Embedded newform 430.2.r.a.323.3

• Label: 430.2.r.a.323.3
• Level: $430$
• Weight: $2$
• Character: 430.323
• 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.r (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.43356728692\)
Analytic rank:	\(0\)
Dimension:	\(264\)
Relative dimension:	\(22\) over \(\Q(\zeta_{28})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{28}]$

Embedding invariants

Embedding label			323.3
Character	\(\chi\)	\(=\)	430.323
Dual form			430.2.r.a.217.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.330279 - 0.943883i) q^{2} +(-0.637714 + 1.82248i) q^{3} +(-0.781831 + 0.623490i) q^{4} +(1.50212 - 1.65639i) q^{5} +1.93083 q^{6} +(0.527834 + 0.527834i) q^{7} +(0.846724 + 0.532032i) q^{8} +(-0.569264 - 0.453973i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 8 q^{6}+O(q^{10}) \)

Character values

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

\(n\)	\(87\)	\(261\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{5}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.330279	−	0.943883i	−0.233543	−	0.667426i
							\(3\)	−0.637714	+	1.82248i	−0.368184	+	1.05221i	0.599572	+	0.800321i	\(0.295338\pi\)
−0.967756	+	0.251889i	\(0.918948\pi\)
\(5\)	1.50212	−	1.65639i	0.671768	−	0.740762i
							\(7\)	0.527834	+	0.527834i	0.199502	+	0.199502i	0.799787	−	0.600284i	\(-0.204946\pi\)
−0.600284	+	0.799787i	\(0.704946\pi\)
\(11\)	−0.835181	+	1.04728i	−0.251817	+	0.315768i	−0.891632	−	0.452760i	\(-0.850439\pi\)
							0.639816	+	0.768528i	\(0.279011\pi\)
\(13\)	1.68362	−	2.67947i	0.466952	−	0.743150i	−0.527362	−	0.849641i	\(-0.676819\pi\)
							0.994314	+	0.106491i	\(0.0339615\pi\)
\(17\)	1.36060	+	2.16539i	0.329994	+	0.525183i	0.969974	−	0.243209i	\(-0.0782000\pi\)
							−0.639979	+	0.768392i	\(0.721057\pi\)
\(19\)	2.35800	+	2.95683i	0.540961	+	0.678344i	0.974912	−	0.222593i	\(-0.0714520\pi\)
							−0.433950	+	0.900937i	\(0.642881\pi\)
\(23\)	−1.13354	+	0.127719i	−0.236359	+	0.0266313i	−0.229350	−	0.973344i	\(-0.573660\pi\)
							−0.00700988	+	0.999975i	\(0.502231\pi\)
\(29\)	8.80088	+	4.23828i	1.63428	+	0.787029i	0.999902	+	0.0140079i	\(0.00445901\pi\)
							0.634380	+	0.773021i	\(0.281255\pi\)
\(31\)	9.22949	+	4.44469i	1.65767	+	0.798290i	0.998946	+	0.0459014i	\(0.0146160\pi\)
							0.658720	+	0.752388i	\(0.271098\pi\)
\(37\)	−0.528686	−	0.528686i	−0.0869155	−	0.0869155i	0.662312	−	0.749228i	\(-0.269575\pi\)
							−0.749228	+	0.662312i	\(0.769575\pi\)
\(41\)	4.08730	+	1.96834i	0.638328	+	0.307403i	0.724912	−	0.688842i	\(-0.241881\pi\)
							−0.0865833	+	0.996245i	\(0.527595\pi\)
\(43\)	6.46583	−	1.09225i	0.986030	−	0.166566i
							\(47\)	−6.07878	−	0.684914i	−0.886681	−	0.0999050i	−0.343137	−	0.939285i	\(-0.611489\pi\)
−0.543544	+	0.839380i	\(0.682918\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	430.2.r.a.323.3	yes	264
5.2	odd	4	inner	430.2.r.a.237.20	yes	264
43.2	odd	14	inner	430.2.r.a.303.20	yes	264
215.2	even	28	inner	430.2.r.a.217.3	✓	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
430.2.r.a.217.3	✓	264	215.2	even	28	inner
430.2.r.a.237.20	yes	264	5.2	odd	4	inner
430.2.r.a.303.20	yes	264	43.2	odd	14	inner
430.2.r.a.323.3	yes	264	1.1	even	1	trivial