LMFDB - Newform orbit 2600.2.d.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2600,2,Mod(1249,2600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2600, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2600.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2600 = 2^{3} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2600.d (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:	$20.7611045255$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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 + i q^{3} - 5 i q^{7} + 2 q^{9} - 2 q^{11} - i q^{13} + 3 i q^{17} + 2 q^{19} + 5 q^{21} + 4 i q^{23} + 5 i q^{27} + 6 q^{29} - 4 q^{31} - 2 i q^{33} - 11 i q^{37} + q^{39} + 8 q^{41} - i q^{43} - 9 i q^{47} + \cdots - 4 q^{99} +O(q^{100}) $ q + i * q^3 - 5i q^7 + 2 * q^9 - 2 * q^11 - i * q^13 + 3i q^17 + 2 * q^19 + 5 * q^21 + 4i q^23 + 5i q^27 + 6 * q^29 - 4 * q^31 - 2i q^33 - 11i q^37 + q^39 + 8 * q^41 - i * q^43 - 9i q^47 - 18 * q^49 - 3 * q^51 - 12i q^53 + 2i q^57 - 6 * q^59 - 10i q^63 - 6i q^67 - 4 * q^69 + 7 * q^71 - 2i q^73 + 10i q^77 - 12 * q^79 + q^81 - 16i q^83 + 6i q^87 + 10 * q^89 - 5 * q^91 - 4i q^93 + 10i q^97 - 4 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{9} - 4 q^{11} + 4 q^{19} + 10 q^{21} + 12 q^{29} - 8 q^{31} + 2 q^{39} + 16 q^{41} - 36 q^{49} - 6 q^{51} - 12 q^{59} - 8 q^{69} + 14 q^{71} - 24 q^{79} + 2 q^{81} + 20 q^{89} - 10 q^{91} - 8 q^{99}+O(q^{100}) $ 2 * q + 4 * q^9 - 4 * q^11 + 4 * q^19 + 10 * q^21 + 12 * q^29 - 8 * q^31 + 2 * q^39 + 16 * q^41 - 36 * q^49 - 6 * q^51 - 12 * q^59 - 8 * q^69 + 14 * q^71 - 24 * q^79 + 2 * q^81 + 20 * q^89 - 10 * q^91 - 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2600\mathbb{Z}\right)^\times$.

$n$	$1301$	$1601$	$1951$	$1977$
$\chi(n)$	$1$	$1$	$1$	$-1$

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} $

1249.1

	−	1.00000i
		1.00000i

−

1.00000i

5.00000i

2.00000

1249.2

1.00000i

−

5.00000i

2.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2600.2.d.f		2
5.b	even	2	1	inner	2600.2.d.f		2
5.c	odd	4	1		104.2.a.a	✓	1
5.c	odd	4	1		2600.2.a.e		1
15.e	even	4	1		936.2.a.f		1
20.e	even	4	1		208.2.a.b		1
20.e	even	4	1		5200.2.a.bb		1
35.f	even	4	1		5096.2.a.c		1
40.i	odd	4	1		832.2.a.c		1
40.k	even	4	1		832.2.a.h		1
60.l	odd	4	1		1872.2.a.l		1
65.f	even	4	1		1352.2.f.b		2
65.h	odd	4	1		1352.2.a.b		1
65.k	even	4	1		1352.2.f.b		2
65.o	even	12	2		1352.2.o.a		4
65.q	odd	12	2		1352.2.i.b		2
65.r	odd	12	2		1352.2.i.c		2
65.t	even	12	2		1352.2.o.a		4
80.i	odd	4	1		3328.2.b.a		2
80.j	even	4	1		3328.2.b.t		2
80.s	even	4	1		3328.2.b.t		2
80.t	odd	4	1		3328.2.b.a		2
120.q	odd	4	1		7488.2.a.u		1
120.w	even	4	1		7488.2.a.x		1
260.l	odd	4	1		2704.2.f.e		2
260.p	even	4	1		2704.2.a.d		1
260.s	odd	4	1		2704.2.f.e		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
104.2.a.a	✓	1	5.c	odd	4	1
208.2.a.b		1	20.e	even	4	1
832.2.a.c		1	40.i	odd	4	1
832.2.a.h		1	40.k	even	4	1
936.2.a.f		1	15.e	even	4	1
1352.2.a.b		1	65.h	odd	4	1
1352.2.f.b		2	65.f	even	4	1
1352.2.f.b		2	65.k	even	4	1
1352.2.i.b		2	65.q	odd	12	2
1352.2.i.c		2	65.r	odd	12	2
1352.2.o.a		4	65.o	even	12	2
1352.2.o.a		4	65.t	even	12	2
1872.2.a.l		1	60.l	odd	4	1
2600.2.a.e		1	5.c	odd	4	1
2600.2.d.f		2	1.a	even	1	1	trivial
2600.2.d.f		2	5.b	even	2	1	inner
2704.2.a.d		1	260.p	even	4	1
2704.2.f.e		2	260.l	odd	4	1
2704.2.f.e		2	260.s	odd	4	1
3328.2.b.a		2	80.i	odd	4	1
3328.2.b.a		2	80.t	odd	4	1
3328.2.b.t		2	80.j	even	4	1
3328.2.b.t		2	80.s	even	4	1
5096.2.a.c		1	35.f	even	4	1
5200.2.a.bb		1	20.e	even	4	1
7488.2.a.u		1	120.q	odd	4	1
7488.2.a.x		1	120.w	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(2600, [\chi])$:

