Embedded newform 430.2.r.a.323.22

• Label: 430.2.r.a.323.22
• 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			323.22
Character	\(\chi\)	\(=\)	430.323
Dual form			430.2.r.a.217.22

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.330279 + 0.943883i) q^{2} +(1.10715 - 3.16404i) q^{3} +(-0.781831 + 0.623490i) q^{4} +(-0.0970149 + 2.23396i) q^{5} +3.35215 q^{6} +(-1.84292 - 1.84292i) q^{7} +(-0.846724 - 0.532032i) q^{8} +(-6.43989 - 5.13564i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 8 q^{6}+O(q^{10}) \)

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
							\(3\)	1.10715	−	3.16404i	0.639211	−	1.82676i	0.0815881	−	0.996666i	\(-0.474001\pi\)
0.557623	−	0.830094i	\(-0.311713\pi\)
\(5\)	−0.0970149	+	2.23396i	−0.0433864	+	0.999058i
							\(7\)	−1.84292	−	1.84292i	−0.696558	−	0.696558i	0.267109	−	0.963666i	\(-0.413932\pi\)
−0.963666	+	0.267109i	\(0.913932\pi\)
\(11\)	3.85587	−	4.83511i	1.16259	−	1.45784i	0.298566	−	0.954389i	\(-0.403492\pi\)
							0.864023	−	0.503452i	\(-0.167937\pi\)
\(13\)	1.19122	−	1.89582i	0.330386	−	0.525806i	−0.639682	−	0.768639i	\(-0.720934\pi\)
							0.970068	+	0.242834i	\(0.0780769\pi\)
\(17\)	−1.14082	−	1.81560i	−0.276689	−	0.440349i	0.679253	−	0.733905i	\(-0.262304\pi\)
							−0.955942	+	0.293556i	\(0.905161\pi\)
\(19\)	1.66667	+	2.08993i	0.382360	+	0.479464i	0.935350	−	0.353724i	\(-0.115085\pi\)
							−0.552990	+	0.833188i	\(0.686513\pi\)
\(23\)	0.580367	−	0.0653917i	0.121015	−	0.0136351i	−0.0512492	−	0.998686i	\(-0.516320\pi\)
							0.172264	+	0.985051i	\(0.444892\pi\)
\(29\)	6.93694	+	3.34065i	1.28816	+	0.620344i	0.947474	−	0.319834i	\(-0.103627\pi\)
							0.340684	+	0.940178i	\(0.389341\pi\)
\(31\)	5.92449	+	2.85308i	1.06407	+	0.512429i	0.882191	−	0.470892i	\(-0.156068\pi\)
							0.181878	+	0.983321i	\(0.441782\pi\)
\(37\)	1.29055	+	1.29055i	0.212165	+	0.212165i	0.805187	−	0.593022i	\(-0.202065\pi\)
							−0.593022	+	0.805187i	\(0.702065\pi\)
\(41\)	−4.52783	−	2.18049i	−0.707129	−	0.340535i	0.0455145	−	0.998964i	\(-0.485507\pi\)
							−0.752643	+	0.658428i	\(0.771222\pi\)
\(43\)	−2.66307	+	5.99233i	−0.406115	+	0.913822i
							\(47\)	−9.65636	−	1.08801i	−1.40852	−	0.158703i	−0.625272	−	0.780407i	\(-0.715012\pi\)
−0.783252	+	0.621704i	\(0.786441\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	430.2.r.a.323.22	yes	264
5.2	odd	4	inner	430.2.r.a.237.1	yes	264
43.2	odd	14	inner	430.2.r.a.303.1	yes	264
215.2	even	28	inner	430.2.r.a.217.22	✓	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
430.2.r.a.217.22	✓	264	215.2	even	28	inner
430.2.r.a.237.1	yes	264	5.2	odd	4	inner
430.2.r.a.303.1	yes	264	43.2	odd	14	inner
430.2.r.a.323.22	yes	264	1.1	even	1	trivial