Embedded newform 430.2.x.a.3.9

• Label: 430.2.x.a.3.9
• Level: $430$
• Weight: $2$
• Character: 430.3
• Analytic conductor: $3.434$
• Analytic rank: $0$
• Dimension: $528$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 430 = 2 \cdot 5 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	430.x (of order \(84\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.43356728692\)
Analytic rank:	\(0\)
Dimension:	\(528\)
Relative dimension:	\(22\) over \(\Q(\zeta_{84})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{84}]$

Embedding invariants

Embedding label			3.9
Character	\(\chi\)	\(=\)	430.3
Dual form			430.2.x.a.287.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.330279 - 0.943883i) q^{2} +(1.23231 + 1.43197i) q^{3} +(-0.781831 + 0.623490i) q^{4} +(1.04170 + 1.97860i) q^{5} +(0.944603 - 1.63610i) q^{6} +(1.31062 - 4.89129i) q^{7} +(0.846724 + 0.532032i) q^{8} +(-0.0848203 + 0.562746i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 528 q - 4 q^{6} - 12 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(87\)	\(261\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{1}{42}\right)\)

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
							\(3\)	1.23231	+	1.43197i	0.711472	+	0.826746i	0.991066	−	0.133373i	\(-0.0425808\pi\)
−0.279594	+	0.960118i	\(0.590200\pi\)
\(5\)	1.04170	+	1.97860i	0.465860	+	0.884858i
							\(7\)	1.31062	−	4.89129i	0.495367	−	1.84873i	−0.0326001	−	0.999468i	\(-0.510379\pi\)
0.527967	−	0.849265i	\(-0.322955\pi\)
\(11\)	1.00366	−	1.25855i	0.302615	−	0.379468i	−0.607152	−	0.794586i	\(-0.707688\pi\)
							0.909768	+	0.415118i	\(0.136260\pi\)
\(13\)	0.679847	+	0.0254381i	0.188556	+	0.00705525i	0.131502	−	0.991316i	\(-0.458020\pi\)
							0.0570534	+	0.998371i	\(0.481829\pi\)
\(17\)	−1.94753	+	3.68491i	−0.472345	+	0.893721i	0.526701	+	0.850051i	\(0.323429\pi\)
							−0.999046	+	0.0436701i	\(0.986095\pi\)
\(19\)	7.19160	−	1.08396i	1.64987	−	0.248677i	0.742754	−	0.669565i	\(-0.233519\pi\)
							0.907113	+	0.420887i	\(0.138281\pi\)
\(23\)	−1.16377	+	2.66739i	−0.242663	+	0.556190i	−0.994687	−	0.102946i	\(-0.967173\pi\)
							0.752024	+	0.659136i	\(0.229078\pi\)
\(29\)	0.563883	+	7.52450i	0.104711	+	1.39726i	0.763923	+	0.645308i	\(0.223271\pi\)
							−0.659212	+	0.751957i	\(0.729110\pi\)
\(31\)	−3.82204	+	2.60582i	−0.686459	+	0.468020i	−0.855644	−	0.517564i	\(-0.826839\pi\)
							0.169185	+	0.985584i	\(0.445886\pi\)
\(37\)	4.29965	−	1.15209i	0.706859	−	0.189402i	0.112558	−	0.993645i	\(-0.464096\pi\)
							0.594301	+	0.804243i	\(0.297429\pi\)
\(41\)	−5.75283	−	2.77041i	−0.898440	−	0.432666i	−0.0731150	−	0.997324i	\(-0.523294\pi\)
							−0.825325	+	0.564657i	\(0.809008\pi\)
\(43\)	−6.48035	−	1.00255i	−0.988244	−	0.152887i
							\(47\)	−6.44537	−	0.726219i	−0.940154	−	0.105930i	−0.371432	−	0.928460i	\(-0.621133\pi\)
−0.568722	+	0.822530i	\(0.692562\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	430.2.x.a.3.9	✓	528
5.2	odd	4	inner	430.2.x.a.347.20	yes	528
43.29	odd	42	inner	430.2.x.a.373.20	yes	528
215.72	even	84	inner	430.2.x.a.287.9	yes	528

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
430.2.x.a.3.9	✓	528	1.1	even	1	trivial
430.2.x.a.287.9	yes	528	215.72	even	84	inner
430.2.x.a.347.20	yes	528	5.2	odd	4	inner
430.2.x.a.373.20	yes	528	43.29	odd	42	inner