Embedded newform 429.2.bn.b.400.1

• Label: 429.2.bn.b.400.1
• Level: $429$
• Weight: $2$
• Character: 429.400
• 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			400.1
Character	\(\chi\)	\(=\)	429.400
Dual form			429.2.bn.b.355.1

$q$-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 \(\left(\mathbb{Z}/429\mathbb{Z}\right)^\times\).

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

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.bn.b.4.1	✓	112	13.4	even	6	inner
429.2.bn.b.322.1	yes	112	11.3	even	5	inner
429.2.bn.b.355.1	yes	112	143.69	even	30	inner
429.2.bn.b.400.1	yes	112	1.1	even	1	trivial