$ T_{3}^{2} + 1 $

$ T_{7}^{2} + 25 $

$ T_{11} + 2 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} $ T^2
$3$	$ T^{2} + 1 $ T^2 + 1
$5$	$ T^{2} $ T^2
$7$	$ T^{2} + 25 $ T^2 + 25
$11$	$ (T + 2)^{2} $ (T + 2)^2
$13$	$ T^{2} + 1 $ T^2 + 1
$17$	$ T^{2} + 9 $ T^2 + 9
$19$	$ (T - 2)^{2} $ (T - 2)^2
$23$	$ T^{2} + 16 $ T^2 + 16
$29$	$ (T - 6)^{2} $ (T - 6)^2
$31$	$ (T + 4)^{2} $ (T + 4)^2
$37$	$ T^{2} + 121 $ T^2 + 121
$41$	$ (T - 8)^{2} $ (T - 8)^2
$43$	$ T^{2} + 1 $ T^2 + 1
$47$	$ T^{2} + 81 $ T^2 + 81
$53$	$ T^{2} + 144 $ T^2 + 144
$59$	$ (T + 6)^{2} $ (T + 6)^2
$61$	$ T^{2} $ T^2
$67$	$ T^{2} + 36 $ T^2 + 36
$71$	$ (T - 7)^{2} $ (T - 7)^2
$73$	$ T^{2} + 4 $ T^2 + 4
$79$	$ (T + 12)^{2} $ (T + 12)^2
$83$	$ T^{2} + 256 $ T^2 + 256
$89$	$ (T - 10)^{2} $ (T - 10)^2
$97$	$ T^{2} + 100 $ T^2 + 100
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Elliptic curves over $\Q$
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Level:	\( N \)	\(=\)	\( 2600 = 2^{3} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2600.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + i q^{3} - 5 i q^{7} + 2 q^{9} - 2 q^{11} - i q^{13} + 3 i q^{17} + 2 q^{19} + 5 q^{21} + 4 i q^{23} + 5 i q^{27} + 6 q^{29} - 4 q^{31} - 2 i q^{33} - 11 i q^{37} + q^{39} + 8 q^{41} - i q^{43} - 9 i q^{47} + \cdots - 4 q^{99} +O(q^{100}) \) q + i * q^3 - 5i q^7 + 2 * q^9 - 2 * q^11 - i * q^13 + 3i q^17 + 2 * q^19 + 5 * q^21 + 4i q^23 + 5i q^27 + 6 * q^29 - 4 * q^31 - 2i q^33 - 11i q^37 + q^39 + 8 * q^41 - i * q^43 - 9i q^47 - 18 * q^49 - 3 * q^51 - 12i q^53 + 2i q^57 - 6 * q^59 - 10i q^63 - 6i q^67 - 4 * q^69 + 7 * q^71 - 2i q^73 + 10i q^77 - 12 * q^79 + q^81 - 16i q^83 + 6i q^87 + 10 * q^89 - 5 * q^91 - 4i q^93 + 10i q^97 - 4 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{9} - 4 q^{11} + 4 q^{19} + 10 q^{21} + 12 q^{29} - 8 q^{31} + 2 q^{39} + 16 q^{41} - 36 q^{49} - 6 q^{51} - 12 q^{59} - 8 q^{69} + 14 q^{71} - 24 q^{79} + 2 q^{81} + 20 q^{89} - 10 q^{91} - 8 q^{99}+O(q^{100}) \) 2 * q + 4 * q^9 - 4 * q^11 + 4 * q^19 + 10 * q^21 + 12 * q^29 - 8 * q^31 + 2 * q^39 + 16 * q^41 - 36 * q^49 - 6 * q^51 - 12 * q^59 - 8 * q^69 + 14 * q^71 - 24 * q^79 + 2 * q^81 + 20 * q^89 - 10 * q^91 - 8 * q^99

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