LMFDB - Newform orbit 576.4.d.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [576,4,Mod(289,576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(576, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("576.289");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	576.d (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:	no
Analytic conductor:	$33.9851001633$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{4} $
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)

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} q^{13} - 7 \beta_1 q^{19} + 125 q^{25} + 5 \beta_{2} q^{31} - 7 \beta_{3} q^{37} - 65 \beta_1 q^{43} + 629 q^{49} + 15 \beta_{3} q^{61} - 110 \beta_1 q^{67} + 1190 q^{73} - 35 \beta_{2} q^{79} - 243 \beta_1 q^{91} + 1330 q^{97}+O(q^{100}) $ q - b2 * q^7 + b3 * q^13 - 7b1 q^19 + 125 * q^25 + 5b2 q^31 - 7b3 q^37 - 65b1 q^43 + 629 * q^49 + 15b3 q^61 - 110b1 q^67 + 1190 * q^73 - 35b2 q^79 - 243b1 q^91 + 1330 * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 500 q^{25} + 2516 q^{49} + 4760 q^{73} + 5320 q^{97}+O(q^{100}) $ 4 * q + 500 * q^25 + 2516 * q^49 + 4760 * q^73 + 5320 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring

$\beta_{1}$	$=$	$ 8\zeta_{12}^{3} $ 8*v^3
$\beta_{2}$	$=$	$ -18\zeta_{12}^{3} + 36\zeta_{12} $ -18v^3 + 36v
$\beta_{3}$	$=$	$ 72\zeta_{12}^{2} - 36 $ 72*v^2 - 36

Display $\zeta_{12}^j$ in terms of $\beta_i$

$\zeta_{12}$	$=$	$ ( 4\beta_{2} + 9\beta_1 ) / 144 $ (4b2 + 9b1) / 144
$\zeta_{12}^{2}$	$=$	$ ( \beta_{3} + 36 ) / 72 $ (b3 + 36) / 72
$\zeta_{12}^{3}$	$=$	$ ( \beta_1 ) / 8 $ (b1) / 8

Display $\beta_i$ in terms of $\zeta_{12}^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$325$
$\chi(n)$	$1$	$1$	$-1$

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} $

289.1

0.866025	−	0.500000i
0.866025	+	0.500000i
−0.866025	+	0.500000i
−0.866025	−	0.500000i

−31.1769

289.2

−31.1769

289.3

31.1769

289.4

31.1769

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3}) $
4.b	odd	2	1	inner
8.b	even	2	1	inner
8.d	odd	2	1	inner
12.b	even	2	1	inner
24.f	even	2	1	inner
24.h	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	576.4.d.d	✓	4
3.b	odd	2	1	CM	576.4.d.d	✓	4
4.b	odd	2	1	inner	576.4.d.d	✓	4
8.b	even	2	1	inner	576.4.d.d	✓	4
8.d	odd	2	1	inner	576.4.d.d	✓	4
12.b	even	2	1	inner	576.4.d.d	✓	4
16.e	even	4	1		2304.4.a.bc		2
16.e	even	4	1		2304.4.a.bd		2
16.f	odd	4	1		2304.4.a.bc		2
16.f	odd	4	1		2304.4.a.bd		2
24.f	even	2	1	inner	576.4.d.d	✓	4
24.h	odd	2	1	inner	576.4.d.d	✓	4
48.i	odd	4	1		2304.4.a.bc		2
48.i	odd	4	1		2304.4.a.bd		2
48.k	even	4	1		2304.4.a.bc		2
48.k	even	4	1		2304.4.a.bd		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
576.4.d.d	✓	4	1.a	even	1	1	trivial
576.4.d.d	✓	4	3.b	odd	2	1	CM
576.4.d.d	✓	4	4.b	odd	2	1	inner
576.4.d.d	✓	4	8.b	even	2	1	inner
576.4.d.d	✓	4	8.d	odd	2	1	inner
576.4.d.d	✓	4	12.b	even	2	1	inner
576.4.d.d	✓	4	24.f	even	2	1	inner
576.4.d.d	✓	4	24.h	odd	2	1	inner
2304.4.a.bc		2	16.e	even	4	1
2304.4.a.bc		2	16.f	odd	4	1
2304.4.a.bc		2	48.i	odd	4	1
2304.4.a.bc		2	48.k	even	4	1
2304.4.a.bd		2	16.e	even	4	1
2304.4.a.bd		2	16.f	odd	4	1
2304.4.a.bd		2	48.i	odd	4	1
2304.4.a.bd		2	48.k	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(576, [\chi])$:

$ T_{5} $

$ T_{7}^{2} - 972 $

$ T_{17} $

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} - 972)^{2} $ (T^2 - 972)^2
$11$	$ T^{4} $ T^4
$13$	$ (T^{2} + 3888)^{2} $ (T^2 + 3888)^2
$17$	$ T^{4} $ T^4
$19$	$ (T^{2} + 3136)^{2} $ (T^2 + 3136)^2
$23$	$ T^{4} $ T^4
$29$	$ T^{4} $ T^4
$31$	$ (T^{2} - 24300)^{2} $ (T^2 - 24300)^2
$37$	$ (T^{2} + 190512)^{2} $ (T^2 + 190512)^2
$41$	$ T^{4} $ T^4
$43$	$ (T^{2} + 270400)^{2} $ (T^2 + 270400)^2
$47$	$ T^{4} $ T^4
$53$	$ T^{4} $ T^4
$59$	$ T^{4} $ T^4
$61$	$ (T^{2} + 874800)^{2} $ (T^2 + 874800)^2
$67$	$ (T^{2} + 774400)^{2} $ (T^2 + 774400)^2
$71$	$ T^{4} $ T^4
$73$	$ (T - 1190)^{4} $ (T - 1190)^4
$79$	$ (T^{2} - 1190700)^{2} $ (T^2 - 1190700)^2
$83$	$ T^{4} $ T^4
$89$	$ T^{4} $ T^4
$97$	$ (T - 1330)^{4} $ (T - 1330)^4
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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:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	576.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{2} q^{7}+O(q^{10}) \) q - b2 * q^7	\( q - \beta_{2} q^{7} + \beta_{3} q^{13} - 7 \beta_1 q^{19} + 125 q^{25} + 5 \beta_{2} q^{31} - 7 \beta_{3} q^{37} - 65 \beta_1 q^{43} + 629 q^{49} + 15 \beta_{3} q^{61} - 110 \beta_1 q^{67} + 1190 q^{73} - 35 \beta_{2} q^{79} - 243 \beta_1 q^{91} + 1330 q^{97}+O(q^{100}) \) q - b2 * q^7 + b3 * q^13 - 7b1 q^19 + 125 * q^25 + 5b2 q^31 - 7b3 q^37 - 65b1 q^43 + 629 * q^49 + 15b3 q^61 - 110b1 q^67 + 1190 * q^73 - 35b2 q^79 - 243b1 q^91 + 1330 * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 500 q^{25} + 2516 q^{49} + 4760 q^{73} + 5320 q^{97}+O(q^{100}) \) 4 * q + 500 * q^25 + 2516 * q^49 + 4760 * q^73 + 5320 * q^97

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