LMFDB - Newform orbit 1280.3.e.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,3,Mod(639,1280)]

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, 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("1280.639");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1280 = 2^{8} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1280.e (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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 20)
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 + 4 i q^{3} + 5 i q^{5} + 4 q^{7} - 7 q^{9}+O(q^{10}) $ q + 4i q^3 + 5i q^5 + 4 * q^7 - 7 * q^9	$ q + 4 i q^{3} + 5 i q^{5} + 4 q^{7} - 7 q^{9} - 20 q^{15} + 16 i q^{21} - 44 q^{23} - 25 q^{25} + 8 i q^{27} - 22 i q^{29} + 20 i q^{35} - 62 q^{41} + 76 i q^{43} - 35 i q^{45} - 4 q^{47} - 33 q^{49} - 58 i q^{61} - 28 q^{63} + 116 i q^{67} - 176 i q^{69} - 100 i q^{75} - 95 q^{81} - 76 i q^{83} + 88 q^{87} + 142 q^{89} +O(q^{100}) $ q + 4i q^3 + 5i q^5 + 4 * q^7 - 7 * q^9 - 20 * q^15 + 16i q^21 - 44 * q^23 - 25 * q^25 + 8i q^27 - 22i q^29 + 20i q^35 - 62 * q^41 + 76i q^43 - 35i q^45 - 4 * q^47 - 33 * q^49 - 58i q^61 - 28 * q^63 + 116i q^67 - 176i q^69 - 100i q^75 - 95 * q^81 - 76i q^83 + 88 * q^87 + 142 * q^89
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{7} - 14 q^{9}+O(q^{10}) $ 2 * q + 8 * q^7 - 14 * q^9	$ 2 q + 8 q^{7} - 14 q^{9} - 40 q^{15} - 88 q^{23} - 50 q^{25} - 124 q^{41} - 8 q^{47} - 66 q^{49} - 56 q^{63} - 190 q^{81} + 176 q^{87} + 284 q^{89}+O(q^{100}) $ 2 * q + 8 * q^7 - 14 * q^9 - 40 * q^15 - 88 * q^23 - 50 * q^25 - 124 * q^41 - 8 * q^47 - 66 * q^49 - 56 * q^63 - 190 * q^81 + 176 * q^87 + 284 * q^89

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} $

639.1

	−	1.00000i
		1.00000i

−

4.00000i

−

5.00000i

4.00000

−7.00000

639.2

4.00000i

5.00000i

4.00000

−7.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
20.d	odd	2	1	CM by $\Q(\sqrt{-5}) $
8.b	even	2	1	inner
40.e	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.e.c		2
4.b	odd	2	1		1280.3.e.b		2
5.b	even	2	1		1280.3.e.b		2
8.b	even	2	1	inner	1280.3.e.c		2
8.d	odd	2	1		1280.3.e.b		2
16.e	even	4	1		20.3.d.a	✓	1
16.e	even	4	1		320.3.h.a		1
16.f	odd	4	1		20.3.d.b	yes	1
16.f	odd	4	1		320.3.h.b		1
20.d	odd	2	1	CM	1280.3.e.c		2
40.e	odd	2	1	inner	1280.3.e.c		2
40.f	even	2	1		1280.3.e.b		2
48.i	odd	4	1		180.3.f.b		1
48.k	even	4	1		180.3.f.a		1
80.i	odd	4	1		100.3.b.c		2
80.i	odd	4	1		1600.3.b.f		2
80.j	even	4	1		100.3.b.c		2
80.j	even	4	1		1600.3.b.f		2
80.k	odd	4	1		20.3.d.a	✓	1
80.k	odd	4	1		320.3.h.a		1
80.q	even	4	1		20.3.d.b	yes	1
80.q	even	4	1		320.3.h.b		1
80.s	even	4	1		100.3.b.c		2
80.s	even	4	1		1600.3.b.f		2
80.t	odd	4	1		100.3.b.c		2
80.t	odd	4	1		1600.3.b.f		2
240.t	even	4	1		180.3.f.b		1
240.z	odd	4	1		900.3.c.h		2
240.bb	even	4	1		900.3.c.h		2
240.bd	odd	4	1		900.3.c.h		2
240.bf	even	4	1		900.3.c.h		2
240.bm	odd	4	1		180.3.f.a		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
20.3.d.a	✓	1	16.e	even	4	1
20.3.d.a	✓	1	80.k	odd	4	1
20.3.d.b	yes	1	16.f	odd	4	1
20.3.d.b	yes	1	80.q	even	4	1
100.3.b.c		2	80.i	odd	4	1
100.3.b.c		2	80.j	even	4	1
100.3.b.c		2	80.s	even	4	1
100.3.b.c		2	80.t	odd	4	1
180.3.f.a		1	48.k	even	4	1
180.3.f.a		1	240.bm	odd	4	1
180.3.f.b		1	48.i	odd	4	1
180.3.f.b		1	240.t	even	4	1
320.3.h.a		1	16.e	even	4	1
320.3.h.a		1	80.k	odd	4	1
320.3.h.b		1	16.f	odd	4	1
320.3.h.b		1	80.q	even	4	1
900.3.c.h		2	240.z	odd	4	1
900.3.c.h		2	240.bb	even	4	1
900.3.c.h		2	240.bd	odd	4	1
900.3.c.h		2	240.bf	even	4	1
1280.3.e.b		2	4.b	odd	2	1
1280.3.e.b		2	5.b	even	2	1
1280.3.e.b		2	8.d	odd	2	1
1280.3.e.b		2	40.f	even	2	1
1280.3.e.c		2	1.a	even	1	1	trivial
1280.3.e.c		2	8.b	even	2	1	inner
1280.3.e.c		2	20.d	odd	2	1	CM
1280.3.e.c		2	40.e	odd	2	1	inner
1600.3.b.f		2	80.i	odd	4	1
1600.3.b.f		2	80.j	even	4	1
1600.3.b.f		2	80.s	even	4	1
1600.3.b.f		2	80.t	odd	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}^{2} + 16 $

