Embedded newform 430.3.o.a.51.14

• Label: 430.3.o.a.51.14
• Level: $430$
• Weight: $3$
• Character: 430.51
• Analytic conductor: $11.717$
• Analytic rank: $0$
• Dimension: $192$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 430 = 2 \cdot 5 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	430.o (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.7166513675\)
Analytic rank:	\(0\)
Dimension:	\(192\)
Relative dimension:	\(32\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			51.14
Character	\(\chi\)	\(=\)	430.51
Dual form			430.3.o.a.371.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.37876 - 0.314692i) q^{2} +(4.18687 - 0.955625i) q^{3} +(1.80194 + 0.867767i) q^{4} +(1.74823 + 1.39417i) q^{5} -6.07340 q^{6} -3.82766i q^{7} +(-2.21135 - 1.76350i) q^{8} +(8.50793 - 4.09720i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q + 64 q^{4} - 8 q^{6} + 72 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(87\)	\(261\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{13}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.37876	−	0.314692i	−0.689378	−	0.157346i
							\(3\)	4.18687	−	0.955625i	1.39562	−	0.318542i	0.542414	−	0.840111i	\(-0.317510\pi\)
0.853209	+	0.521569i	\(0.174653\pi\)
\(5\)	1.74823	+	1.39417i	0.349646	+	0.278833i
							\(7\)		−	3.82766i		−	0.546809i	−0.961899	−	0.273404i	\(-0.911850\pi\)
0.961899	−	0.273404i	\(-0.0881497\pi\)
\(11\)	2.41922	−	1.16503i	0.219929	−	0.105912i	−0.320675	−	0.947189i	\(-0.603910\pi\)
							0.540604	+	0.841277i	\(0.318196\pi\)
\(13\)	2.44769	−	3.06930i	0.188284	−	0.236100i	−0.678726	−	0.734392i	\(-0.737468\pi\)
							0.867010	+	0.498291i	\(0.166039\pi\)
\(17\)	−10.0982	−	12.6628i	−0.594014	−	0.744870i	0.390418	−	0.920638i	\(-0.372331\pi\)
							−0.984432	+	0.175768i	\(0.943759\pi\)
\(19\)	8.04631	−	16.7083i	0.423490	−	0.879386i	−0.574649	−	0.818400i	\(-0.694861\pi\)
							0.998139	−	0.0609856i	\(-0.0194244\pi\)
\(23\)	41.2975	−	19.8878i	1.79554	−	0.864688i	0.861254	−	0.508175i	\(-0.169680\pi\)
							0.934290	−	0.356514i	\(-0.116035\pi\)
\(29\)	29.8896	+	6.82211i	1.03068	+	0.235245i	0.704231	−	0.709971i	\(-0.251292\pi\)
							0.326446	+	0.945216i	\(0.394149\pi\)
\(31\)	−6.43376	+	28.1881i	−0.207541	+	0.909295i	0.758657	+	0.651491i	\(0.225856\pi\)
							−0.966197	+	0.257804i	\(0.917001\pi\)
\(37\)			8.07212i			0.218166i	0.994033	+	0.109083i	\(0.0347914\pi\)
							−0.994033	+	0.109083i	\(0.965209\pi\)
\(41\)	−3.95614	+	17.3330i	−0.0964912	+	0.422756i	−0.999983	−	0.00585002i	\(-0.998138\pi\)
							0.903492	+	0.428606i	\(0.140995\pi\)
\(43\)	−33.5700	+	26.8711i	−0.780698	+	0.624909i
							\(47\)	−7.23317	−	3.48331i	−0.153897	−	0.0741130i	0.355350	−	0.934733i	\(-0.384362\pi\)
−0.509247	+	0.860620i	\(0.670076\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	430.3.o.a.51.14	✓	192
43.27	odd	14	inner	430.3.o.a.371.14	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
430.3.o.a.51.14	✓	192	1.1	even	1	trivial
430.3.o.a.371.14	yes	192	43.27	odd	14	inner