LMFDB - Newform orbit 1280.3.h.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,3,Mod(1279,1280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

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

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

chi := DirichletCharacter("1280.1279");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1280 = 2^{8} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1280.h (of order $2$, degree $1$, not 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:	no
Analytic conductor:	$34.8774738381$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-1}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 40)
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)

Coefficients of the $q$-expansion are expressed in terms of $i = \sqrt{-1}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + 5 i q^{5} - 6 q^{7} - 9 q^{9}+O(q^{10}) $ q + 5i q^5 - 6 * q^7 - 9 * q^9	$ q + 5 i q^{5} - 6 q^{7} - 9 q^{9} - 18 i q^{11} + 6 i q^{13} + 2 i q^{19} + 26 q^{23} - 25 q^{25} - 30 i q^{35} - 54 i q^{37} + 78 q^{41} - 45 i q^{45} + 86 q^{47} - 13 q^{49} + 74 i q^{53} + 90 q^{55} + 78 i q^{59} + 54 q^{63} - 30 q^{65} + 108 i q^{77} + 81 q^{81} - 18 q^{89} - 36 i q^{91} - 10 q^{95} + 162 i q^{99} +O(q^{100}) $ q + 5i q^5 - 6 * q^7 - 9 * q^9 - 18i q^11 + 6i q^13 + 2i q^19 + 26 * q^23 - 25 * q^25 - 30i q^35 - 54i q^37 + 78 * q^41 - 45i q^45 + 86 * q^47 - 13 * q^49 + 74i q^53 + 90 * q^55 + 78i q^59 + 54 * q^63 - 30 * q^65 + 108i q^77 + 81 * q^81 - 18 * q^89 - 36i q^91 - 10 * q^95 + 162i q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 12 q^{7} - 18 q^{9}+O(q^{10}) $ 2 * q - 12 * q^7 - 18 * q^9	$ 2 q - 12 q^{7} - 18 q^{9} + 52 q^{23} - 50 q^{25} + 156 q^{41} + 172 q^{47} - 26 q^{49} + 180 q^{55} + 108 q^{63} - 60 q^{65} + 162 q^{81} - 36 q^{89} - 20 q^{95}+O(q^{100}) $ 2 * q - 12 * q^7 - 18 * q^9 + 52 * q^23 - 50 * q^25 + 156 * q^41 + 172 * q^47 - 26 * q^49 + 180 * q^55 + 108 * q^63 - 60 * q^65 + 162 * q^81 - 36 * q^89 - 20 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times$.

$n$	$257$	$261$	$511$
$\chi(n)$	$-1$	$1$	$-1$

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} $

1279.1

	−	1.00000i
		1.00000i

−

5.00000i

−6.00000

−9.00000

1279.2

5.00000i

−6.00000

−9.00000

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1280.3.h.b		2
4.b	odd	2	1		1280.3.h.c		2
5.b	even	2	1		1280.3.h.c		2
8.b	even	2	1	inner	1280.3.h.b		2
8.d	odd	2	1		1280.3.h.c		2
16.e	even	4	1		40.3.e.a	✓	1
16.e	even	4	1		160.3.e.a		1
16.f	odd	4	1		40.3.e.b	yes	1
16.f	odd	4	1		160.3.e.b		1
20.d	odd	2	1	inner	1280.3.h.b		2
40.e	odd	2	1	CM	1280.3.h.b		2
40.f	even	2	1		1280.3.h.c		2
48.i	odd	4	1		360.3.p.b		1
48.i	odd	4	1		1440.3.p.b		1
48.k	even	4	1		360.3.p.a		1
48.k	even	4	1		1440.3.p.a		1
80.i	odd	4	1		200.3.g.c		2
80.i	odd	4	1		800.3.g.c		2
80.j	even	4	1		200.3.g.c		2
80.j	even	4	1		800.3.g.c		2
80.k	odd	4	1		40.3.e.a	✓	1
80.k	odd	4	1		160.3.e.a		1
80.q	even	4	1		40.3.e.b	yes	1
80.q	even	4	1		160.3.e.b		1
80.s	even	4	1		200.3.g.c		2
80.s	even	4	1		800.3.g.c		2
80.t	odd	4	1		200.3.g.c		2
80.t	odd	4	1		800.3.g.c		2
240.t	even	4	1		360.3.p.b		1
240.t	even	4	1		1440.3.p.b		1
240.bm	odd	4	1		360.3.p.a		1
240.bm	odd	4	1		1440.3.p.a		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
40.3.e.a	✓	1	16.e	even	4	1
40.3.e.a	✓	1	80.k	odd	4	1
40.3.e.b	yes	1	16.f	odd	4	1
40.3.e.b	yes	1	80.q	even	4	1
160.3.e.a		1	16.e	even	4	1
160.3.e.a		1	80.k	odd	4	1
160.3.e.b		1	16.f	odd	4	1
160.3.e.b		1	80.q	even	4	1
200.3.g.c		2	80.i	odd	4	1
200.3.g.c		2	80.j	even	4	1
200.3.g.c		2	80.s	even	4	1
200.3.g.c		2	80.t	odd	4	1
360.3.p.a		1	48.k	even	4	1
360.3.p.a		1	240.bm	odd	4	1
360.3.p.b		1	48.i	odd	4	1
360.3.p.b		1	240.t	even	4	1
800.3.g.c		2	80.i	odd	4	1
800.3.g.c		2	80.j	even	4	1
800.3.g.c		2	80.s	even	4	1
800.3.g.c		2	80.t	odd	4	1
1280.3.h.b		2	1.a	even	1	1	trivial
1280.3.h.b		2	8.b	even	2	1	inner
1280.3.h.b		2	20.d	odd	2	1	inner
1280.3.h.b		2	40.e	odd	2	1	CM
1280.3.h.c		2	4.b	odd	2	1
1280.3.h.c		2	5.b	even	2	1
1280.3.h.c		2	8.d	odd	2	1
1280.3.h.c		2	40.f	even	2	1
1440.3.p.a		1	48.k	even	4	1
1440.3.p.a		1	240.bm	odd	4	1
1440.3.p.b		1	48.i	odd	4	1
1440.3.p.b		1	240.t	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{3}^{\mathrm{new}}(1280, [\chi])$:

