LMFDB - Newform orbit 40.3.e.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [40,3,Mod(19,40)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(40, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("40.19");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 40 = 2^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	40.e (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:	$1.08992105744$
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^{4} - 5 q^{5} - 6 q^{7} + 8 q^{8} + 9 q^{9}+O(q^{10}) $

$ q + 2 q^{2} + 4 q^{4} - 5 q^{5} - 6 q^{7} + 8 q^{8} + 9 q^{9} - 10 q^{10} - 18 q^{11} + 6 q^{13} - 12 q^{14} + 16 q^{16} + 18 q^{18} - 2 q^{19} - 20 q^{20} - 36 q^{22} + 26 q^{23} + 25 q^{25} + 12 q^{26} - 24 q^{28} + 32 q^{32} + 30 q^{35} + 36 q^{36} + 54 q^{37} - 4 q^{38} - 40 q^{40} - 78 q^{41} - 72 q^{44} - 45 q^{45} + 52 q^{46} - 86 q^{47} - 13 q^{49} + 50 q^{50} + 24 q^{52} - 74 q^{53} + 90 q^{55} - 48 q^{56} + 78 q^{59} - 54 q^{63} + 64 q^{64} - 30 q^{65} + 60 q^{70} + 72 q^{72} + 108 q^{74} - 8 q^{76} + 108 q^{77} - 80 q^{80} + 81 q^{81} - 156 q^{82} - 144 q^{88} + 18 q^{89} - 90 q^{90} - 36 q^{91} + 104 q^{92} - 172 q^{94} + 10 q^{95} - 26 q^{98} - 162 q^{99}+O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/40\mathbb{Z}\right)^\times$.

$n$	$17$	$21$	$31$
$\chi(n)$	$-1$	$-1$	$-1$

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

−6.00000

8.00000

9.00000

−10.0000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
40.e	odd	2	1	CM by $\Q(\sqrt{-10}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	40.3.e.b	yes	1
3.b	odd	2	1		360.3.p.a		1
4.b	odd	2	1		160.3.e.a		1
5.b	even	2	1		40.3.e.a	✓	1
5.c	odd	4	2		200.3.g.c		2
8.b	even	2	1		160.3.e.b		1
8.d	odd	2	1		40.3.e.a	✓	1
12.b	even	2	1		1440.3.p.b		1
15.d	odd	2	1		360.3.p.b		1
16.e	even	4	2		1280.3.h.c		2
16.f	odd	4	2		1280.3.h.b		2
20.d	odd	2	1		160.3.e.b		1
20.e	even	4	2		800.3.g.c		2
24.f	even	2	1		360.3.p.b		1
24.h	odd	2	1		1440.3.p.a		1
40.e	odd	2	1	CM	40.3.e.b	yes	1
40.f	even	2	1		160.3.e.a		1
40.i	odd	4	2		800.3.g.c		2
40.k	even	4	2		200.3.g.c		2
60.h	even	2	1		1440.3.p.a		1
80.k	odd	4	2		1280.3.h.c		2
80.q	even	4	2		1280.3.h.b		2
120.i	odd	2	1		1440.3.p.b		1
120.m	even	2	1		360.3.p.a		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
40.3.e.a	✓	1	5.b	even	2	1
40.3.e.a	✓	1	8.d	odd	2	1
40.3.e.b	yes	1	1.a	even	1	1	trivial
40.3.e.b	yes	1	40.e	odd	2	1	CM
160.3.e.a		1	4.b	odd	2	1
160.3.e.a		1	40.f	even	2	1
160.3.e.b		1	8.b	even	2	1
160.3.e.b		1	20.d	odd	2	1
200.3.g.c		2	5.c	odd	4	2
200.3.g.c		2	40.k	even	4	2
360.3.p.a		1	3.b	odd	2	1
360.3.p.a		1	120.m	even	2	1
360.3.p.b		1	15.d	odd	2	1
360.3.p.b		1	24.f	even	2	1
800.3.g.c		2	20.e	even	4	2
800.3.g.c		2	40.i	odd	4	2
1280.3.h.b		2	16.f	odd	4	2
1280.3.h.b		2	80.q	even	4	2
1280.3.h.c		2	16.e	even	4	2
1280.3.h.c		2	80.k	odd	4	2
1440.3.p.a		1	24.h	odd	2	1
1440.3.p.a		1	60.h	even	2	1
1440.3.p.b		1	12.b	even	2	1
1440.3.p.b		1	120.i	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{3}^{\mathrm{new}}(40, [\chi])$:

$ T_{3} $

$ T_{7} + 6 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 2 $ T - 2
$3$	$ T $ T
$5$	$ T + 5 $ T + 5
$7$	$ T + 6 $ T + 6
$11$	$ T + 18 $ T + 18
$13$	$ T - 6 $ T - 6
$17$	$ T $ T
$19$	$ T + 2 $ T + 2
$23$	$ T - 26 $ T - 26
$29$	$ T $ T
$31$	$ T $ T
$37$	$ T - 54 $ T - 54
$41$	$ T + 78 $ T + 78
$43$	$ T $ T
$47$	$ T + 86 $ T + 86
$53$	$ T + 74 $ T + 74
$59$	$ T - 78 $ T - 78
$61$	$ T $ T
$67$	$ T $ T
$71$	$ T $ T
$73$	$ T $ T
$79$	$ T $ T
$83$	$ T $ T
$89$	$ T - 18 $ T - 18
$97$	$ T $ T
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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 40 = 2^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	40.e (of order \(2\), degree \(1\), minimal)

\(n\)	\(17\)	\(21\)	\(31\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)

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