - 18 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.41409	+	0.0183407i	0.999916	+	0.0129688i
							\(3\)	1.46853	+	0.918372i	0.847859	+	0.530222i
\(5\)	−1.31986	+	0.924175i	−0.590259	+	0.413304i								−0.830185	−	0.557487i	\(-0.811765\pi\)
							0.239927	+	0.970791i	\(0.422877\pi\)
\(7\)	0.654358	−	0.549071i	0.247324	−	0.207530i	−0.510695	−	0.859762i	\(-0.670612\pi\)
							0.758019	+	0.652232i	\(0.226167\pi\)
\(11\)	0.656679	+	0.459812i	0.197996	+	0.138638i	0.668370	−	0.743829i	\(-0.266992\pi\)
							−0.470374	+	0.882467i	\(0.655881\pi\)
\(13\)	−5.34200	−	2.49101i	−1.48160	−	0.690883i	−0.497335	−	0.867559i	\(-0.665688\pi\)
							−0.984269	+	0.176676i	\(0.943466\pi\)
\(17\)	−0.0328561	−	0.0189695i	−0.00796878	−	0.00460078i	0.496010	−	0.868317i	\(-0.334798\pi\)
							−0.503979	+	0.863716i	\(0.668131\pi\)
\(19\)	−4.65206	−	1.24652i	−1.06726	−	0.285970i	−0.317891	−	0.948127i	\(-0.602974\pi\)
							−0.749365	+	0.662157i	\(0.769641\pi\)
\(23\)	5.92419	−	7.06017i	1.23528	−	1.47215i	0.405461	−	0.914112i	\(-0.367111\pi\)
							0.829817	−	0.558035i	\(-0.188445\pi\)
\(29\)	5.01111	−	2.33672i	0.930539	−	0.433918i	0.102592	−	0.994724i	\(-0.467286\pi\)
							0.827947	+	0.560806i	\(0.189509\pi\)
\(31\)	−2.98008	+	3.55152i	−0.535237	+	0.637871i	−0.964113	−	0.265493i	\(-0.914465\pi\)
							0.428876	+	0.903364i	\(0.358910\pi\)
\(37\)	−2.14212	−	7.99449i	−0.352162	−	1.31429i	−0.884019	−	0.467452i	\(-0.845172\pi\)
							0.531857	−	0.846834i	\(-0.321495\pi\)
\(41\)	2.16164	−	0.786771i	0.337591	−	0.122873i	−0.167662	−	0.985845i	\(-0.553622\pi\)
							0.505252	+	0.862972i	\(0.331399\pi\)
\(43\)	−0.666961	+	0.952519i	−0.101711	+	0.145258i	−0.866781	−	0.498689i	\(-0.833815\pi\)
							0.765071	+	0.643946i	\(0.222704\pi\)
\(47\)	−5.94208	+	4.98599i	−0.866741	+	0.727282i	−0.963409	−	0.268035i	\(-0.913626\pi\)
							0.0966684	+	0.995317i	\(0.469181\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.2.bj.a.131.70	✓	840
16.11	odd	4	inner	432.2.bj.a.347.16	yes	840
27.20	odd	18	inner	432.2.bj.a.371.16	yes	840
432.155	even	36	inner	432.2.bj.a.155.70	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.2.bj.a.131.70	✓	840	1.1	even	1	trivial
432.2.bj.a.155.70	yes	840	432.155	even	36	inner
432.2.bj.a.347.16	yes	840	16.11	odd	4	inner
432.2.bj.a.371.16	yes	840	27.20	odd	18	inner