LMFDB - Newform orbit 4851.2.a.bk

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4851,2,Mod(1,4851)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4851, 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("4851.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$4851 = 3^{2} \cdot 7^{2} \cdot 11$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	4851.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:	$38.7354300205$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{3} - 3x - 1$ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 77)
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 $1,\beta_1,\beta_2$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - \beta_1 q^{2} + \beta_{2} q^{4} + (\beta_{2} + 2) q^{5} + (\beta_1 - 1) q^{8} + ( - 3 \beta_1 - 1) q^{10} - q^{11} + ( - \beta_{2} - \beta_1 - 1) q^{13} + ( - 3 \beta_{2} + \beta_1 - 2) q^{16} + ( - 4 \beta_{2} + \beta_1 - 1) q^{17}+ \cdots + (\beta_{2} - \beta_1 - 15) q^{97}+O(q^{100})$ q - b1 * q^2 + b2 * q^4 + (b2 + 2) * q^5 + (b1 - 1) * q^8 + (-3b1 - 1) q^10 - q^11 + (-b2 - b1 - 1) * q^13 + (-3b2 + b1 - 2) q^16 + (-4b2 + b1 - 1) q^17 + (2b2 - b1 - 3) q^19 + (b2 + b1 + 2) * q^20 + b1 * q^22 + (-2b2 + 5b1) * q^23 + (3b2 + b1 + 1) q^25 + (b2 + 2b1 + 3) q^26 + (3b2 - 4b1 + 1) * q^29 + (b2 - 3) * q^31 + (-b2 + 3b1 + 3) q^32 + (-b2 + 5b1 + 2) q^34 + (-2b2 + 4b1) * q^37 + (b2 + b1) * q^38 + (-b2 + 3b1 - 1) q^40 + (-3b2 + b1 + 3) q^41 + (-b2 - b1) * q^43 - b2 * q^44 + (-5b2 + 2b1 - 8) * q^46 + (-6b2 + 3b1 - 1) * q^47 + (-b2 - 4b1 - 5) q^50 + (-2b1 - 3) q^52 + (3b2 + 3b1 - 3) * q^53 + (-b2 - 2) * q^55 + (4b2 - 4b1 + 5) * q^58 + (5b2 + b1) q^59 + (-4b2 + 6b1 - 4) * q^61 + (2b1 - 1) q^62 + (3b2 - 4b1 - 1) * q^64 + (-2b2 - 4b1 - 5) * q^65 + (2b2 - 2b1) * q^67 + (3b2 - 3b1 - 7) * q^68 + (-2b2 - 5b1 + 3) * q^71 + (b2 - b1 - 2) * q^73 + (-4b2 + 2b1 - 6) * q^74 + (-5b2 + b1 + 3) q^76 + (-5b2 - 2b1 - 1) * q^79 + (-5b2 - 9) q^80 + (-b2 + 1) * q^82 + (2b2 - 5b1 - 5) * q^83 + (-5b2 - b1 - 9) q^85 + (b2 + b1 + 3) * q^86 + (-b1 + 1) * q^88 + (b2 - 5) * q^89 + (2b2 + 3b1 + 1) * q^92 + (-3b2 + 7b1) * q^94 + (-b2 - b1 - 3) * q^95 + (b2 - b1 - 15) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$3 q + 6 q^{5} - 3 q^{8} - 3 q^{10} - 3 q^{11} - 3 q^{13} - 6 q^{16} - 3 q^{17} - 9 q^{19} + 6 q^{20} + 3 q^{25} + 9 q^{26} + 3 q^{29} - 9 q^{31} + 9 q^{32} + 6 q^{34} - 3 q^{40} + 9 q^{41} - 24 q^{46} - 3 q^{47}+ \cdots - 45 q^{97}+O(q^{100})$ 3 * q + 6 * q^5 - 3 * q^8 - 3 * q^10 - 3 * q^11 - 3 * q^13 - 6 * q^16 - 3 * q^17 - 9 * q^19 + 6 * q^20 + 3 * q^25 + 9 * q^26 + 3 * q^29 - 9 * q^31 + 9 * q^32 + 6 * q^34 - 3 * q^40 + 9 * q^41 - 24 * q^46 - 3 * q^47 - 15 * q^50 - 9 * q^52 - 9 * q^53 - 6 * q^55 + 15 * q^58 - 12 * q^61 - 3 * q^62 - 3 * q^64 - 15 * q^65 - 21 * q^68 + 9 * q^71 - 6 * q^73 - 18 * q^74 + 9 * q^76 - 3 * q^79 - 27 * q^80 + 3 * q^82 - 15 * q^83 - 27 * q^85 + 9 * q^86 + 3 * q^88 - 15 * q^89 + 3 * q^92 - 9 * q^95 - 45 * q^97

