Embedded newform 430.2.x.a.3.1

• Label: 430.2.x.a.3.1
• 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			3.1
Character	\(\chi\)	\(=\)	430.3
Dual form			430.2.x.a.287.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.330279 - 0.943883i) q^{2} +(-2.12618 - 2.47066i) q^{3} +(-0.781831 + 0.623490i) q^{4} +(-1.43957 + 1.71103i) q^{5} +(-1.62979 + 2.82287i) q^{6} +(0.781116 - 2.91516i) q^{7} +(0.846724 + 0.532032i) q^{8} +(-1.13642 + 7.53964i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 528 q - 4 q^{6} - 12 q^{7}+O(q^{10}) \)

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
							\(3\)	−2.12618	−	2.47066i	−1.22755	−	1.42644i	−0.870368	−	0.492403i	\(-0.836119\pi\)
−0.357181	−	0.934035i	\(-0.616262\pi\)
\(5\)	−1.43957	+	1.71103i	−0.643797	+	0.765196i
							\(7\)	0.781116	−	2.91516i	0.295234	−	1.10183i	−0.645797	−	0.763509i	\(-0.723475\pi\)
0.941031	−	0.338320i	\(-0.109859\pi\)
\(11\)	−2.77358	+	3.47796i	−0.836265	+	1.04864i	0.161822	+	0.986820i	\(0.448263\pi\)
							−0.998087	+	0.0618235i	\(0.980308\pi\)
\(13\)	−1.84011	−	0.0688522i	−0.510356	−	0.0190962i	−0.219026	−	0.975719i	\(-0.570288\pi\)
							−0.291329	+	0.956623i	\(0.594098\pi\)
\(17\)	−0.290756	+	0.550137i	−0.0705186	+	0.133428i	−0.916854	−	0.399223i	\(-0.869280\pi\)
							0.846335	+	0.532650i	\(0.178804\pi\)
\(19\)	7.10261	−	1.07055i	1.62945	−	0.245600i	0.730264	−	0.683165i	\(-0.239397\pi\)
							0.899187	+	0.437564i	\(0.144159\pi\)
\(23\)	−1.29953	+	2.97855i	−0.270970	+	0.621071i	−0.997839	−	0.0657057i	\(-0.979070\pi\)
							0.726869	+	0.686777i	\(0.240975\pi\)
\(29\)	−0.0637998	−	0.851349i	−0.0118473	−	0.158092i	−0.999990	−	0.00455086i	\(-0.998551\pi\)
							0.988142	−	0.153541i	\(-0.0490676\pi\)
\(31\)	−0.677716	+	0.462059i	−0.121721	+	0.0829883i	−0.622650	−	0.782500i	\(-0.713944\pi\)
							0.500929	+	0.865488i	\(0.332992\pi\)
\(37\)	−8.94727	+	2.39741i	−1.47092	+	0.394132i	−0.903247	−	0.429121i	\(-0.858824\pi\)
							−0.567675	+	0.823253i	\(0.692157\pi\)
\(41\)	5.82192	+	2.80369i	0.909231	+	0.437862i	0.829214	−	0.558931i	\(-0.188788\pi\)
							0.0800166	+	0.996794i	\(0.474503\pi\)
\(43\)	−0.0851440	+	6.55689i	−0.0129843	+	0.999916i
							\(47\)	7.37114	+	0.830528i	1.07519	+	0.121145i	0.631760	−	0.775164i	\(-0.282333\pi\)
0.443431	+	0.896309i	\(0.353761\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	430.2.x.a.3.1	✓	528
5.2	odd	4	inner	430.2.x.a.347.12	yes	528
43.29	odd	42	inner	430.2.x.a.373.12	yes	528
215.72	even	84	inner	430.2.x.a.287.1	yes	528

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
430.2.x.a.3.1	✓	528	1.1	even	1	trivial
430.2.x.a.287.1	yes	528	215.72	even	84	inner
430.2.x.a.347.12	yes	528	5.2	odd	4	inner
430.2.x.a.373.12	yes	528	43.29	odd	42	inner