$ T_{3} $

$ T_{7} + 6 $

$ T_{29} $

$ T_{61} $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} $ T^2
$3$	$ T^{2} $ T^2
$5$	$ T^{2} + 25 $ T^2 + 25
$7$	$ (T + 6)^{2} $ (T + 6)^2
$11$	$ T^{2} + 324 $ T^2 + 324
$13$	$ T^{2} + 36 $ T^2 + 36
$17$	$ T^{2} $ T^2
$19$	$ T^{2} + 4 $ T^2 + 4
$23$	$ (T - 26)^{2} $ (T - 26)^2
$29$	$ T^{2} $ T^2
$31$	$ T^{2} $ T^2
$37$	$ T^{2} + 2916 $ T^2 + 2916
$41$	$ (T - 78)^{2} $ (T - 78)^2
$43$	$ T^{2} $ T^2
$47$	$ (T - 86)^{2} $ (T - 86)^2
$53$	$ T^{2} + 5476 $ T^2 + 5476
$59$	$ T^{2} + 6084 $ T^2 + 6084
$61$	$ T^{2} $ T^2
$67$	$ T^{2} $ T^2
$71$	$ T^{2} $ T^2
$73$	$ T^{2} $ T^2
$79$	$ T^{2} $ T^2
$83$	$ T^{2} $ T^2
$89$	$ (T + 18)^{2} $ (T + 18)^2
$97$	$ T^{2} $ T^2
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Classical	Maass
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1280 = 2^{8} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1280.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + 5 i q^{5} - 6 q^{7} - 9 q^{9}+O(q^{10}) \) q + 5i q^5 - 6 * q^7 - 9 * q^9	\( q + 5 i q^{5} - 6 q^{7} - 9 q^{9} - 18 i q^{11} + 6 i q^{13} + 2 i q^{19} + 26 q^{23} - 25 q^{25} - 30 i q^{35} - 54 i q^{37} + 78 q^{41} - 45 i q^{45} + 86 q^{47} - 13 q^{49} + 74 i q^{53} + 90 q^{55} + 78 i q^{59} + 54 q^{63} - 30 q^{65} + 108 i q^{77} + 81 q^{81} - 18 q^{89} - 36 i q^{91} - 10 q^{95} + 162 i q^{99} +O(q^{100}) \) q + 5i q^5 - 6 * q^7 - 9 * q^9 - 18i q^11 + 6i q^13 + 2i q^19 + 26 * q^23 - 25 * q^25 - 30i q^35 - 54i q^37 + 78 * q^41 - 45i q^45 + 86 * q^47 - 13 * q^49 + 74i q^53 + 90 * q^55 + 78i q^59 + 54 * q^63 - 30 * q^65 + 108i q^77 + 81 * q^81 - 18 * q^89 - 36i q^91 - 10 * q^95 + 162i q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 12 q^{7} - 18 q^{9}+O(q^{10}) \) 2 * q - 12 * q^7 - 18 * q^9	\( 2 q - 12 q^{7} - 18 q^{9} + 52 q^{23} - 50 q^{25} + 156 q^{41} + 172 q^{47} - 26 q^{49} + 180 q^{55} + 108 q^{63} - 60 q^{65} + 162 q^{81} - 36 q^{89} - 20 q^{95}+O(q^{100}) \) 2 * q - 12 * q^7 - 18 * q^9 + 52 * q^23 - 50 * q^25 + 156 * q^41 + 172 * q^47 - 26 * q^49 + 180 * q^55 + 108 * q^63 - 60 * q^65 + 162 * q^81 - 36 * q^89 - 20 * q^95

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