Embedded newform 62.3.h.a.43.4

• Label: 62.3.h.a.43.4
• 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			43.4
Character	\(\chi\)	\(=\)	62.43
Dual form			62.3.h.a.13.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.14412 + 0.831254i) q^{2} +(-1.14843 + 2.57943i) q^{3} +(0.618034 + 1.90211i) q^{4} +(-1.15365 + 1.99818i) q^{5} +(-3.45811 + 1.99654i) q^{6} +(1.42640 - 1.58417i) q^{7} +(-0.874032 + 2.68999i) q^{8} +(0.687638 + 0.763699i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 6 q^{3} - 24 q^{4} + 6 q^{5} + 18 q^{7} - 44 q^{9}+O(q^{10}) \)

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
							\(3\)	−1.14843	+	2.57943i	−0.382812	+	0.859809i	0.614663	+	0.788789i	\(0.289292\pi\)
−0.997475	+	0.0710192i	\(0.977375\pi\)
\(5\)	−1.15365	+	1.99818i	−0.230730	+	0.399637i	−0.958023	−	0.286691i	\(-0.907445\pi\)
							0.727293	+	0.686327i	\(0.240778\pi\)
\(7\)	1.42640	−	1.58417i	0.203771	−	0.226310i	−0.632594	−	0.774484i	\(-0.718010\pi\)
							0.836365	+	0.548173i	\(0.184676\pi\)
\(11\)	1.85202	−	8.71305i	0.168365	−	0.792095i	−0.810199	−	0.586154i	\(-0.800641\pi\)
							0.978564	−	0.205941i	\(-0.0660254\pi\)
\(13\)	13.8836	−	1.45922i	1.06797	−	0.112248i	0.445800	−	0.895133i	\(-0.352919\pi\)
							0.622166	+	0.782885i	\(0.286253\pi\)
\(17\)	−1.35243	−	6.36268i	−0.0795546	−	0.374275i	0.920302	−	0.391209i	\(-0.127943\pi\)
							−0.999857	+	0.0169336i	\(0.994610\pi\)
\(19\)	2.04971	−	19.5017i	0.107880	−	1.02641i	−0.797938	−	0.602739i	\(-0.794076\pi\)
							0.905818	−	0.423667i	\(-0.139257\pi\)
\(23\)	−8.52311	−	2.76932i	−0.370570	−	0.120405i	0.117811	−	0.993036i	\(-0.462412\pi\)
							−0.488381	+	0.872631i	\(0.662412\pi\)
\(29\)	3.48021	−	4.79010i	0.120007	−	0.165176i	−0.744787	−	0.667303i	\(-0.767449\pi\)
							0.864794	+	0.502126i	\(0.167449\pi\)
\(31\)	−26.7127	−	15.7300i	−0.861700	−	0.507418i
							\(37\)	11.4645	−	6.61903i	0.309851	−	0.178893i	−0.337009	−	0.941502i	\(-0.609415\pi\)
0.646860	+	0.762609i	\(0.276082\pi\)
\(41\)	−27.5093	+	12.2479i	−0.670959	+	0.298730i	−0.713798	−	0.700352i	\(-0.753027\pi\)
							0.0428386	+	0.999082i	\(0.486360\pi\)
\(43\)	45.8933	+	4.82358i	1.06729	+	0.112176i	0.621849	−	0.783137i	\(-0.286382\pi\)
							0.445436	+	0.895314i	\(0.353049\pi\)
\(47\)	−73.2505	+	53.2196i	−1.55852	+	1.13233i	−0.621313	+	0.783563i	\(0.713400\pi\)
							−0.937209	+	0.348769i	\(0.886600\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	62.3.h.a.43.4	yes	48
31.13	odd	30	inner	62.3.h.a.13.4	✓	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
62.3.h.a.13.4	✓	48	31.13	odd	30	inner
62.3.h.a.43.4	yes	48	1.1	even	1	trivial