Embedded newform 429.2.bn.b.49.3

• Label: 429.2.bn.b.49.3
• Level: $429$
• Weight: $2$
• Character: 429.49
• 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			49.3
Character	\(\chi\)	\(=\)	429.49
Dual form			429.2.bn.b.394.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.60557 - 1.44566i) q^{2} +(-0.104528 + 0.994522i) q^{3} +(0.278861 + 2.65319i) q^{4} +(-0.655963 - 0.213135i) q^{5} +(1.60557 - 1.44566i) q^{6} +(0.714969 - 0.0751462i) q^{7} +(0.848055 - 1.16725i) q^{8} +(-0.978148 - 0.207912i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q + 14 q^{3} - 8 q^{4} + 14 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(67\)	\(79\)	\(287\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.60557	−	1.44566i	−1.13531	−	1.02224i	−0.999504	−	0.0315032i	\(-0.989971\pi\)
							−0.135807	−	0.990735i	\(-0.543363\pi\)
\(3\)	−0.104528	+	0.994522i	−0.0603495	+	0.574187i
							\(5\)	−0.655963	−	0.213135i	−0.293355	−	0.0953169i	0.158642	−	0.987336i	\(-0.449288\pi\)
−0.451997	+	0.892019i	\(0.649288\pi\)
\(7\)	0.714969	−	0.0751462i	0.270233	−	0.0284026i	0.0315571	−	0.999502i	\(-0.489953\pi\)
							0.238676	+	0.971099i	\(0.423287\pi\)
\(11\)	−2.01875	+	2.63147i	−0.608675	+	0.793420i
							\(13\)	0.399630	−	3.58334i	0.110837	−	0.993839i
\(17\)	−4.26105	−	4.73238i	−1.03346	−	1.14777i								−0.988873	−	0.148765i	\(-0.952470\pi\)
							−0.0445849	−	0.999006i	\(-0.514197\pi\)
\(19\)	1.10039	+	2.47152i	0.252447	+	0.567006i	0.994666	−	0.103152i	\(-0.0328929\pi\)
							−0.742218	+	0.670158i	\(0.766226\pi\)
\(23\)	−0.496171	−	0.859393i	−0.103459	−	0.179196i	0.809649	−	0.586915i	\(-0.199658\pi\)
							−0.913107	+	0.407719i	\(0.866324\pi\)
\(29\)	−8.32933	−	3.70846i	−1.54672	−	0.688643i	−0.556848	−	0.830615i	\(-0.687989\pi\)
							−0.989871	+	0.141971i	\(0.954656\pi\)
\(31\)	−8.34253	+	2.71065i	−1.49836	+	0.486848i	−0.939540	−	0.342439i	\(-0.888747\pi\)
							−0.558823	+	0.829287i	\(0.688747\pi\)
\(37\)	−2.23912	+	5.02915i	−0.368110	+	0.826788i	0.630605	+	0.776104i	\(0.282807\pi\)
							−0.998715	+	0.0506839i	\(0.983860\pi\)
\(41\)	11.0506	+	1.16147i	1.72582	+	0.181391i	0.914876	−	0.403734i	\(-0.132288\pi\)
							0.810943	+	0.585125i	\(0.198955\pi\)
\(43\)	2.08606	−	3.61316i	0.318121	−	0.551002i	−0.661975	−	0.749526i	\(-0.730282\pi\)
							0.980096	+	0.198524i	\(0.0636149\pi\)
\(47\)	−2.65171	+	3.64977i	−0.386792	+	0.532374i	−0.957368	−	0.288871i	\(-0.906720\pi\)
							0.570576	+	0.821245i	\(0.306720\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.bn.b.49.3	✓	112
11.9	even	5	inner	429.2.bn.b.361.12	yes	112
13.4	even	6	inner	429.2.bn.b.82.12	yes	112
143.108	even	30	inner	429.2.bn.b.394.3	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.bn.b.49.3	✓	112	1.1	even	1	trivial
429.2.bn.b.82.12	yes	112	13.4	even	6	inner
429.2.bn.b.361.12	yes	112	11.9	even	5	inner
429.2.bn.b.394.3	yes	112	143.108	even	30	inner