Embedded newform 429.2.be.a.89.2

• Label: 429.2.be.a.89.2
• 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			89.2
Character	\(\chi\)	\(=\)	429.89
Dual form			429.2.be.a.188.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.48352 - 0.665458i) q^{2} +(-1.51130 - 0.846155i) q^{3} +(3.99300 + 2.30536i) q^{4} +(-0.0572884 + 0.0572884i) q^{5} +(3.19026 + 3.10715i) q^{6} +(-0.793036 - 2.95965i) q^{7} +(-4.74647 - 4.74647i) q^{8} +(1.56804 + 2.55758i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 12 q^{6} + 12 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.48352	−	0.665458i	−1.75612	−	0.470550i	−0.770202	−	0.637800i	\(-0.779845\pi\)
							−0.985914	+	0.167250i	\(0.946511\pi\)
\(3\)	−1.51130	−	0.846155i	−0.872548	−	0.488528i
							\(5\)	−0.0572884	+	0.0572884i	−0.0256201	+	0.0256201i	−0.719801	−	0.694181i	\(-0.755767\pi\)
0.694181	+	0.719801i	\(0.255767\pi\)
\(7\)	−0.793036	−	2.95965i	−0.299740	−	1.11864i	−0.937380	−	0.348309i	\(-0.886756\pi\)
							0.637640	−	0.770334i	\(-0.279911\pi\)
\(11\)	−0.258819	+	0.965926i	−0.0780369	+	0.291238i
							\(13\)	1.72949	+	3.16368i	0.479674	+	0.877447i
\(17\)	3.66470	−	6.34745i	0.888820	−	1.53948i								0.0475485	−	0.998869i	\(-0.484859\pi\)
							0.841272	−	0.540613i	\(-0.181808\pi\)
\(19\)	4.42309	−	1.18516i	1.01473	−	0.271895i	0.287124	−	0.957894i	\(-0.407301\pi\)
							0.727603	+	0.685998i	\(0.240634\pi\)
\(23\)	0.845925	+	1.46518i	0.176388	+	0.305512i	0.940641	−	0.339404i	\(-0.110225\pi\)
							−0.764253	+	0.644916i	\(0.776892\pi\)
\(29\)	3.95328	−	2.28243i	0.734105	−	0.423836i	−0.0858169	−	0.996311i	\(-0.527350\pi\)
							0.819922	+	0.572475i	\(0.194017\pi\)
\(31\)	−4.48676	−	4.48676i	−0.805846	−	0.805846i	0.178156	−	0.984002i	\(-0.442987\pi\)
							−0.984002	+	0.178156i	\(0.942987\pi\)
\(37\)	−5.09060	−	1.36402i	−0.836890	−	0.224244i	−0.185173	−	0.982706i	\(-0.559284\pi\)
							−0.651717	+	0.758462i	\(0.725951\pi\)
\(41\)	−6.92027	−	1.85428i	−1.08076	−	0.289590i	−0.325856	−	0.945419i	\(-0.605652\pi\)
							−0.754909	+	0.655829i	\(0.772319\pi\)
\(43\)	−4.74662	−	2.74046i	−0.723853	−	0.417917i	0.0923163	−	0.995730i	\(-0.470573\pi\)
							−0.816169	+	0.577813i	\(0.803906\pi\)
\(47\)	−7.99097	−	7.99097i	−1.16560	−	1.16560i	−0.983229	−	0.182374i	\(-0.941622\pi\)
							−0.182374	−	0.983229i	\(-0.558378\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.be.a.89.2	✓	184
3.2	odd	2	inner	429.2.be.a.89.45	yes	184
13.6	odd	12	inner	429.2.be.a.188.45	yes	184
39.32	even	12	inner	429.2.be.a.188.2	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.be.a.89.2	✓	184	1.1	even	1	trivial
429.2.be.a.89.45	yes	184	3.2	odd	2	inner
429.2.be.a.188.2	yes	184	39.32	even	12	inner
429.2.be.a.188.45	yes	184	13.6	odd	12	inner