Embedded newform 20.10.e.b.3.17

• Label: 20.10.e.b.3.17
• 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			3.17
Character	\(\chi\)	\(=\)	20.3
Dual form			20.10.e.b.7.17

$q$-expansion

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	13.1360	+	18.4240i	0.580535	+	0.814236i
							\(3\)	30.0552	−	30.0552i	0.214227	−	0.214227i	−0.591834	−	0.806060i	\(-0.701596\pi\)
0.806060	+	0.591834i	\(0.201596\pi\)
\(5\)	965.172	−	1010.73i	0.690621	−	0.723217i
							\(7\)	6801.74	+	6801.74i	1.07073	+	1.07073i	0.997301	+	0.0734264i	\(0.0233934\pi\)
0.0734264	+	0.997301i	\(0.476607\pi\)
\(11\)		−	19925.4i		−	0.410336i	−0.978727	−	0.205168i	\(-0.934226\pi\)
							0.978727	−	0.205168i	\(-0.0657741\pi\)
\(13\)	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	\(-0.272651\pi\)
							0.655042	+	0.755593i	\(0.272651\pi\)
\(23\)	685876.	−	685876.i	0.511058	−	0.511058i	−0.403792	−	0.914851i	\(-0.632308\pi\)
							0.914851	+	0.403792i	\(0.132308\pi\)
\(29\)		−	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	\(-0.659045\pi\)
							0.479123	−	0.877748i	\(-0.340955\pi\)
\(37\)	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	−0.719075	−	0.694933i	\(-0.755434\pi\)
							0.694933	+	0.719075i	\(0.255434\pi\)
\(47\)	2.77864e6	+	2.77864e6i	0.0830599	+	0.0830599i	0.747416	−	0.664356i	\(-0.231294\pi\)
							−0.664356	+	0.747416i	\(0.731294\pi\)

(See \(a_n\) instead)

Twists

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

    

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