Embedded newform 432.2.bj.a.155.8

• Label: 432.2.bj.a.155.8
• Level: $432$
• Weight: $2$
• Character: 432.155
• 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			155.8
Character	\(\chi\)	\(=\)	432.155
Dual form			432.2.bj.a.131.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.33001 - 0.480714i) q^{2} +(-1.50171 + 0.863051i) q^{3} +(1.53783 + 1.27870i) q^{4} +(1.54953 + 1.08500i) q^{5} +(2.41217 - 0.425969i) q^{6} +(3.35737 + 2.81717i) q^{7} +(-1.43063 - 2.43994i) q^{8} +(1.51029 - 2.59211i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 24 q^{7} - 18 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(271\)	\(325\)	\(353\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{7}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.33001	−	0.480714i	−0.940456	−	0.339916i
							\(3\)	−1.50171	+	0.863051i	−0.867015	+	0.498283i
\(5\)	1.54953	+	1.08500i	0.692973	+	0.485225i								0.866226	−	0.499652i	\(-0.166539\pi\)
							−0.173253	+	0.984877i	\(0.555428\pi\)
\(7\)	3.35737	+	2.81717i	1.26897	+	1.06479i	0.994666	+	0.103147i	\(0.0328913\pi\)
							0.274302	+	0.961644i	\(0.411553\pi\)
\(11\)	3.87498	−	2.71329i	1.16835	−	0.818089i	0.182091	−	0.983282i	\(-0.441714\pi\)
							0.986261	+	0.165193i	\(0.0528247\pi\)
\(13\)	−5.67478	+	2.64619i	−1.57390	+	0.733922i	−0.996583	−	0.0825972i	\(-0.973679\pi\)
							−0.577318	+	0.816519i	\(0.695901\pi\)
\(17\)	1.62009	−	0.935357i	0.392928	−	0.226857i	−0.290500	−	0.956875i	\(-0.593822\pi\)
							0.683428	+	0.730018i	\(0.260488\pi\)
\(19\)	3.06478	−	0.821205i	0.703108	−	0.188397i	0.110486	−	0.993878i	\(-0.464759\pi\)
							0.592622	+	0.805480i	\(0.298093\pi\)
\(23\)	1.11622	+	1.33026i	0.232748	+	0.277379i	0.869759	−	0.493476i	\(-0.164274\pi\)
							−0.637011	+	0.770855i	\(0.719829\pi\)
\(29\)	−2.47785	−	1.15544i	−0.460126	−	0.214560i	0.178712	−	0.983901i	\(-0.442807\pi\)
							−0.638838	+	0.769341i	\(0.720585\pi\)
\(31\)	5.77789	+	6.88583i	1.03774	+	1.23673i	0.971030	+	0.238959i	\(0.0768060\pi\)
							0.0667108	+	0.997772i	\(0.478750\pi\)
\(37\)	0.743566	−	2.77503i	0.122242	−	0.456212i	−0.877485	−	0.479604i	\(-0.840780\pi\)
							0.999726	+	0.0233928i	\(0.00744683\pi\)
\(41\)	3.28003	+	1.19383i	0.512254	+	0.186445i	0.585197	−	0.810891i	\(-0.301017\pi\)
							−0.0729435	+	0.997336i	\(0.523239\pi\)
\(43\)	2.16510	+	3.09209i	0.330175	+	0.471539i	0.949576	−	0.313535i	\(-0.101513\pi\)
							−0.619401	+	0.785075i	\(0.712625\pi\)
\(47\)	−2.39354	−	2.00842i	−0.349134	−	0.292959i	0.451308	−	0.892368i	\(-0.350958\pi\)
							−0.800442	+	0.599410i	\(0.795402\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.2.bj.a.155.8	yes	840
16.3	odd	4	inner	432.2.bj.a.371.47	yes	840
27.23	odd	18	inner	432.2.bj.a.347.47	yes	840
432.131	even	36	inner	432.2.bj.a.131.8	✓	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.2.bj.a.131.8	✓	840	432.131	even	36	inner
432.2.bj.a.155.8	yes	840	1.1	even	1	trivial
432.2.bj.a.347.47	yes	840	27.23	odd	18	inner
432.2.bj.a.371.47	yes	840	16.3	odd	4	inner