Embedded newform 20.10.e.b.3.11

• Label: 20.10.e.b.3.11
• 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			3.11
Character	\(\chi\)	\(=\)	20.3
Dual form			20.10.e.b.7.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.66069 - 21.6249i) q^{2} +(40.3330 - 40.3330i) q^{3} +(-423.270 + 288.073i) q^{4} +(1192.73 + 728.376i) q^{5} +(-1140.84 - 603.551i) q^{6} +(3888.44 + 3888.44i) q^{7} +(9048.82 + 7234.41i) q^{8} +16429.5i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 30 q^{2} - 1440 q^{5} + 6152 q^{6} - 17100 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.66069	−	21.6249i	−0.294364	−	0.955694i
							\(3\)	40.3330	−	40.3330i	0.287485	−	0.287485i	−0.548600	−	0.836085i	\(-0.684839\pi\)
0.836085	+	0.548600i	\(0.184839\pi\)
\(5\)	1192.73	+	728.376i	0.853445	+	0.521183i
							\(7\)	3888.44	+	3888.44i	0.612116	+	0.612116i	0.943497	−	0.331381i	\(-0.107514\pi\)
−0.331381	+	0.943497i	\(0.607514\pi\)
\(11\)		−	47923.9i		−	0.986928i	−0.869766	−	0.493464i	\(-0.835730\pi\)
							0.869766	−	0.493464i	\(-0.164270\pi\)
\(13\)	34179.1	+	34179.1i	0.331906	+	0.331906i	0.853310	−	0.521404i	\(-0.174591\pi\)
							−0.521404	+	0.853310i	\(0.674591\pi\)
\(17\)	244799.	−	244799.i	0.710870	−	0.710870i	−0.255848	−	0.966717i	\(-0.582354\pi\)
							0.966717	+	0.255848i	\(0.0823545\pi\)
\(19\)	525921.			0.925827			0.462913	−	0.886404i	\(-0.346804\pi\)
							0.462913	+	0.886404i	\(0.346804\pi\)
\(23\)	−1.02588e6	+	1.02588e6i	−0.764398	+	0.764398i	−0.977114	−	0.212716i	\(-0.931769\pi\)
							0.212716	+	0.977114i	\(0.431769\pi\)
\(29\)		−	4.16677e6i		−	1.09398i	−0.837140	−	0.546989i	\(-0.815774\pi\)
							0.837140	−	0.546989i	\(-0.184226\pi\)
\(31\)			9.41184e6i			1.83040i	0.402995	+	0.915202i	\(0.367969\pi\)
							−0.402995	+	0.915202i	\(0.632031\pi\)
\(37\)	606097.	−	606097.i	0.0531660	−	0.0531660i	−0.680024	−	0.733190i	\(-0.738031\pi\)
							0.733190	+	0.680024i	\(0.238031\pi\)
\(41\)	2.53858e7			1.40302			0.701509	−	0.712660i	\(-0.252510\pi\)
							0.701509	+	0.712660i	\(0.252510\pi\)
\(43\)	5.84539e6	−	5.84539e6i	0.260739	−	0.260739i	−0.564615	−	0.825354i	\(-0.690975\pi\)
							0.825354	+	0.564615i	\(0.190975\pi\)
\(47\)	−1.42267e7	−	1.42267e7i	−0.425268	−	0.425268i	0.461745	−	0.887013i	\(-0.347224\pi\)
							−0.887013	+	0.461745i	\(0.847224\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.10.e.b.3.11	✓	48
4.3	odd	2	inner	20.10.e.b.3.23	yes	48
5.2	odd	4	inner	20.10.e.b.7.23	yes	48
5.3	odd	4		100.10.e.e.7.2		48
5.4	even	2		100.10.e.e.43.14		48
20.3	even	4		100.10.e.e.7.14		48
20.7	even	4	inner	20.10.e.b.7.11	yes	48
20.19	odd	2		100.10.e.e.43.2		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.10.e.b.3.11	✓	48	1.1	even	1	trivial
20.10.e.b.3.23	yes	48	4.3	odd	2	inner
20.10.e.b.7.11	yes	48	20.7	even	4	inner
20.10.e.b.7.23	yes	48	5.2	odd	4	inner
100.10.e.e.7.2		48	5.3	odd	4
100.10.e.e.7.14		48	20.3	even	4
100.10.e.e.43.2		48	20.19	odd	2
100.10.e.e.43.14		48	5.4	even	2