Embedded newform 432.2.bg.a.13.19

• Label: 432.2.bg.a.13.19
• Level: $432$
• Weight: $2$
• Character: 432.13
• 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.bg (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			13.19
Character	\(\chi\)	\(=\)	432.13
Dual form			432.2.bg.a.133.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.881475 - 1.10589i) q^{2} +(-0.338579 - 1.69864i) q^{3} +(-0.446003 + 1.94964i) q^{4} +(0.423518 - 0.0370531i) q^{5} +(-1.58006 + 1.87174i) q^{6} +(0.916962 - 2.51933i) q^{7} +(2.54923 - 1.22532i) q^{8} +(-2.77073 + 1.15024i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 6 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(271\)	\(325\)	\(353\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{4}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.881475	−	1.10589i	−0.623297	−	0.781985i
							\(3\)	−0.338579	−	1.69864i	−0.195479	−	0.980708i
\(5\)	0.423518	−	0.0370531i	0.189403	−	0.0165706i								0.00794058	−	0.999968i	\(-0.497472\pi\)
							0.181463	+	0.983398i	\(0.441917\pi\)
\(7\)	0.916962	−	2.51933i	0.346579	−	0.952218i	−0.636860	−	0.770979i	\(-0.719767\pi\)
							0.983439	−	0.181238i	\(-0.0580106\pi\)
\(11\)	0.274781	−	3.14076i	0.0828495	−	0.946974i	−0.835054	−	0.550169i	\(-0.814563\pi\)
							0.917903	−	0.396805i	\(-0.129881\pi\)
\(13\)	−1.55009	+	1.08538i	−0.429917	+	0.301031i	−0.768434	−	0.639929i	\(-0.778964\pi\)
							0.338517	+	0.940960i	\(0.390075\pi\)
\(17\)	1.61788	−	2.80224i	0.392392	−	0.679644i	−0.600372	−	0.799721i	\(-0.704981\pi\)
							0.992765	+	0.120077i	\(0.0383142\pi\)
\(19\)	−0.408402	+	1.52418i	−0.0936938	+	0.349670i	−0.996818	−	0.0797091i	\(-0.974601\pi\)
							0.903124	+	0.429379i	\(0.141268\pi\)
\(23\)	−2.45757	−	6.75212i	−0.512439	−	1.40791i	−0.878688	−	0.477397i	\(-0.841580\pi\)
							0.366249	−	0.930517i	\(-0.380642\pi\)
\(29\)	0.795476	−	1.13606i	0.147716	−	0.210961i	−0.738474	−	0.674282i	\(-0.764453\pi\)
							0.886191	+	0.463321i	\(0.153342\pi\)
\(31\)	−2.03530	+	0.740788i	−0.365550	+	0.133049i	−0.518263	−	0.855221i	\(-0.673421\pi\)
							0.152713	+	0.988271i	\(0.451199\pi\)
\(37\)	0.727455	+	2.71490i	0.119593	+	0.446326i	0.999589	−	0.0286535i	\(-0.00912195\pi\)
							−0.879997	+	0.474980i	\(0.842455\pi\)
\(41\)	−10.1355	−	1.78716i	−1.58290	−	0.279107i	−0.688112	−	0.725604i	\(-0.741560\pi\)
							−0.894785	+	0.446497i	\(0.852671\pi\)
\(43\)	−0.694914	+	7.94290i	−0.105973	+	1.21128i	0.737964	+	0.674840i	\(0.235787\pi\)
							−0.843938	+	0.536441i	\(0.819768\pi\)
\(47\)	6.32597	+	2.30246i	0.922737	+	0.335849i	0.759327	−	0.650709i	\(-0.225528\pi\)
							0.163410	+	0.986558i	\(0.447751\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.2.bg.a.13.19	✓	840
16.5	even	4	inner	432.2.bg.a.229.42	yes	840
27.25	even	9	inner	432.2.bg.a.349.42	yes	840
432.133	even	36	inner	432.2.bg.a.133.19	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.2.bg.a.13.19	✓	840	1.1	even	1	trivial
432.2.bg.a.133.19	yes	840	432.133	even	36	inner
432.2.bg.a.229.42	yes	840	16.5	even	4	inner
432.2.bg.a.349.42	yes	840	27.25	even	9	inner