$ T_{7} - 4 $

$ T_{11} $

$ T_{13} $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} $ T^2
$3$	$ T^{2} + 16 $ T^2 + 16
$5$	$ T^{2} + 25 $ T^2 + 25
$7$	$ (T - 4)^{2} $ (T - 4)^2
$11$	$ T^{2} $ T^2
$13$	$ T^{2} $ T^2
$17$	$ T^{2} $ T^2
$19$	$ T^{2} $ T^2
$23$	$ (T + 44)^{2} $ (T + 44)^2
$29$	$ T^{2} + 484 $ T^2 + 484
$31$	$ T^{2} $ T^2
$37$	$ T^{2} $ T^2
$41$	$ (T + 62)^{2} $ (T + 62)^2
$43$	$ T^{2} + 5776 $ T^2 + 5776
$47$	$ (T + 4)^{2} $ (T + 4)^2
$53$	$ T^{2} $ T^2
$59$	$ T^{2} $ T^2
$61$	$ T^{2} + 3364 $ T^2 + 3364
$67$	$ T^{2} + 13456 $ T^2 + 13456
$71$	$ T^{2} $ T^2
$73$	$ T^{2} $ T^2
$79$	$ T^{2} $ T^2
$83$	$ T^{2} + 5776 $ T^2 + 5776
$89$	$ (T - 142)^{2} $ (T - 142)^2
$97$	$ T^{2} $ T^2
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Elliptic curves over $\Q$
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Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\( q + 4 i q^{3} + 5 i q^{5} + 4 q^{7} - 7 q^{9}+O(q^{10}) \) q + 4i q^3 + 5i q^5 + 4 * q^7 - 7 * q^9	\( q + 4 i q^{3} + 5 i q^{5} + 4 q^{7} - 7 q^{9} - 20 q^{15} + 16 i q^{21} - 44 q^{23} - 25 q^{25} + 8 i q^{27} - 22 i q^{29} + 20 i q^{35} - 62 q^{41} + 76 i q^{43} - 35 i q^{45} - 4 q^{47} - 33 q^{49} - 58 i q^{61} - 28 q^{63} + 116 i q^{67} - 176 i q^{69} - 100 i q^{75} - 95 q^{81} - 76 i q^{83} + 88 q^{87} + 142 q^{89} +O(q^{100}) \) q + 4i q^3 + 5i q^5 + 4 * q^7 - 7 * q^9 - 20 * q^15 + 16i q^21 - 44 * q^23 - 25 * q^25 + 8i q^27 - 22i q^29 + 20i q^35 - 62 * q^41 + 76i q^43 - 35i q^45 - 4 * q^47 - 33 * q^49 - 58i q^61 - 28 * q^63 + 116i q^67 - 176i q^69 - 100i q^75 - 95 * q^81 - 76i q^83 + 88 * q^87 + 142 * q^89
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{7} - 14 q^{9}+O(q^{10}) \) 2 * q + 8 * q^7 - 14 * q^9	\( 2 q + 8 q^{7} - 14 q^{9} - 40 q^{15} - 88 q^{23} - 50 q^{25} - 124 q^{41} - 8 q^{47} - 66 q^{49} - 56 q^{63} - 190 q^{81} + 176 q^{87} + 284 q^{89}+O(q^{100}) \) 2 * q + 8 * q^7 - 14 * q^9 - 40 * q^15 - 88 * q^23 - 50 * q^25 - 124 * q^41 - 8 * q^47 - 66 * q^49 - 56 * q^63 - 190 * q^81 + 176 * q^87 + 284 * q^89

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