Embedded newform 430.2.r.a.323.12

• Label: 430.2.r.a.323.12
• 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			323.12
Character	\(\chi\)	\(=\)	430.323
Dual form			430.2.r.a.217.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.330279 + 0.943883i) q^{2} +(-0.995949 + 2.84626i) q^{3} +(-0.781831 + 0.623490i) q^{4} +(1.90328 + 1.17368i) q^{5} -3.01548 q^{6} +(-2.21017 - 2.21017i) q^{7} +(-0.846724 - 0.532032i) q^{8} +(-4.76378 - 3.79899i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 8 q^{6}+O(q^{10}) \)

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
							\(3\)	−0.995949	+	2.84626i	−0.575011	+	1.64329i	0.177055	+	0.984201i	\(0.443343\pi\)
−0.752067	+	0.659087i	\(0.770943\pi\)
\(5\)	1.90328	+	1.17368i	0.851171	+	0.524888i
							\(7\)	−2.21017	−	2.21017i	−0.835367	−	0.835367i	0.152878	−	0.988245i	\(-0.451146\pi\)
−0.988245	+	0.152878i	\(0.951146\pi\)
\(11\)	−3.58685	+	4.49776i	−1.08148	+	1.35613i	−0.151520	+	0.988454i	\(0.548417\pi\)
							−0.929955	+	0.367673i	\(0.880155\pi\)
\(13\)	1.03804	−	1.65204i	0.287901	−	0.458192i	−0.671192	−	0.741284i	\(-0.734217\pi\)
							0.959093	+	0.283092i	\(0.0913601\pi\)
\(17\)	−1.28718	−	2.04854i	−0.312187	−	0.496843i	0.653368	−	0.757040i	\(-0.273355\pi\)
							−0.965556	+	0.260197i	\(0.916212\pi\)
\(19\)	4.37657	+	5.48805i	1.00405	+	1.25904i	0.965667	+	0.259782i	\(0.0836506\pi\)
							0.0383872	+	0.999263i	\(0.487778\pi\)
\(23\)	−3.11019	+	0.350435i	−0.648520	+	0.0730707i	−0.430096	−	0.902783i	\(-0.641520\pi\)
							−0.218424	+	0.975854i	\(0.570092\pi\)
\(29\)	−2.97291	−	1.43168i	−0.552056	−	0.265856i	0.136992	−	0.990572i	\(-0.456257\pi\)
							−0.689048	+	0.724716i	\(0.741971\pi\)
\(31\)	1.93165	+	0.930235i	0.346935	+	0.167075i	0.599233	−	0.800574i	\(-0.295472\pi\)
							−0.252298	+	0.967650i	\(0.581186\pi\)
\(37\)	8.41759	+	8.41759i	1.38384	+	1.38384i	0.837676	+	0.546168i	\(0.183914\pi\)
							0.546168	+	0.837676i	\(0.316086\pi\)
\(41\)	2.16252	+	1.04142i	0.337729	+	0.162642i	0.595059	−	0.803682i	\(-0.297129\pi\)
							−0.257330	+	0.966324i	\(0.582843\pi\)
\(43\)	−3.10053	−	5.77812i	−0.472826	−	0.881156i
							\(47\)	−5.92965	−	0.668111i	−0.864929	−	0.0974540i	−0.331659	−	0.943399i	\(-0.607608\pi\)
−0.533270	+	0.845945i	\(0.679037\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	430.2.r.a.323.12	yes	264
5.2	odd	4	inner	430.2.r.a.237.11	yes	264
43.2	odd	14	inner	430.2.r.a.303.11	yes	264
215.2	even	28	inner	430.2.r.a.217.12	✓	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
430.2.r.a.217.12	✓	264	215.2	even	28	inner
430.2.r.a.237.11	yes	264	5.2	odd	4	inner
430.2.r.a.303.11	yes	264	43.2	odd	14	inner
430.2.r.a.323.12	yes	264	1.1	even	1	trivial