Embedded newform 62.3.h.a.11.1

• Label: 62.3.h.a.11.1
• Level: $62$
• Weight: $3$
• Character: 62.11
• 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			11.1
Character	\(\chi\)	\(=\)	62.11
Dual form			62.3.h.a.17.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.437016 - 1.34500i) q^{2} +(-2.55630 + 2.30170i) q^{3} +(-1.61803 + 1.17557i) q^{4} +(2.12476 + 3.68018i) q^{5} +(4.21292 + 2.43233i) q^{6} +(0.717825 + 6.82965i) q^{7} +(2.28825 + 1.66251i) q^{8} +(0.296073 - 2.81694i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 6 q^{3} - 24 q^{4} + 6 q^{5} + 18 q^{7} - 44 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{23}{30}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.437016	−	1.34500i	−0.218508	−	0.672499i
							\(3\)	−2.55630	+	2.30170i	−0.852098	+	0.767233i	−0.974296	−	0.225273i	\(-0.927673\pi\)
0.122197	+	0.992506i	\(0.461006\pi\)
\(5\)	2.12476	+	3.68018i	0.424951	+	0.736037i	0.996416	−	0.0845897i	\(-0.0269579\pi\)
							−0.571465	+	0.820627i	\(0.693625\pi\)
\(7\)	0.717825	+	6.82965i	0.102546	+	0.975664i	0.917930	+	0.396743i	\(0.129859\pi\)
							−0.815383	+	0.578921i	\(0.803474\pi\)
\(11\)	−3.13172	+	7.03396i	−0.284702	+	0.639451i	−0.998120	−	0.0612874i	\(-0.980479\pi\)
							0.713418	+	0.700739i	\(0.247146\pi\)
\(13\)	−2.78561	−	13.1052i	−0.214277	−	1.00810i	−0.945416	−	0.325867i	\(-0.894344\pi\)
							0.731138	−	0.682229i	\(-0.238989\pi\)
\(17\)	0.130279	+	0.292611i	0.00766345	+	0.0172124i	0.917337	−	0.398112i	\(-0.130335\pi\)
							−0.909673	+	0.415325i	\(0.863668\pi\)
\(19\)	9.39073	+	1.99606i	0.494249	+	0.105056i	0.448291	−	0.893888i	\(-0.352033\pi\)
							0.0459575	+	0.998943i	\(0.485366\pi\)
\(23\)	−16.2727	+	22.3974i	−0.707507	+	0.973800i	0.292340	+	0.956314i	\(0.405566\pi\)
							−0.999847	+	0.0174855i	\(0.994434\pi\)
\(29\)	48.3796	−	15.7195i	1.66826	−	0.542051i	0.685683	−	0.727900i	\(-0.259503\pi\)
							0.982578	+	0.185849i	\(0.0595035\pi\)
\(31\)	30.7095	+	4.23389i	0.990629	+	0.136577i
							\(37\)	49.0344	+	28.3100i	1.32525	+	0.765136i	0.984562	−	0.175038i	\(-0.0560050\pi\)
0.340693	+	0.940175i	\(0.389338\pi\)
\(41\)	−19.7421	+	21.9258i	−0.481514	+	0.534775i	−0.934131	−	0.356929i	\(-0.883824\pi\)
							0.452617	+	0.891705i	\(0.350490\pi\)
\(43\)	14.3478	−	67.5012i	0.333670	−	1.56979i	−0.416868	−	0.908967i	\(-0.636872\pi\)
							0.750538	−	0.660827i	\(-0.229794\pi\)
\(47\)	−19.6761	+	60.5567i	−0.418640	+	1.28844i	0.490315	+	0.871546i	\(0.336882\pi\)
							−0.908954	+	0.416895i	\(0.863118\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	62.3.h.a.11.1	✓	48
31.17	odd	30	inner	62.3.h.a.17.1	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
62.3.h.a.11.1	✓	48	1.1	even	1	trivial
62.3.h.a.17.1	yes	48	31.17	odd	30	inner