Embedded newform 432.2.bj.a.155.22

• Label: 432.2.bj.a.155.22
• 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			155.22
Character	\(\chi\)	\(=\)	432.155
Dual form			432.2.bj.a.131.22

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.775984 - 1.18231i) q^{2} +(-0.162320 + 1.72443i) q^{3} +(-0.795697 + 1.83490i) q^{4} +(2.44275 + 1.71043i) q^{5} +(2.16476 - 1.14622i) q^{6} +(2.32820 + 1.95359i) q^{7} +(2.78687 - 0.483097i) q^{8} +(-2.94730 - 0.559820i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 24 q^{7} - 18 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.775984	−	1.18231i	−0.548704	−	0.836017i
							\(3\)	−0.162320	+	1.72443i	−0.0937157	+	0.995599i
\(5\)	2.44275	+	1.71043i	1.09243	+	0.764929i								0.973663	−	0.227993i	\(-0.0732162\pi\)
							0.118769	+	0.992922i	\(0.462105\pi\)
\(7\)	2.32820	+	1.95359i	0.879978	+	0.738389i	0.966175	−	0.257889i	\(-0.0830268\pi\)
							−0.0861964	+	0.996278i	\(0.527471\pi\)
\(11\)	−3.36801	+	2.35830i	−1.01549	+	0.711055i	−0.957846	−	0.287282i	\(-0.907248\pi\)
							−0.0576462	+	0.998337i	\(0.518360\pi\)
\(13\)	3.58238	−	1.67049i	0.993573	−	0.463311i	0.143315	−	0.989677i	\(-0.454224\pi\)
							0.850258	+	0.526367i	\(0.176446\pi\)
\(17\)	−1.29180	+	0.745824i	−0.313309	+	0.180889i	−0.648406	−	0.761295i	\(-0.724564\pi\)
							0.335097	+	0.942183i	\(0.391231\pi\)
\(19\)	−2.53524	+	0.679316i	−0.581624	+	0.155846i	−0.537622	−	0.843186i	\(-0.680677\pi\)
							−0.0440017	+	0.999031i	\(0.514011\pi\)
\(23\)	−4.76124	−	5.67422i	−0.992787	−	1.18316i	−0.983075	−	0.183204i	\(-0.941353\pi\)
							−0.00971201	−	0.999953i	\(-0.503091\pi\)
\(29\)	4.74511	+	2.21268i	0.881146	+	0.410885i	0.809896	−	0.586574i	\(-0.199524\pi\)
							0.0712497	+	0.997459i	\(0.477301\pi\)
\(31\)	6.48198	+	7.72493i	1.16420	+	1.38744i	0.907027	+	0.421072i	\(0.138346\pi\)
							0.257171	+	0.966366i	\(0.417210\pi\)
\(37\)	0.325155	−	1.21349i	0.0534551	−	0.199497i	−0.934034	−	0.357184i	\(-0.883737\pi\)
							0.987489	+	0.157687i	\(0.0504036\pi\)
\(41\)	−6.38780	−	2.32497i	−0.997607	−	0.363099i	−0.208946	−	0.977927i	\(-0.567003\pi\)
							−0.788661	+	0.614828i	\(0.789225\pi\)
\(43\)	3.88932	+	5.55453i	0.593116	+	0.847058i	0.997724	−	0.0674328i	\(-0.0214808\pi\)
							−0.404608	+	0.914490i	\(0.632592\pi\)
\(47\)	0.0868981	+	0.0729161i	0.0126754	+	0.0106359i	0.649103	−	0.760700i	\(-0.275144\pi\)
							−0.636428	+	0.771336i	\(0.719589\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.2.bj.a.155.22	yes	840
16.3	odd	4	inner	432.2.bj.a.371.33	yes	840
27.23	odd	18	inner	432.2.bj.a.347.33	yes	840
432.131	even	36	inner	432.2.bj.a.131.22	✓	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.2.bj.a.131.22	✓	840	432.131	even	36	inner
432.2.bj.a.155.22	yes	840	1.1	even	1	trivial
432.2.bj.a.347.33	yes	840	27.23	odd	18	inner
432.2.bj.a.371.33	yes	840	16.3	odd	4	inner