Embedded newform 430.2.t.a.109.8

• Label: 430.2.t.a.109.8
• Level: $430$
• Weight: $2$
• Character: 430.109
• Analytic conductor: $3.434$
• Analytic rank: $0$
• Dimension: $264$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 430 = 2 \cdot 5 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	430.t (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.43356728692\)
Analytic rank:	\(0\)
Dimension:	\(264\)
Relative dimension:	\(22\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			109.8
Character	\(\chi\)	\(=\)	430.109
Dual form			430.2.t.a.359.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.781831 + 0.623490i) q^{2} +(1.31281 - 0.515238i) q^{3} +(0.222521 - 0.974928i) q^{4} +(-2.23600 - 0.0174191i) q^{5} +(-0.705147 + 1.22135i) q^{6} +(-2.14802 + 1.24016i) q^{7} +(0.433884 + 0.900969i) q^{8} +(-0.741167 + 0.687703i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q + 44 q^{4} + 4 q^{5} - 2 q^{6} - 6 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/430\mathbb{Z}\right)^\times\).

\(n\)	\(87\)	\(261\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{8}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.781831	+	0.623490i	−0.552838	+	0.440874i
							\(3\)	1.31281	−	0.515238i	0.757949	−	0.297473i	0.0452806	−	0.998974i	\(-0.485582\pi\)
0.712668	+	0.701501i	\(0.247487\pi\)
\(5\)	−2.23600	−	0.0174191i	−0.999970	−	0.00779008i
							\(7\)	−2.14802	+	1.24016i	−0.811875	+	0.468737i	−0.847607	−	0.530625i	\(-0.821957\pi\)
0.0357313	+	0.999361i	\(0.488624\pi\)
\(11\)	0.259836	+	1.13842i	0.0783436	+	0.343246i	0.998875	−	0.0474230i	\(-0.0151009\pi\)
							−0.920531	+	0.390669i	\(0.872244\pi\)
\(13\)	−0.426395	+	0.625407i	−0.118261	+	0.173457i	−0.880805	−	0.473480i	\(-0.842998\pi\)
							0.762544	+	0.646936i	\(0.223950\pi\)
\(17\)	−6.18633	+	0.463602i	−1.50041	+	0.112440i	−0.799436	−	0.600751i	\(-0.794868\pi\)
							−0.700970	+	0.713191i	\(0.747249\pi\)
\(19\)	−0.738105	−	0.684861i	−0.169333	−	0.157118i	0.590954	−	0.806705i	\(-0.298751\pi\)
							−0.760287	+	0.649587i	\(0.774942\pi\)
\(23\)	−1.05169	+	3.40948i	−0.219291	+	0.710926i	0.777352	+	0.629065i	\(0.216562\pi\)
							−0.996644	+	0.0818601i	\(0.973914\pi\)
\(29\)	−0.858999	+	2.18869i	−0.159512	+	0.406430i	−0.988178	−	0.153314i	\(-0.951005\pi\)
							0.828665	+	0.559744i	\(0.189101\pi\)
\(31\)	−1.05658	+	0.159254i	−0.189768	+	0.0286029i	−0.243239	−	0.969967i	\(-0.578210\pi\)
							0.0534705	+	0.998569i	\(0.482972\pi\)
\(37\)	−2.11817	−	1.22293i	−0.348225	−	0.201048i	0.315678	−	0.948866i	\(-0.397768\pi\)
							−0.663903	+	0.747818i	\(0.731101\pi\)
\(41\)	−0.468527	−	0.587515i	−0.0731717	−	0.0917544i	0.743898	−	0.668293i	\(-0.232975\pi\)
							−0.817070	+	0.576538i	\(0.804403\pi\)
\(43\)	6.53897	+	0.491744i	0.997184	+	0.0749902i
							\(47\)	−2.11599	−	0.482962i	−0.308649	−	0.0704472i	0.0653907	−	0.997860i	\(-0.479171\pi\)
−0.374040	+	0.927413i	\(0.622028\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	430.2.t.a.109.8	✓	264
5.4	even	2	inner	430.2.t.a.109.15	yes	264
43.15	even	21	inner	430.2.t.a.359.15	yes	264
215.144	even	42	inner	430.2.t.a.359.8	yes	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
430.2.t.a.109.8	✓	264	1.1	even	1	trivial
430.2.t.a.109.15	yes	264	5.4	even	2	inner
430.2.t.a.359.8	yes	264	215.144	even	42	inner
430.2.t.a.359.15	yes	264	43.15	even	21	inner