Embedded newform 429.2.be.a.89.16

• Label: 429.2.be.a.89.16
• Level: $429$
• Weight: $2$
• Character: 429.89
• Analytic conductor: $3.426$
• Analytic rank: $0$
• Dimension: $184$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 429 = 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	429.be (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.42558224671\)
Analytic rank:	\(0\)
Dimension:	\(184\)
Relative dimension:	\(46\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			89.16
Character	\(\chi\)	\(=\)	429.89
Dual form			429.2.be.a.188.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.954637 - 0.255794i) q^{2} +(1.35903 - 1.07380i) q^{3} +(-0.886149 - 0.511619i) q^{4} +(-1.59904 + 1.59904i) q^{5} +(-1.57205 + 0.677454i) q^{6} +(0.743538 + 2.77492i) q^{7} +(2.11277 + 2.11277i) q^{8} +(0.693924 - 2.91864i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 12 q^{6} + 12 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(67\)	\(79\)	\(287\)
\(\chi(n)\)	\(e\left(\frac{7}{12}\right)\)	\(1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.954637	−	0.255794i	−0.675030	−	0.180874i	−0.0950107	−	0.995476i	\(-0.530289\pi\)
							−0.580020	+	0.814602i	\(0.696955\pi\)
\(3\)	1.35903	−	1.07380i	0.784636	−	0.619956i
							\(5\)	−1.59904	+	1.59904i	−0.715113	+	0.715113i	−0.967600	−	0.252487i	\(-0.918751\pi\)
0.252487	+	0.967600i	\(0.418751\pi\)
\(7\)	0.743538	+	2.77492i	0.281031	+	1.04882i	0.951691	+	0.307057i	\(0.0993442\pi\)
							−0.670660	+	0.741765i	\(0.733989\pi\)
\(11\)	0.258819	−	0.965926i	0.0780369	−	0.291238i
							\(13\)	3.55503	−	0.601447i	0.985989	−	0.166811i
\(17\)	−4.05092	+	7.01639i	−0.982491	+	1.70173i								−0.329898	+	0.944016i	\(0.607014\pi\)
							−0.652593	+	0.757709i	\(0.726319\pi\)
\(19\)	7.47126	−	2.00192i	1.71402	−	0.459271i	0.737619	−	0.675217i	\(-0.235950\pi\)
							0.976405	+	0.215946i	\(0.0692836\pi\)
\(23\)	1.16765	+	2.02243i	0.243473	+	0.421707i	0.961701	−	0.274100i	\(-0.0883801\pi\)
							−0.718229	+	0.695807i	\(0.755047\pi\)
\(29\)	4.91814	−	2.83949i	0.913276	−	0.527280i	0.0317924	−	0.999494i	\(-0.489878\pi\)
							0.881484	+	0.472214i	\(0.156545\pi\)
\(31\)	2.30878	+	2.30878i	0.414669	+	0.414669i	0.883362	−	0.468692i	\(-0.155275\pi\)
							−0.468692	+	0.883362i	\(0.655275\pi\)
\(37\)	4.35492	+	1.16690i	0.715944	+	0.191837i	0.598361	−	0.801226i	\(-0.295819\pi\)
							0.117583	+	0.993063i	\(0.462485\pi\)
\(41\)	3.97348	+	1.06469i	0.620553	+	0.166277i	0.555379	−	0.831597i	\(-0.312573\pi\)
							0.0651736	+	0.997874i	\(0.479240\pi\)
\(43\)	−7.54230	−	4.35455i	−1.15019	−	0.664063i	−0.201257	−	0.979538i	\(-0.564503\pi\)
							−0.948934	+	0.315476i	\(0.897836\pi\)
\(47\)	−3.89690	−	3.89690i	−0.568421	−	0.568421i	0.363265	−	0.931686i	\(-0.381662\pi\)
							−0.931686	+	0.363265i	\(0.881662\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.be.a.89.16	✓	184
3.2	odd	2	inner	429.2.be.a.89.31	yes	184
13.6	odd	12	inner	429.2.be.a.188.31	yes	184
39.32	even	12	inner	429.2.be.a.188.16	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.be.a.89.16	✓	184	1.1	even	1	trivial
429.2.be.a.89.31	yes	184	3.2	odd	2	inner
429.2.be.a.188.16	yes	184	39.32	even	12	inner
429.2.be.a.188.31	yes	184	13.6	odd	12	inner