LMFDB - Newform orbit 4608.2.a.bb

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4608,2,Mod(1,4608)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4608, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4608.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$4608 = 2^{9} \cdot 3^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	4608.a (trivial)

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:	yes
Analytic conductor:	$36.7950652514$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - 4x^{2} + 1$ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$2^{3}$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$ -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.

$f(q)$	$=$	$q + \beta_1 q^{5} + \beta_1 q^{7} + 2 q^{11} - \beta_{2} q^{13} + \beta_{3} q^{17} + \beta_{3} q^{19} - 2 \beta_{2} q^{23} + q^{25} + \beta_1 q^{29} + 3 \beta_1 q^{31} + 6 q^{35} + 3 \beta_{2} q^{37}+ \cdots - 6 q^{97}+O(q^{100})$ q + b1 * q^5 + b1 * q^7 + 2 * q^11 - b2 * q^13 + b3 * q^17 + b3 * q^19 - 2b2 q^23 + q^25 + b1 * q^29 + 3b1 q^31 + 6 * q^35 + 3b2 q^37 - 3b3 q^41 - b3 * q^43 + 2b2 q^47 - q^49 + b1 * q^53 + 2b1 q^55 + 8 * q^59 - b2 * q^61 + 3b3 q^65 - 4b3 q^67 + 4b2 q^71 + 2b1 q^77 + b1 * q^79 + 14 * q^83 - 2b2 q^85 - 4b3 q^89 + 3b3 q^91 - 2b2 q^95 - 6 * q^97
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 8 q^{11} + 4 q^{25} + 24 q^{35} - 4 q^{49} + 32 q^{59} + 56 q^{83} - 24 q^{97}+O(q^{100})$ 4 * q + 8 * q^11 + 4 * q^25 + 24 * q^35 - 4 * q^49 + 32 * q^59 + 56 * q^83 - 24 * q^97

Basis of coefficient ring in terms of $\nu = \zeta_{24} + \zeta_{24}^{-1}$ :

$\beta_{1}$	$=$	$\nu^{3} - 5\nu$ v^3 - 5*v
$\beta_{2}$	$=$	$2\nu^{2} - 4$ 2*v^2 - 4
$\beta_{3}$	$=$	$2\nu^{3} - 6\nu$ 2v^3 - 6v

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

$\nu$	$=$	$( \beta_{3} - 2\beta_1 ) / 4$ (b3 - 2*b1) / 4
$\nu^{2}$	$=$	$( \beta_{2} + 4 ) / 2$ (b2 + 4) / 2
$\nu^{3}$	$=$	$( 5\beta_{3} - 6\beta_1 ) / 4$ (5b3 - 6b1) / 4

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

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}

1.1

1.93185
0.517638
−1.93185
−0.517638

−2.44949

1.2

−2.44949

1.3

2.44949

1.4

2.44949

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$3$	$+1$

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	inner
12.b	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	4608.2.a.bb	yes	4
3.b	odd	2	1		4608.2.a.u	✓	4
4.b	odd	2	1		4608.2.a.u	✓	4
8.b	even	2	1		4608.2.a.u	✓	4
8.d	odd	2	1	inner	4608.2.a.bb	yes	4
12.b	even	2	1	inner	4608.2.a.bb	yes	4
16.e	even	4	2		4608.2.d.r		8
16.f	odd	4	2		4608.2.d.r		8
24.f	even	2	1		4608.2.a.u	✓	4
24.h	odd	2	1	inner	4608.2.a.bb	yes	4
48.i	odd	4	2		4608.2.d.r		8
48.k	even	4	2		4608.2.d.r		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
4608.2.a.u	✓	4	3.b	odd	2	1
4608.2.a.u	✓	4	4.b	odd	2	1
4608.2.a.u	✓	4	8.b	even	2	1
4608.2.a.u	✓	4	24.f	even	2	1
4608.2.a.bb	yes	4	1.a	even	1	1	trivial
4608.2.a.bb	yes	4	8.d	odd	2	1	inner
4608.2.a.bb	yes	4	12.b	even	2	1	inner
4608.2.a.bb	yes	4	24.h	odd	2	1	inner
4608.2.d.r		8	16.e	even	4	2
4608.2.d.r		8	16.f	odd	4	2
4608.2.d.r		8	48.i	odd	4	2
4608.2.d.r		8	48.k	even	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(4608))$ :

T_{5}^{2} - 6

T_{7}^{2} - 6

T_{11} - 2

T_{17}^{2} - 8

T_{23}^{2} - 48

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{4}$ T^4
$3$	$T^{4}$ T^4
$5$	$(T^{2} - 6)^{2}$ (T^2 - 6)^2
$7$	$(T^{2} - 6)^{2}$ (T^2 - 6)^2
$11$	$(T - 2)^{4}$ (T - 2)^4
$13$	$(T^{2} - 12)^{2}$ (T^2 - 12)^2
$17$	$(T^{2} - 8)^{2}$ (T^2 - 8)^2
$19$	$(T^{2} - 8)^{2}$ (T^2 - 8)^2
$23$	$(T^{2} - 48)^{2}$ (T^2 - 48)^2
$29$	$(T^{2} - 6)^{2}$ (T^2 - 6)^2
$31$	$(T^{2} - 54)^{2}$ (T^2 - 54)^2
$37$	$(T^{2} - 108)^{2}$ (T^2 - 108)^2
$41$	$(T^{2} - 72)^{2}$ (T^2 - 72)^2
$43$	$(T^{2} - 8)^{2}$ (T^2 - 8)^2
$47$	$(T^{2} - 48)^{2}$ (T^2 - 48)^2
$53$	$(T^{2} - 6)^{2}$ (T^2 - 6)^2
$59$	$(T - 8)^{4}$ (T - 8)^4
$61$	$(T^{2} - 12)^{2}$ (T^2 - 12)^2
$67$	$(T^{2} - 128)^{2}$ (T^2 - 128)^2
$71$	$(T^{2} - 192)^{2}$ (T^2 - 192)^2
$73$	$T^{4}$ T^4
$79$	$(T^{2} - 6)^{2}$ (T^2 - 6)^2
$83$	$(T - 14)^{4}$ (T - 14)^4
$89$	$(T^{2} - 128)^{2}$ (T^2 - 128)^2
$97$	$(T + 6)^{4}$ (T + 6)^4
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