Embedded newform 429.2.p.b.230.1

• Label: 429.2.p.b.230.1
• 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			230.1
Character	\(\chi\)	\(=\)	429.230
Dual form			429.2.p.b.263.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.38419 + 2.39749i) q^{2} +(-1.43448 - 0.970706i) q^{3} +(-2.83198 - 4.90513i) q^{4} +1.85441i q^{5} +(4.31285 - 2.09551i) q^{6} +(1.32496 - 0.764967i) q^{7} +10.1432 q^{8} +(1.11546 + 2.78491i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 2 q^{3} - 60 q^{4} - 2 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.38419	+	2.39749i	−0.978772	+	1.69528i	−0.311891	+	0.950118i	\(0.600962\pi\)
							−0.666881	+	0.745164i	\(0.732371\pi\)
\(3\)	−1.43448	−	0.970706i	−0.828197	−	0.560437i
							\(5\)			1.85441i			0.829316i	0.909977	+	0.414658i	\(0.136099\pi\)
−0.909977	+	0.414658i	\(0.863901\pi\)
\(7\)	1.32496	−	0.764967i	0.500788	−	0.289130i	−0.228251	−	0.973602i	\(-0.573301\pi\)
							0.729039	+	0.684472i	\(0.239967\pi\)
\(11\)	−1.39971	−	3.00679i	−0.422030	−	0.906582i
							\(13\)	−0.573902	+	3.55958i	−0.159172	+	0.987251i
\(17\)	2.42727	+	4.20416i	0.588700	+	1.01966i								0.994403	+	0.105653i	\(0.0336933\pi\)
							−0.405703	+	0.914005i	\(0.632973\pi\)
\(19\)	−1.55221	+	0.896172i	−0.356102	+	0.205596i	−0.667370	−	0.744727i	\(-0.732580\pi\)
							0.311267	+	0.950322i	\(0.399247\pi\)
\(23\)	−4.53161	−	2.61633i	−0.944906	−	0.545542i	−0.0534111	−	0.998573i	\(-0.517009\pi\)
							−0.891495	+	0.453031i	\(0.850343\pi\)
\(29\)	−1.94343	+	3.36612i	−0.360886	+	0.625074i	−0.988107	−	0.153768i	\(-0.950859\pi\)
							0.627221	+	0.778842i	\(0.284192\pi\)
\(31\)	−6.14424			−1.10354			−0.551769	−	0.833997i	\(-0.686047\pi\)
							−0.551769	+	0.833997i	\(0.686047\pi\)
\(37\)	2.45823	−	4.25777i	0.404130	−	0.699974i	−0.590090	−	0.807338i	\(-0.700908\pi\)
							0.994220	+	0.107364i	\(0.0342410\pi\)
\(41\)	−1.69933	+	2.94333i	−0.265391	+	0.459671i	−0.967666	−	0.252235i	\(-0.918835\pi\)
							0.702275	+	0.711906i	\(0.252168\pi\)
\(43\)	−6.80493	+	3.92883i	−1.03774	+	0.599141i	−0.919193	−	0.393808i	\(-0.871158\pi\)
							−0.118549	+	0.992948i	\(0.537824\pi\)
\(47\)			4.95477i			0.722728i	0.932425	+	0.361364i	\(0.117689\pi\)
							−0.932425	+	0.361364i	\(0.882311\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.p.b.230.1	✓	96
3.2	odd	2	inner	429.2.p.b.230.48	yes	96
11.10	odd	2	inner	429.2.p.b.230.47	yes	96
13.3	even	3	inner	429.2.p.b.263.2	yes	96
33.32	even	2	inner	429.2.p.b.230.2	yes	96
39.29	odd	6	inner	429.2.p.b.263.47	yes	96
143.120	odd	6	inner	429.2.p.b.263.48	yes	96
429.263	even	6	inner	429.2.p.b.263.1	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.p.b.230.1	✓	96	1.1	even	1	trivial
429.2.p.b.230.2	yes	96	33.32	even	2	inner
429.2.p.b.230.47	yes	96	11.10	odd	2	inner
429.2.p.b.230.48	yes	96	3.2	odd	2	inner
429.2.p.b.263.1	yes	96	429.263	even	6	inner
429.2.p.b.263.2	yes	96	13.3	even	3	inner
429.2.p.b.263.47	yes	96	39.29	odd	6	inner
429.2.p.b.263.48	yes	96	143.120	odd	6	inner