Embedded newform 20.10.e.b.3.14

• Label: 20.10.e.b.3.14
• Level: $20$
• Weight: $10$
• Character: 20.3
• Analytic conductor: $10.301$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 20 = 2^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	20.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.3007167233\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(24\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			3.14
Character	\(\chi\)	\(=\)	20.3
Dual form			20.10.e.b.7.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.62751 + 22.1492i) q^{2} +(-107.646 + 107.646i) q^{3} +(-469.172 + 204.991i) q^{4} +(-103.512 + 1393.70i) q^{5} +(-2882.39 - 1886.13i) q^{6} +(1033.09 + 1033.09i) q^{7} +(-6711.48 - 9443.19i) q^{8} -3492.17i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 30 q^{2} - 1440 q^{5} + 6152 q^{6} - 17100 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/20\mathbb{Z}\right)^\times\).

\(n\)	\(11\)	\(17\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	4.62751	+	22.1492i	0.204509	+	0.978865i
							\(3\)	−107.646	+	107.646i	−0.767275	+	0.767275i	−0.977626	−	0.210351i	\(-0.932539\pi\)
0.210351	+	0.977626i	\(0.432539\pi\)
\(5\)	−103.512	+	1393.70i	−0.0740673	+	0.997253i
							\(7\)	1033.09	+	1033.09i	0.162629	+	0.162629i	0.783730	−	0.621101i	\(-0.213314\pi\)
−0.621101	+	0.783730i	\(0.713314\pi\)
\(11\)		−	55858.4i		−	1.15033i	−0.818039	−	0.575163i	\(-0.804939\pi\)
							0.818039	−	0.575163i	\(-0.195061\pi\)
\(13\)	77057.7	+	77057.7i	0.748292	+	0.748292i	0.974158	−	0.225867i	\(-0.0725213\pi\)
							−0.225867	+	0.974158i	\(0.572521\pi\)
\(17\)	132442.	−	132442.i	0.384597	−	0.384597i	−0.488158	−	0.872755i	\(-0.662331\pi\)
							0.872755	+	0.488158i	\(0.162331\pi\)
\(19\)	−683935.			−1.20399			−0.601996	−	0.798499i	\(-0.705628\pi\)
							−0.601996	+	0.798499i	\(0.705628\pi\)
\(23\)	−498319.	+	498319.i	−0.371306	+	0.371306i	−0.867953	−	0.496647i	\(-0.834564\pi\)
							0.496647	+	0.867953i	\(0.334564\pi\)
\(29\)			6.68026e6i			1.75389i	0.480590	+	0.876945i	\(0.340423\pi\)
							−0.480590	+	0.876945i	\(0.659577\pi\)
\(31\)			3.14047e6i			0.610754i	0.952232	+	0.305377i	\(0.0987825\pi\)
							−0.952232	+	0.305377i	\(0.901217\pi\)
\(37\)	3.82144e6	−	3.82144e6i	0.335211	−	0.335211i	−0.519350	−	0.854562i	\(-0.673826\pi\)
							0.854562	+	0.519350i	\(0.173826\pi\)
\(41\)	−2.79544e7			−1.54498			−0.772489	−	0.635028i	\(-0.780988\pi\)
							−0.772489	+	0.635028i	\(0.780988\pi\)
\(43\)	6.41468e6	−	6.41468e6i	0.286132	−	0.286132i	−0.549416	−	0.835549i	\(-0.685150\pi\)
							0.835549	+	0.549416i	\(0.185150\pi\)
\(47\)	−3.41562e7	−	3.41562e7i	−1.02101	−	1.02101i	−0.999775	−	0.0212331i	\(-0.993241\pi\)
							−0.0212331	−	0.999775i	\(-0.506759\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.10.e.b.3.14	yes	48
4.3	odd	2	inner	20.10.e.b.3.2	✓	48
5.2	odd	4	inner	20.10.e.b.7.2	yes	48
5.3	odd	4		100.10.e.e.7.23		48
5.4	even	2		100.10.e.e.43.11		48
20.3	even	4		100.10.e.e.7.11		48
20.7	even	4	inner	20.10.e.b.7.14	yes	48
20.19	odd	2		100.10.e.e.43.23		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.10.e.b.3.2	✓	48	4.3	odd	2	inner
20.10.e.b.3.14	yes	48	1.1	even	1	trivial
20.10.e.b.7.2	yes	48	5.2	odd	4	inner
20.10.e.b.7.14	yes	48	20.7	even	4	inner
100.10.e.e.7.11		48	20.3	even	4
100.10.e.e.7.23		48	5.3	odd	4
100.10.e.e.43.11		48	5.4	even	2
100.10.e.e.43.23		48	20.19	odd	2