Embedded newform 429.2.be.a.89.7

• Label: 429.2.be.a.89.7
• 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			89.7
Character	\(\chi\)	\(=\)	429.89
Dual form			429.2.be.a.188.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.20878 - 0.591842i) q^{2} +(-1.22709 - 1.22240i) q^{3} +(2.79640 + 1.61450i) q^{4} +(-2.05423 + 2.05423i) q^{5} +(1.98690 + 3.42626i) q^{6} +(0.902198 + 3.36705i) q^{7} +(-1.98722 - 1.98722i) q^{8} +(0.0114791 + 2.99998i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 12 q^{6} + 12 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.20878	−	0.591842i	−1.56185	−	0.418495i	−0.628598	−	0.777730i	\(-0.716371\pi\)
							−0.933247	+	0.359235i	\(0.883038\pi\)
\(3\)	−1.22709	−	1.22240i	−0.708458	−	0.705753i
							\(5\)	−2.05423	+	2.05423i	−0.918680	+	0.918680i	−0.996933	−	0.0782537i	\(-0.975066\pi\)
0.0782537	+	0.996933i	\(0.475066\pi\)
\(7\)	0.902198	+	3.36705i	0.340999	+	1.27263i	0.897217	+	0.441589i	\(0.145585\pi\)
							−0.556218	+	0.831036i	\(0.687748\pi\)
\(11\)	0.258819	−	0.965926i	0.0780369	−	0.291238i
							\(13\)	−0.622186	−	3.55146i	−0.172563	−	0.984998i
\(17\)	0.120970	−	0.209526i	0.0293395	−	0.0508176i								−0.850983	−	0.525194i	\(-0.823993\pi\)
							0.880322	+	0.474376i	\(0.157326\pi\)
\(19\)	−4.02709	+	1.07905i	−0.923877	+	0.247552i	−0.689242	−	0.724532i	\(-0.742056\pi\)
							−0.234635	+	0.972084i	\(0.575389\pi\)
\(23\)	−1.09210	−	1.89156i	−0.227718	−	0.394418i	0.729414	−	0.684073i	\(-0.239793\pi\)
							−0.957131	+	0.289654i	\(0.906460\pi\)
\(29\)	−0.957468	+	0.552795i	−0.177797	+	0.102651i	−0.586257	−	0.810125i	\(-0.699399\pi\)
							0.408460	+	0.912776i	\(0.366066\pi\)
\(31\)	5.46680	+	5.46680i	0.981866	+	0.981866i	0.999838	−	0.0179726i	\(-0.00572115\pi\)
							−0.0179726	+	0.999838i	\(0.505721\pi\)
\(37\)	−7.70627	−	2.06489i	−1.26690	−	0.339466i	−0.438059	−	0.898946i	\(-0.644334\pi\)
							−0.828843	+	0.559481i	\(0.811001\pi\)
\(41\)	6.57908	+	1.76286i	1.02748	+	0.275313i	0.732916	−	0.680319i	\(-0.238159\pi\)
							0.294564	+	0.955632i	\(0.404825\pi\)
\(43\)	−7.24530	−	4.18308i	−1.10490	−	0.637913i	−0.167395	−	0.985890i	\(-0.553536\pi\)
							−0.937503	+	0.347977i	\(0.886869\pi\)
\(47\)	−8.02518	−	8.02518i	−1.17059	−	1.17059i	−0.982069	−	0.188524i	\(-0.939630\pi\)
							−0.188524	−	0.982069i	\(-0.560370\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.be.a.89.7	✓	184
3.2	odd	2	inner	429.2.be.a.89.40	yes	184
13.6	odd	12	inner	429.2.be.a.188.40	yes	184
39.32	even	12	inner	429.2.be.a.188.7	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.be.a.89.7	✓	184	1.1	even	1	trivial
429.2.be.a.89.40	yes	184	3.2	odd	2	inner
429.2.be.a.188.7	yes	184	39.32	even	12	inner
429.2.be.a.188.40	yes	184	13.6	odd	12	inner