Embedded newform 429.2.bn.a.4.4

• Label: 429.2.bn.a.4.4
• 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			4.4
Character	\(\chi\)	\(=\)	429.4
Dual form			429.2.bn.a.322.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.688193 - 1.54571i) q^{2} +(-0.669131 - 0.743145i) q^{3} +(-0.577339 + 0.641200i) q^{4} +(1.57124 + 2.16263i) q^{5} +(-0.688193 + 1.54571i) q^{6} +(1.25124 + 1.12663i) q^{7} +(-1.82992 - 0.594578i) q^{8} +(-0.104528 + 0.994522i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 14 q^{3} - 20 q^{4} + 14 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.688193	−	1.54571i	−0.486626	−	1.09298i	−0.975375	−	0.220555i	\(-0.929213\pi\)
							0.488749	−	0.872425i	\(-0.337454\pi\)
\(3\)	−0.669131	−	0.743145i	−0.386323	−	0.429055i
							\(5\)	1.57124	+	2.16263i	0.702682	+	0.967158i	0.999924	+	0.0123560i	\(0.00393313\pi\)
−0.297242	+	0.954802i	\(0.596067\pi\)
\(7\)	1.25124	+	1.12663i	0.472926	+	0.425824i	0.870816	−	0.491609i	\(-0.163591\pi\)
							−0.397890	+	0.917433i	\(0.630258\pi\)
\(11\)	0.0935446	+	3.31531i	0.0282048	+	0.999602i
							\(13\)	3.35166	+	1.32905i	0.929583	+	0.368612i
\(17\)	6.50219	+	2.89496i	1.57701	+	0.702132i								0.993903	−	0.110261i	\(-0.0351685\pi\)
							0.583111	+	0.812392i	\(0.301835\pi\)
\(19\)	−0.874531	−	4.11434i	−0.200631	−	0.943895i	−0.957076	−	0.289837i	\(-0.906399\pi\)
							0.756445	−	0.654057i	\(-0.226935\pi\)
\(23\)	−3.30144	+	5.71827i	−0.688398	+	1.19234i	0.283957	+	0.958837i	\(0.408353\pi\)
							−0.972356	+	0.233504i	\(0.924981\pi\)
\(29\)	−3.03590	−	0.645300i	−0.563752	−	0.119829i	−0.0827842	−	0.996567i	\(-0.526381\pi\)
							−0.480968	+	0.876738i	\(0.659715\pi\)
\(31\)	1.68673	−	2.32159i	0.302946	−	0.416969i	−0.630219	−	0.776417i	\(-0.717035\pi\)
							0.933165	+	0.359448i	\(0.117035\pi\)
\(37\)	0.769524	−	3.62033i	0.126509	−	0.595178i	−0.868525	−	0.495645i	\(-0.834932\pi\)
							0.995034	−	0.0995330i	\(-0.0317349\pi\)
\(41\)	5.10705	−	4.59841i	0.797587	−	0.718150i	−0.165831	−	0.986154i	\(-0.553031\pi\)
							0.963418	+	0.268004i	\(0.0863640\pi\)
\(43\)	0.646286	+	1.11940i	0.0985576	+	0.170707i	0.911088	−	0.412212i	\(-0.135244\pi\)
							−0.812530	+	0.582919i	\(0.801910\pi\)
\(47\)	−11.9215	−	3.87353i	−1.73893	−	0.565012i	−0.744236	−	0.667917i	\(-0.767186\pi\)
							−0.994691	+	0.102905i	\(0.967186\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.bn.a.4.4	✓	112
11.3	even	5	inner	429.2.bn.a.355.4	yes	112
13.10	even	6	inner	429.2.bn.a.400.4	yes	112
143.36	even	30	inner	429.2.bn.a.322.4	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.bn.a.4.4	✓	112	1.1	even	1	trivial
429.2.bn.a.322.4	yes	112	143.36	even	30	inner
429.2.bn.a.355.4	yes	112	11.3	even	5	inner
429.2.bn.a.400.4	yes	112	13.10	even	6	inner