Embedded newform 62.3.h.a.43.6

• Label: 62.3.h.a.43.6
• Level: $62$
• Weight: $3$
• Character: 62.43
• Analytic conductor: $1.689$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 62 = 2 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	62.h (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.68937763903\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(6\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			43.6
Character	\(\chi\)	\(=\)	62.43
Dual form			62.3.h.a.13.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.14412 + 0.831254i) q^{2} +(1.87649 - 4.21466i) q^{3} +(0.618034 + 1.90211i) q^{4} +(-2.55443 + 4.42440i) q^{5} +(5.65038 - 3.26225i) q^{6} +(6.21587 - 6.90342i) q^{7} +(-0.874032 + 2.68999i) q^{8} +(-8.21998 - 9.12921i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 6 q^{3} - 24 q^{4} + 6 q^{5} + 18 q^{7} - 44 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/62\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{19}{30}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.14412	+	0.831254i	0.572061	+	0.415627i
							\(3\)	1.87649	−	4.21466i	0.625496	−	1.40489i	−0.271323	−	0.962488i	\(-0.587461\pi\)
0.896819	−	0.442398i	\(-0.145872\pi\)
\(5\)	−2.55443	+	4.42440i	−0.510886	+	0.884881i	0.489034	+	0.872265i	\(0.337349\pi\)
							−0.999920	+	0.0126162i	\(0.995984\pi\)
\(7\)	6.21587	−	6.90342i	0.887982	−	0.986203i	−0.111990	−	0.993709i	\(-0.535723\pi\)
							0.999972	+	0.00750588i	\(0.00238922\pi\)
\(11\)	−2.05481	+	9.66710i	−0.186800	+	0.878827i	0.780492	+	0.625165i	\(0.214968\pi\)
							−0.967293	+	0.253662i	\(0.918365\pi\)
\(13\)	−18.5648	+	1.95124i	−1.42806	+	0.150095i	−0.786792	−	0.617219i	\(-0.788259\pi\)
							−0.641271	+	0.767314i	\(0.721593\pi\)
\(17\)	4.21814	+	19.8448i	0.248126	+	1.16734i	0.908985	+	0.416830i	\(0.136859\pi\)
							−0.660859	+	0.750510i	\(0.729808\pi\)
\(19\)	0.156709	−	1.49098i	0.00824783	−	0.0784729i	−0.989624	−	0.143684i	\(-0.954105\pi\)
							0.997872	+	0.0652106i	\(0.0207719\pi\)
\(23\)	−30.0446	−	9.76210i	−1.30629	−	0.424439i	−0.428524	−	0.903530i	\(-0.640966\pi\)
							−0.877765	+	0.479092i	\(0.840966\pi\)
\(29\)	26.1300	−	35.9649i	0.901036	−	1.24017i	−0.0691009	−	0.997610i	\(-0.522013\pi\)
							0.970137	−	0.242560i	\(-0.0779870\pi\)
\(31\)	20.7644	−	23.0182i	0.669820	−	0.742524i
							\(37\)	4.87272	−	2.81326i	0.131695	−	0.0760342i	−0.432705	−	0.901536i	\(-0.642441\pi\)
0.564400	+	0.825501i	\(0.309108\pi\)
\(41\)	34.5373	−	15.3770i	0.842372	−	0.375048i	0.0602576	−	0.998183i	\(-0.480808\pi\)
							0.782115	+	0.623135i	\(0.214141\pi\)
\(43\)	−33.2021	−	3.48968i	−0.772142	−	0.0811554i	−0.289740	−	0.957105i	\(-0.593569\pi\)
							−0.482402	+	0.875950i	\(0.660236\pi\)
\(47\)	34.9925	−	25.4235i	0.744521	−	0.540926i	−0.149603	−	0.988746i	\(-0.547800\pi\)
							0.894124	+	0.447820i	\(0.147800\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	62.3.h.a.43.6	yes	48
31.13	odd	30	inner	62.3.h.a.13.6	✓	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
62.3.h.a.13.6	✓	48	31.13	odd	30	inner
62.3.h.a.43.6	yes	48	1.1	even	1	trivial