Embedded newform 429.2.bn.b.355.1

• Label: 429.2.bn.b.355.1
• Level: $429$
• Weight: $2$
• Character: 429.355
• Analytic conductor: $3.426$
• Analytic rank: $0$
• Dimension: $112$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 429 = 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	429.bn (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.42558224671\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(14\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			355.1
Character	\(\chi\)	\(=\)	429.355
Dual form			429.2.bn.b.400.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.65684 + 0.279245i) q^{2} +(-0.978148 - 0.207912i) q^{3} +(5.02453 - 1.06800i) q^{4} +(1.96001 - 2.69773i) q^{5} +(2.65684 + 0.279245i) q^{6} +(0.736140 + 3.46327i) q^{7} +(-7.96970 + 2.58951i) q^{8} +(0.913545 + 0.406737i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q + 14 q^{3} - 8 q^{4} + 14 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(67\)	\(79\)	\(287\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.65684	+	0.279245i	−1.87867	+	0.197456i	−0.973671	−	0.227957i	\(-0.926795\pi\)
							−0.904999	+	0.425413i	\(0.860129\pi\)
\(3\)	−0.978148	−	0.207912i	−0.564734	−	0.120038i
							\(5\)	1.96001	−	2.69773i	0.876545	−	1.20646i	−0.100821	−	0.994905i	\(-0.532147\pi\)
0.977366	−	0.211556i	\(-0.0678530\pi\)
\(7\)	0.736140	+	3.46327i	0.278235	+	1.30899i	0.866034	+	0.499985i	\(0.166661\pi\)
							−0.587800	+	0.809007i	\(0.700006\pi\)
\(11\)	2.68820	+	1.94257i	0.810522	+	0.585708i
							\(13\)	−0.259934	+	3.59617i	−0.0720928	+	0.997398i
\(17\)	0.163572	−	1.55628i	0.0396720	−	0.377454i								−0.956615	−	0.291356i	\(-0.905894\pi\)
							0.996287	−	0.0860977i	\(-0.0274397\pi\)
\(19\)	−1.24234	−	1.11861i	−0.285012	−	0.256626i	0.514207	−	0.857666i	\(-0.328086\pi\)
							−0.799219	+	0.601040i	\(0.794753\pi\)
\(23\)	−3.45629	+	5.98647i	−0.720687	+	1.24827i	0.240038	+	0.970763i	\(0.422840\pi\)
							−0.960725	+	0.277503i	\(0.910493\pi\)
\(29\)	−2.29069	−	2.54407i	−0.425370	−	0.472422i	0.491920	−	0.870640i	\(-0.336295\pi\)
							−0.917291	+	0.398219i	\(0.869629\pi\)
\(31\)	5.14025	+	7.07495i	0.923216	+	1.27070i	0.962448	+	0.271468i	\(0.0875090\pi\)
							−0.0392317	+	0.999230i	\(0.512491\pi\)
\(37\)	2.07902	−	1.87195i	0.341788	−	0.307747i	−0.480304	−	0.877102i	\(-0.659474\pi\)
							0.822092	+	0.569355i	\(0.192807\pi\)
\(41\)	−2.32620	+	10.9439i	−0.363291	+	1.70915i	0.294248	+	0.955729i	\(0.404931\pi\)
							−0.657539	+	0.753421i	\(0.728402\pi\)
\(43\)	0.619072	+	1.07226i	0.0944076	+	0.163519i	0.909361	−	0.416007i	\(-0.136571\pi\)
							−0.814954	+	0.579526i	\(0.803238\pi\)
\(47\)	7.34566	−	2.38675i	1.07147	−	0.348143i	0.280413	−	0.959880i	\(-0.409529\pi\)
							0.791062	+	0.611736i	\(0.209529\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.bn.b.355.1	yes	112
11.4	even	5	inner	429.2.bn.b.4.1	✓	112
13.10	even	6	inner	429.2.bn.b.322.1	yes	112
143.114	even	30	inner	429.2.bn.b.400.1	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.bn.b.4.1	✓	112	11.4	even	5	inner
429.2.bn.b.322.1	yes	112	13.10	even	6	inner
429.2.bn.b.355.1	yes	112	1.1	even	1	trivial
429.2.bn.b.400.1	yes	112	143.114	even	30	inner