LMFDB - Newform orbit 20.3.d.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [20,3,Mod(19,20)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(20, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("20.19");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 20 = 2^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	20.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$0.544960528721$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$

$=$

$ q - 2 q^{2} + 4 q^{3} + 4 q^{4} - 5 q^{5} - 8 q^{6} - 4 q^{7} - 8 q^{8} + 7 q^{9}+O(q^{10}) $

$ q - 2 q^{2} + 4 q^{3} + 4 q^{4} - 5 q^{5} - 8 q^{6} - 4 q^{7} - 8 q^{8} + 7 q^{9} + 10 q^{10} + 16 q^{12} + 8 q^{14} - 20 q^{15} + 16 q^{16} - 14 q^{18} - 20 q^{20} - 16 q^{21} + 44 q^{23} - 32 q^{24} + 25 q^{25} - 8 q^{27} - 16 q^{28} - 22 q^{29} + 40 q^{30} - 32 q^{32} + 20 q^{35} + 28 q^{36} + 40 q^{40} + 62 q^{41} + 32 q^{42} - 76 q^{43} - 35 q^{45} - 88 q^{46} - 4 q^{47} + 64 q^{48} - 33 q^{49} - 50 q^{50} + 16 q^{54} + 32 q^{56} + 44 q^{58} - 80 q^{60} - 58 q^{61} - 28 q^{63} + 64 q^{64} + 116 q^{67} + 176 q^{69} - 40 q^{70} - 56 q^{72} + 100 q^{75} - 80 q^{80} - 95 q^{81} - 124 q^{82} - 76 q^{83} - 64 q^{84} + 152 q^{86} - 88 q^{87} - 142 q^{89} + 70 q^{90} + 176 q^{92} + 8 q^{94} - 128 q^{96} + 66 q^{98}+O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

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

$n$	$11$	$17$
$\chi(n)$	$-1$	$-1$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

19.1

−2.00000

4.00000

−5.00000

−8.00000

−4.00000

−8.00000

7.00000

10.0000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
20.d	odd	2	1	CM by $\Q(\sqrt{-5}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	20.3.d.a	✓	1
3.b	odd	2	1		180.3.f.b		1
4.b	odd	2	1		20.3.d.b	yes	1
5.b	even	2	1		20.3.d.b	yes	1
5.c	odd	4	2		100.3.b.c		2
8.b	even	2	1		320.3.h.a		1
8.d	odd	2	1		320.3.h.b		1
12.b	even	2	1		180.3.f.a		1
15.d	odd	2	1		180.3.f.a		1
15.e	even	4	2		900.3.c.h		2
16.e	even	4	2		1280.3.e.c		2
16.f	odd	4	2		1280.3.e.b		2
20.d	odd	2	1	CM	20.3.d.a	✓	1
20.e	even	4	2		100.3.b.c		2
40.e	odd	2	1		320.3.h.a		1
40.f	even	2	1		320.3.h.b		1
40.i	odd	4	2		1600.3.b.f		2
40.k	even	4	2		1600.3.b.f		2
60.h	even	2	1		180.3.f.b		1
60.l	odd	4	2		900.3.c.h		2
80.k	odd	4	2		1280.3.e.c		2
80.q	even	4	2		1280.3.e.b		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
20.3.d.a	✓	1	1.a	even	1	1	trivial
20.3.d.a	✓	1	20.d	odd	2	1	CM
20.3.d.b	yes	1	4.b	odd	2	1
20.3.d.b	yes	1	5.b	even	2	1
100.3.b.c		2	5.c	odd	4	2
100.3.b.c		2	20.e	even	4	2
180.3.f.a		1	12.b	even	2	1
180.3.f.a		1	15.d	odd	2	1
180.3.f.b		1	3.b	odd	2	1
180.3.f.b		1	60.h	even	2	1
320.3.h.a		1	8.b	even	2	1
320.3.h.a		1	40.e	odd	2	1
320.3.h.b		1	8.d	odd	2	1
320.3.h.b		1	40.f	even	2	1
900.3.c.h		2	15.e	even	4	2
900.3.c.h		2	60.l	odd	4	2
1280.3.e.b		2	16.f	odd	4	2
1280.3.e.b		2	80.q	even	4	2
1280.3.e.c		2	16.e	even	4	2
1280.3.e.c		2	80.k	odd	4	2
1600.3.b.f		2	40.i	odd	4	2
1600.3.b.f		2	40.k	even	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3} - 4 $ T3 - 4 acting on $S_{3}^{\mathrm{new}}(20, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 2 $ T + 2
$3$	$ T - 4 $ T - 4
$5$	$ T + 5 $ T + 5
$7$	$ T + 4 $ T + 4
$11$	$ T $ T
$13$	$ T $ T
$17$	$ T $ T
$19$	$ T $ T
$23$	$ T - 44 $ T - 44
$29$	$ T + 22 $ T + 22
$31$	$ T $ T
$37$	$ T $ T
$41$	$ T - 62 $ T - 62
$43$	$ T + 76 $ T + 76
$47$	$ T + 4 $ T + 4
$53$	$ T $ T
$59$	$ T $ T
$61$	$ T + 58 $ T + 58
$67$	$ T - 116 $ T - 116
$71$	$ T $ T
$73$	$ T $ T
$79$	$ T $ T
$83$	$ T + 76 $ T + 76
$89$	$ T + 142 $ T + 142
$97$	$ T $ T
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Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 20 = 2^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	20.d (of order \(2\), degree \(1\), minimal)

\(n\)	\(11\)	\(17\)
\(\chi(n)\)	\(-1\)	\(-1\)

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