Embedded newform 429.2.be.a.89.17

• Label: 429.2.be.a.89.17
• 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			89.17
Character	\(\chi\)	\(=\)	429.89
Dual form			429.2.be.a.188.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.810471 - 0.217165i) q^{2} +(-1.43084 - 0.976061i) q^{3} +(-1.12235 - 0.647988i) q^{4} +(1.96327 - 1.96327i) q^{5} +(0.947689 + 1.10180i) q^{6} +(1.13192 + 4.22438i) q^{7} +(1.95552 + 1.95552i) q^{8} +(1.09461 + 2.79318i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 12 q^{6} + 12 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.810471	−	0.217165i	−0.573090	−	0.153559i	−0.0393764	−	0.999224i	\(-0.512537\pi\)
							−0.533713	+	0.845666i	\(0.679204\pi\)
\(3\)	−1.43084	−	0.976061i	−0.826096	−	0.563529i
							\(5\)	1.96327	−	1.96327i	0.878003	−	0.878003i	−0.115325	−	0.993328i	\(-0.536791\pi\)
0.993328	+	0.115325i	\(0.0367909\pi\)
\(7\)	1.13192	+	4.22438i	0.427825	+	1.59666i	0.757676	+	0.652631i	\(0.226335\pi\)
							−0.329851	+	0.944033i	\(0.606998\pi\)
\(11\)	0.258819	−	0.965926i	0.0780369	−	0.291238i
							\(13\)	−2.05606	+	2.96186i	−0.570249	+	0.821472i
\(17\)	0.941202	−	1.63021i	0.228275	−	0.395384i								−0.729022	−	0.684490i	\(-0.760025\pi\)
							0.957297	+	0.289106i	\(0.0933582\pi\)
\(19\)	4.92521	−	1.31971i	1.12992	−	0.302762i	0.355029	−	0.934855i	\(-0.384471\pi\)
							0.774892	+	0.632094i	\(0.217804\pi\)
\(23\)	2.71310	+	4.69923i	0.565721	+	0.979857i	0.996982	+	0.0776299i	\(0.0247352\pi\)
							−0.431262	+	0.902227i	\(0.641931\pi\)
\(29\)	7.04358	−	4.06661i	1.30796	−	0.755151i	0.326204	−	0.945299i	\(-0.394230\pi\)
							0.981755	+	0.190149i	\(0.0608970\pi\)
\(31\)	0.898617	+	0.898617i	0.161396	+	0.161396i	0.783185	−	0.621789i	\(-0.213594\pi\)
							−0.621789	+	0.783185i	\(0.713594\pi\)
\(37\)	−9.68354	−	2.59470i	−1.59196	−	0.426565i	−0.649361	−	0.760481i	\(-0.724963\pi\)
							−0.942603	+	0.333915i	\(0.891630\pi\)
\(41\)	−4.41339	−	1.18256i	−0.689255	−	0.184685i	−0.102842	−	0.994698i	\(-0.532794\pi\)
							−0.586413	+	0.810012i	\(0.699460\pi\)
\(43\)	6.94722	+	4.01098i	1.05944	+	0.611668i	0.925277	−	0.379291i	\(-0.123832\pi\)
							0.134163	+	0.990959i	\(0.457165\pi\)
\(47\)	6.62414	+	6.62414i	0.966231	+	0.966231i	0.999448	−	0.0332175i	\(-0.0105754\pi\)
							−0.0332175	+	0.999448i	\(0.510575\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.be.a.89.17	✓	184
3.2	odd	2	inner	429.2.be.a.89.30	yes	184
13.6	odd	12	inner	429.2.be.a.188.30	yes	184
39.32	even	12	inner	429.2.be.a.188.17	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.be.a.89.17	✓	184	1.1	even	1	trivial
429.2.be.a.89.30	yes	184	3.2	odd	2	inner
429.2.be.a.188.17	yes	184	39.32	even	12	inner
429.2.be.a.188.30	yes	184	13.6	odd	12	inner