Embedded newform 429.2.p.b.230.9

• Label: 429.2.p.b.230.9
• Level: $429$
• Weight: $2$
• Character: 429.230
• Analytic conductor: $3.426$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 429 = 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	429.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.42558224671\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(48\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			230.9
Character	\(\chi\)	\(=\)	429.230
Dual form			429.2.p.b.263.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.00971 + 1.74887i) q^{2} +(-1.56002 + 0.752553i) q^{3} +(-1.03904 - 1.79966i) q^{4} +1.23164i q^{5} +(0.259053 - 3.48814i) q^{6} +(3.16404 - 1.82676i) q^{7} +0.157663 q^{8} +(1.86733 - 2.34800i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 2 q^{3} - 60 q^{4} - 2 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/429\mathbb{Z}\right)^\times\).

\(n\)	\(67\)	\(79\)	\(287\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00971	+	1.74887i	−0.713974	+	1.23664i	0.249380	+	0.968406i	\(0.419773\pi\)
							−0.963354	+	0.268234i	\(0.913560\pi\)
\(3\)	−1.56002	+	0.752553i	−0.900678	+	0.434487i
							\(5\)			1.23164i			0.550804i	0.961329	+	0.275402i	\(0.0888110\pi\)
−0.961329	+	0.275402i	\(0.911189\pi\)
\(7\)	3.16404	−	1.82676i	1.19590	−	0.690451i	0.236258	−	0.971690i	\(-0.424079\pi\)
							0.959638	+	0.281239i	\(0.0907455\pi\)
\(11\)	1.57963	−	2.91630i	0.476275	−	0.879296i
							\(13\)	−0.0284714	−	3.60544i	−0.00789654	−	0.999969i
\(17\)	−2.71093	−	4.69548i	−0.657498	−	1.13882i								−0.981261	−	0.192682i	\(-0.938281\pi\)
							0.323763	−	0.946138i	\(-0.395052\pi\)
\(19\)	2.97206	−	1.71592i	0.681836	−	0.393658i	−0.118710	−	0.992929i	\(-0.537876\pi\)
							0.800547	+	0.599271i	\(0.204543\pi\)
\(23\)	1.77660	+	1.02572i	0.370447	+	0.213877i	0.673654	−	0.739047i	\(-0.264724\pi\)
							−0.303207	+	0.952925i	\(0.598057\pi\)
\(29\)	−1.18572	+	2.05372i	−0.220182	+	0.381367i	−0.954863	−	0.297046i	\(-0.903998\pi\)
							0.734681	+	0.678413i	\(0.237332\pi\)
\(31\)	−9.52758			−1.71120			−0.855602	−	0.517635i	\(-0.826813\pi\)
							−0.855602	+	0.517635i	\(0.826813\pi\)
\(37\)	−0.866359	+	1.50058i	−0.142429	+	0.246694i	−0.928411	−	0.371556i	\(-0.878824\pi\)
							0.785982	+	0.618249i	\(0.212158\pi\)
\(41\)	3.06857	−	5.31492i	0.479230	−	0.830051i	−0.520486	−	0.853870i	\(-0.674249\pi\)
							0.999716	+	0.0238193i	\(0.00758262\pi\)
\(43\)	−6.10208	+	3.52304i	−0.930558	+	0.537258i	−0.886988	−	0.461793i	\(-0.847206\pi\)
							−0.0435699	+	0.999050i	\(0.513873\pi\)
\(47\)		−	10.3955i		−	1.51635i	−0.652053	−	0.758173i	\(-0.726092\pi\)
							0.652053	−	0.758173i	\(-0.273908\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.p.b.230.9	✓	96
3.2	odd	2	inner	429.2.p.b.230.40	yes	96
11.10	odd	2	inner	429.2.p.b.230.39	yes	96
13.3	even	3	inner	429.2.p.b.263.10	yes	96
33.32	even	2	inner	429.2.p.b.230.10	yes	96
39.29	odd	6	inner	429.2.p.b.263.39	yes	96
143.120	odd	6	inner	429.2.p.b.263.40	yes	96
429.263	even	6	inner	429.2.p.b.263.9	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.p.b.230.9	✓	96	1.1	even	1	trivial
429.2.p.b.230.10	yes	96	33.32	even	2	inner
429.2.p.b.230.39	yes	96	11.10	odd	2	inner
429.2.p.b.230.40	yes	96	3.2	odd	2	inner
429.2.p.b.263.9	yes	96	429.263	even	6	inner
429.2.p.b.263.10	yes	96	13.3	even	3	inner
429.2.p.b.263.39	yes	96	39.29	odd	6	inner
429.2.p.b.263.40	yes	96	143.120	odd	6	inner