Embedded newform 429.2.bn.b.4.1

• Label: 429.2.bn.b.4.1
• Level: $429$
• Weight: $2$
• Character: 429.4
• 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			4.1
Character	\(\chi\)	\(=\)	429.4
Dual form			429.2.bn.b.322.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.08659 - 2.44051i) q^{2} +(0.669131 + 0.743145i) q^{3} +(-3.43718 + 3.81737i) q^{4} +(-1.96001 - 2.69773i) q^{5} +(1.08659 - 2.44051i) q^{6} +(-2.63121 - 2.36915i) q^{7} +(7.96970 + 2.58951i) q^{8} +(-0.104528 + 0.994522i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q + 14 q^{3} - 8 q^{4} + 14 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.08659	−	2.44051i	−0.768333	−	1.72570i	−0.684252	−	0.729245i	\(-0.739871\pi\)
							−0.0840809	−	0.996459i	\(-0.526795\pi\)
\(3\)	0.669131	+	0.743145i	0.386323	+	0.429055i
							\(5\)	−1.96001	−	2.69773i	−0.876545	−	1.20646i	−0.977366	−	0.211556i	\(-0.932147\pi\)
0.100821	−	0.994905i	\(-0.467853\pi\)
\(7\)	−2.63121	−	2.36915i	−0.994502	−	0.895454i	1.77865e−5	−	1.00000i	\(-0.499994\pi\)
							−0.994520	+	0.104546i	\(0.966661\pi\)
\(11\)	3.02642	+	1.35676i	0.912499	+	0.409079i
							\(13\)	−3.50048	+	0.864065i	−0.970860	+	0.239649i
\(17\)	−1.42956	−	0.636483i	−0.346720	−	0.154370i								0.225986	−	0.974131i	\(-0.427440\pi\)
							−0.572706	+	0.819761i	\(0.694106\pi\)
\(19\)	0.347572	+	1.63520i	0.0797385	+	0.375140i	0.999864	−	0.0164831i	\(-0.00524696\pi\)
							−0.920126	+	0.391623i	\(0.871914\pi\)
\(23\)	−3.45629	+	5.98647i	−0.720687	+	1.24827i	0.240038	+	0.970763i	\(0.422840\pi\)
							−0.960725	+	0.277503i	\(0.910493\pi\)
\(29\)	3.34857	+	0.711761i	0.621814	+	0.132171i	0.508037	−	0.861335i	\(-0.330371\pi\)
							0.113778	+	0.993506i	\(0.463705\pi\)
\(31\)	−5.14025	+	7.07495i	−0.923216	+	1.27070i	0.0392317	+	0.999230i	\(0.487509\pi\)
							−0.962448	+	0.271468i	\(0.912491\pi\)
\(37\)	−0.581652	+	2.73646i	−0.0956230	+	0.449871i	0.904122	+	0.427275i	\(0.140526\pi\)
							−0.999745	+	0.0225958i	\(0.992807\pi\)
\(41\)	8.31460	−	7.48650i	1.29852	−	1.16919i	0.323712	−	0.946156i	\(-0.395069\pi\)
							0.974810	−	0.223039i	\(-0.0715977\pi\)
\(43\)	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	\(-0.590471\pi\)
							−0.791062	+	0.611736i	\(0.790471\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.bn.b.4.1	✓	112	1.1	even	1	trivial
429.2.bn.b.322.1	yes	112	143.36	even	30	inner
429.2.bn.b.355.1	yes	112	11.3	even	5	inner
429.2.bn.b.400.1	yes	112	13.10	even	6	inner