Embedded newform 429.2.p.b.230.2

• Label: 429.2.p.b.230.2
• 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			230.2
Character	\(\chi\)	\(=\)	429.230
Dual form			429.2.p.b.263.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.38419 + 2.39749i) q^{2} +(1.55790 + 0.756943i) q^{3} +(-2.83198 - 4.90513i) q^{4} -1.85441i q^{5} +(-3.97119 + 2.68729i) q^{6} +(-1.32496 + 0.764967i) q^{7} +10.1432 q^{8} +(1.85408 + 2.35847i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 2 q^{3} - 60 q^{4} - 2 q^{9}+O(q^{10}) \)

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	\(0.732371\pi\)
\(3\)	1.55790	+	0.756943i	0.899451	+	0.437021i
							\(5\)		−	1.85441i		−	0.829316i	−0.909977	−	0.414658i	\(-0.863901\pi\)
0.909977	−	0.414658i	\(-0.136099\pi\)
\(7\)	−1.32496	+	0.764967i	−0.500788	+	0.289130i	−0.729039	−	0.684472i	\(-0.760033\pi\)
							0.228251	+	0.973602i	\(0.426699\pi\)
\(11\)	3.30382	−	0.291209i	0.996138	−	0.0878027i
							\(13\)	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	−0.311267	−	0.950322i	\(-0.600753\pi\)
							0.667370	+	0.744727i	\(0.267420\pi\)
\(23\)	4.53161	+	2.61633i	0.944906	+	0.545542i	0.891495	−	0.453031i	\(-0.149657\pi\)
							0.0534111	+	0.998573i	\(0.482991\pi\)
\(29\)	−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	0.118549	−	0.992948i	\(-0.462176\pi\)
							0.919193	+	0.393808i	\(0.128842\pi\)
\(47\)		−	4.95477i		−	0.722728i	−0.932425	−	0.361364i	\(-0.882311\pi\)
							0.932425	−	0.361364i	\(-0.117689\pi\)

(See \(a_n\) instead)

Twists

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

    

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