LMFDB - Newform orbit 1008.2.k.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,2,Mod(881,1008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1008.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1008.k (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:	$8.04892052375$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 8x^{2} + 9 $ x^4 + 8*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 63)
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 a basis $1,\beta_1,\beta_2,\beta_3$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{2} q^{7}+O(q^{10}) $ q - b2 * q^7	$ q - \beta_{2} q^{7} + ( - \beta_{3} + \beta_1) q^{11} + (\beta_{3} + \beta_1) q^{23} - 5 q^{25} + ( - \beta_{3} + 2 \beta_1) q^{29} - 4 \beta_{2} q^{37} - 2 \beta_{2} q^{43} + 7 q^{49} + (\beta_{3} + 2 \beta_1) q^{53} + 4 q^{67} + ( - 3 \beta_{3} + \beta_1) q^{71} + ( - 3 \beta_{3} + 2 \beta_1) q^{77} - 8 q^{79}+O(q^{100}) $ q - b2 * q^7 + (-b3 + b1) * q^11 + (b3 + b1) * q^23 - 5 * q^25 + (-b3 + 2b1) q^29 - 4b2 q^37 - 2b2 q^43 + 7 * q^49 + (b3 + 2b1) q^53 + 4 * q^67 + (-3b3 + b1) q^71 + (-3b3 + 2b1) * q^77 - 8 * q^79
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 20 q^{25} + 28 q^{49} + 16 q^{67} - 32 q^{79}+O(q^{100}) $ 4 * q - 20 * q^25 + 28 * q^49 + 16 * q^67 - 32 * q^79

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{4} + 8x^{2} + 9 $ x^4 + 8*x^2 + 9 :

$\beta_{1}$	$=$	$ ( -2\nu^{3} - 16\nu ) / 3 $ (-2v^3 - 16v) / 3
$\beta_{2}$	$=$	$ \nu^{2} + 4 $ v^2 + 4
$\beta_{3}$	$=$	$ -\nu^{3} - 5\nu $ -v^3 - 5*v

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ ( 2\beta_{3} - 3\beta_1 ) / 6 $ (2b3 - 3b1) / 6
$\nu^{2}$	$=$	$ \beta_{2} - 4 $ b2 - 4
$\nu^{3}$	$=$	$ ( -16\beta_{3} + 15\beta_1 ) / 6 $ (-16b3 + 15b1) / 6

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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} $

