Embedded newform 20.10.e.b.3.6

• Label: 20.10.e.b.3.6
• 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			3.6
Character	\(\chi\)	\(=\)	20.3
Dual form			20.10.e.b.7.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-18.4240 - 13.1360i) q^{2} +(-30.0552 + 30.0552i) q^{3} +(166.891 + 484.037i) q^{4} +(965.172 - 1010.73i) q^{5} +(948.542 - 158.933i) q^{6} +(-6801.74 - 6801.74i) q^{7} +(3283.50 - 11110.2i) q^{8} +17876.4i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 30 q^{2} - 1440 q^{5} + 6152 q^{6} - 17100 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−18.4240	−	13.1360i	−0.814236	−	0.580535i
							\(3\)	−30.0552	+	30.0552i	−0.214227	+	0.214227i	−0.806060	−	0.591834i	\(-0.798404\pi\)
0.591834	+	0.806060i	\(0.298404\pi\)
\(5\)	965.172	−	1010.73i	0.690621	−	0.723217i
							\(7\)	−6801.74	−	6801.74i	−1.07073	−	1.07073i	−0.997301	−	0.0734264i	\(-0.976607\pi\)
−0.0734264	−	0.997301i	\(-0.523393\pi\)
\(11\)			19925.4i			0.410336i	0.978727	+	0.205168i	\(0.0657741\pi\)
							−0.978727	+	0.205168i	\(0.934226\pi\)
\(13\)	29376.5	+	29376.5i	0.285269	+	0.285269i	0.835206	−	0.549937i	\(-0.185348\pi\)
							−0.549937	+	0.835206i	\(0.685348\pi\)
\(17\)	−398158.	+	398158.i	−1.15621	+	1.15621i	−0.170921	+	0.985285i	\(0.554674\pi\)
							−0.985285	+	0.170921i	\(0.945326\pi\)
\(19\)	−744201.			−1.31008			−0.655042	−	0.755593i	\(-0.727349\pi\)
							−0.655042	+	0.755593i	\(0.727349\pi\)
\(23\)	−685876.	+	685876.i	−0.511058	+	0.511058i	−0.914851	−	0.403792i	\(-0.867692\pi\)
							0.403792	+	0.914851i	\(0.367692\pi\)
\(29\)		−	875168.i		−	0.229774i	−0.993379	−	0.114887i	\(-0.963349\pi\)
							0.993379	−	0.114887i	\(-0.0366506\pi\)
\(31\)			9.02667e6i			1.75550i	0.479123	+	0.877748i	\(0.340955\pi\)
							−0.479123	+	0.877748i	\(0.659045\pi\)
\(37\)	8.16404e6	−	8.16404e6i	0.716139	−	0.716139i	−0.251674	−	0.967812i	\(-0.580981\pi\)
							0.967812	+	0.251674i	\(0.0809809\pi\)
\(41\)	−2.35447e7			−1.30126			−0.650632	−	0.759393i	\(-0.725496\pi\)
							−0.650632	+	0.759393i	\(0.725496\pi\)
\(43\)	541228.	−	541228.i	0.0241419	−	0.0241419i	−0.694933	−	0.719075i	\(-0.744566\pi\)
							0.719075	+	0.694933i	\(0.244566\pi\)
\(47\)	−2.77864e6	−	2.77864e6i	−0.0830599	−	0.0830599i	0.664356	−	0.747416i	\(-0.268706\pi\)
							−0.747416	+	0.664356i	\(0.768706\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.10.e.b.3.6	✓	48
4.3	odd	2	inner	20.10.e.b.3.17	yes	48
5.2	odd	4	inner	20.10.e.b.7.17	yes	48
5.3	odd	4		100.10.e.e.7.8		48
5.4	even	2		100.10.e.e.43.19		48
20.3	even	4		100.10.e.e.7.19		48
20.7	even	4	inner	20.10.e.b.7.6	yes	48
20.19	odd	2		100.10.e.e.43.8		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.10.e.b.3.6	✓	48	1.1	even	1	trivial
20.10.e.b.3.17	yes	48	4.3	odd	2	inner
20.10.e.b.7.6	yes	48	20.7	even	4	inner
20.10.e.b.7.17	yes	48	5.2	odd	4	inner
100.10.e.e.7.8		48	5.3	odd	4
100.10.e.e.7.19		48	20.3	even	4
100.10.e.e.43.8		48	20.19	odd	2
100.10.e.e.43.19		48	5.4	even	2