881.1

		1.16372i
	−	1.16372i
		2.57794i
	−	2.57794i

−2.64575

881.2

−2.64575

881.3

2.64575

881.4

2.64575

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by $\Q(\sqrt{-7}) $
3.b	odd	2	1	inner
21.c	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1008.2.k.a		4
3.b	odd	2	1	inner	1008.2.k.a		4
4.b	odd	2	1		63.2.c.a	✓	4
7.b	odd	2	1	CM	1008.2.k.a		4
8.b	even	2	1		4032.2.k.b		4
8.d	odd	2	1		4032.2.k.c		4
12.b	even	2	1		63.2.c.a	✓	4
20.d	odd	2	1		1575.2.b.a		4
20.e	even	4	2		1575.2.g.d		8
21.c	even	2	1	inner	1008.2.k.a		4
24.f	even	2	1		4032.2.k.c		4
24.h	odd	2	1		4032.2.k.b		4
28.d	even	2	1		63.2.c.a	✓	4
28.f	even	6	2		441.2.p.b		8
28.g	odd	6	2		441.2.p.b		8
36.f	odd	6	2		567.2.o.f		8
36.h	even	6	2		567.2.o.f		8
56.e	even	2	1		4032.2.k.c		4
56.h	odd	2	1		4032.2.k.b		4
60.h	even	2	1		1575.2.b.a		4
60.l	odd	4	2		1575.2.g.d		8
84.h	odd	2	1		63.2.c.a	✓	4
84.j	odd	6	2		441.2.p.b		8
84.n	even	6	2		441.2.p.b		8
140.c	even	2	1		1575.2.b.a		4
140.j	odd	4	2		1575.2.g.d		8
168.e	odd	2	1		4032.2.k.c		4
168.i	even	2	1		4032.2.k.b		4
252.s	odd	6	2		567.2.o.f		8
252.bi	even	6	2		567.2.o.f		8
420.o	odd	2	1		1575.2.b.a		4
420.w	even	4	2		1575.2.g.d		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
63.2.c.a	✓	4	4.b	odd	2	1
63.2.c.a	✓	4	12.b	even	2	1
63.2.c.a	✓	4	28.d	even	2	1
63.2.c.a	✓	4	84.h	odd	2	1
441.2.p.b		8	28.f	even	6	2
441.2.p.b		8	28.g	odd	6	2
441.2.p.b		8	84.j	odd	6	2
441.2.p.b		8	84.n	even	6	2
567.2.o.f		8	36.f	odd	6	2
567.2.o.f		8	36.h	even	6	2
567.2.o.f		8	252.s	odd	6	2
567.2.o.f		8	252.bi	even	6	2
1008.2.k.a		4	1.a	even	1	1	trivial
1008.2.k.a		4	3.b	odd	2	1	inner
1008.2.k.a		4	7.b	odd	2	1	CM
1008.2.k.a		4	21.c	even	2	1	inner
1575.2.b.a		4	20.d	odd	2	1
1575.2.b.a		4	60.h	even	2	1
1575.2.b.a		4	140.c	even	2	1
1575.2.b.a		4	420.o	odd	2	1
1575.2.g.d		8	20.e	even	4	2
1575.2.g.d		8	60.l	odd	4	2
1575.2.g.d		8	140.j	odd	4	2
1575.2.g.d		8	420.w	even	4	2
4032.2.k.b		4	8.b	even	2	1
4032.2.k.b		4	24.h	odd	2	1
4032.2.k.b		4	56.h	odd	2	1
4032.2.k.b		4	168.i	even	2	1
4032.2.k.c		4	8.d	odd	2	1
4032.2.k.c		4	24.f	even	2	1
4032.2.k.c		4	56.e	even	2	1
4032.2.k.c		4	168.e	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5} $ T5 acting on $S_{2}^{\mathrm{new}}(1008, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{4} $ T^4
$3$	$ T^{4} $ T^4
$5$	$ T^{4} $ T^4
$7$	$ (T^{2} - 7)^{2} $ (T^2 - 7)^2
$11$	$ T^{4} + 44T^{2} + 36 $ T^4 + 44*T^2 + 36
$13$	$ T^{4} $ T^4
$17$	$ T^{4} $ T^4
$19$	$ T^{4} $ T^4
$23$	$ T^{4} + 92T^{2} + 324 $ T^4 + 92*T^2 + 324
$29$	$ T^{4} + 116T^{2} + 2916 $ T^4 + 116*T^2 + 2916
$31$	$ T^{4} $ T^4
$37$	$ (T^{2} - 112)^{2} $ (T^2 - 112)^2
$41$	$ T^{4} $ T^4
$43$	$ (T^{2} - 28)^{2} $ (T^2 - 28)^2
$47$	$ T^{4} $ T^4
$53$	$ T^{4} + 212T^{2} + 36 $ T^4 + 212*T^2 + 36
$59$	$ T^{4} $ T^4
$61$	$ T^{4} $ T^4
$67$	$ (T - 4)^{4} $ (T - 4)^4
$71$	$ T^{4} + 284 T^{2} + 12996 $ T^4 + 284*T^2 + 12996
$73$	$ T^{4} $ T^4
$79$	$ (T + 8)^{4} $ (T + 8)^4
$83$	$ T^{4} $ T^4
$89$	$ T^{4} $ T^4
$97$	$ T^{4} $ T^4
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Elliptic curves over $\Q$
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Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1008.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q - \beta_{2} q^{7}+O(q^{10}) \) q - b2 * q^7	\( q - \beta_{2} q^{7} + ( - \beta_{3} + \beta_1) q^{11} + (\beta_{3} + \beta_1) q^{23} - 5 q^{25} + ( - \beta_{3} + 2 \beta_1) q^{29} - 4 \beta_{2} q^{37} - 2 \beta_{2} q^{43} + 7 q^{49} + (\beta_{3} + 2 \beta_1) q^{53} + 4 q^{67} + ( - 3 \beta_{3} + \beta_1) q^{71} + ( - 3 \beta_{3} + 2 \beta_1) q^{77} - 8 q^{79}+O(q^{100}) \) q - b2 * q^7 + (-b3 + b1) * q^11 + (b3 + b1) * q^23 - 5 * q^25 + (-b3 + 2b1) q^29 - 4b2 q^37 - 2b2 q^43 + 7 * q^49 + (b3 + 2b1) q^53 + 4 * q^67 + (-3b3 + b1) q^71 + (-3b3 + 2b1) * q^77 - 8 * q^79
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 20 q^{25} + 28 q^{49} + 16 q^{67} - 32 q^{79}+O(q^{100}) \) 4 * q - 20 * q^25 + 28 * q^49 + 16 * q^67 - 32 * q^79

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