Basis of coefficient ring in terms of $\nu = \zeta_{18} + \zeta_{18}^{-1}$ :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$\nu^{2} - 2$ v^2 - 2

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{2} + 2$ b2 + 2

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

1.87939
−0.347296
−1.53209

−1.87939

1.53209

3.53209

0.879385

−6.63816

1.2

0.347296

−1.87939

0.120615

−1.34730

0.0418891

1.3

1.53209

0.347296

2.34730

−2.53209

3.59627

Atkin-Lehner signs

$p$	Sign
$3$	$-1$
$7$	$-1$
$11$	$+1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	4851.2.a.bk		3
3.b	odd	2	1		539.2.a.g		3
7.b	odd	2	1		4851.2.a.bj		3
7.d	odd	6	2		693.2.i.h		6
12.b	even	2	1		8624.2.a.co		3
21.c	even	2	1		539.2.a.j		3
21.g	even	6	2		77.2.e.a	✓	6
21.h	odd	6	2		539.2.e.m		6
33.d	even	2	1		5929.2.a.u		3
84.h	odd	2	1		8624.2.a.ch		3
84.j	odd	6	2		1232.2.q.m		6
231.h	odd	2	1		5929.2.a.x		3
231.k	odd	6	2		847.2.e.c		6
231.bc	even	30	8		847.2.n.g		24
231.bf	odd	30	8		847.2.n.f		24

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
77.2.e.a	✓	6	21.g	even	6	2
539.2.a.g		3	3.b	odd	2	1
539.2.a.j		3	21.c	even	2	1
539.2.e.m		6	21.h	odd	6	2
693.2.i.h		6	7.d	odd	6	2
847.2.e.c		6	231.k	odd	6	2
847.2.n.f		24	231.bf	odd	30	8
847.2.n.g		24	231.bc	even	30	8
1232.2.q.m		6	84.j	odd	6	2
4851.2.a.bj		3	7.b	odd	2	1
4851.2.a.bk		3	1.a	even	1	1	trivial
5929.2.a.u		3	33.d	even	2	1
5929.2.a.x		3	231.h	odd	2	1
8624.2.a.ch		3	84.h	odd	2	1
8624.2.a.co		3	12.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(4851))$ :

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

T_{5}^{3} - 6T_{5}^{2} + 9T_{5} - 1

T_{13}^{3} + 3T_{13}^{2} - 6T_{13} + 1

T_{17}^{3} + 3T_{17}^{2} - 36T_{17} - 127

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{3} - 3T + 1$ T^3 - 3*T + 1
$3$	$T^{3}$ T^3
$5$	$T^{3} - 6 T^{2} + \cdots - 1$ T^3 - 6T^2 + 9T - 1
$7$	$T^{3}$ T^3
$11$	$(T + 1)^{3}$ (T + 1)^3
$13$	$T^{3} + 3 T^{2} + \cdots + 1$ T^3 + 3T^2 - 6T + 1
$17$	$T^{3} + 3 T^{2} + \cdots - 127$ T^3 + 3T^2 - 36T - 127
$19$	$T^{3} + 9 T^{2} + \cdots + 9$ T^3 + 9T^2 + 18T + 9
$23$	$T^{3} - 57T + 107$ T^3 - 57*T + 107
$29$	$T^{3} - 3 T^{2} + \cdots - 51$ T^3 - 3T^2 - 36T - 51
$31$	$T^{3} + 9 T^{2} + \cdots + 19$ T^3 + 9T^2 + 24T + 19
$37$	$T^{3} - 36T + 72$ T^3 - 36*T + 72
$41$	$T^{3} - 9 T^{2} + \cdots - 1$ T^3 - 9T^2 + 6T - 1
$43$	$T^{3} - 9T + 9$ T^3 - 9*T + 9
$47$	$T^{3} + 3 T^{2} + \cdots - 323$ T^3 + 3T^2 - 78T - 323
$53$	$T^{3} + 9 T^{2} + \cdots - 459$ T^3 + 9T^2 - 54T - 459
$59$	$T^{3} - 93T + 19$ T^3 - 93*T + 19
$61$	$T^{3} + 12 T^{2} + \cdots + 24$ T^3 + 12T^2 - 36T + 24
$67$	$T^{3} - 12T - 8$ T^3 - 12*T - 8
$71$	$T^{3} - 9 T^{2} + \cdots + 801$ T^3 - 9T^2 - 90T + 801
$73$	$T^{3} + 6 T^{2} + \cdots + 1$ T^3 + 6T^2 + 9T + 1
$79$	$T^{3} + 3 T^{2} + \cdots + 37$ T^3 + 3T^2 - 114T + 37
$83$	$T^{3} + 15 T^{2} + \cdots - 267$ T^3 + 15T^2 + 18T - 267
$89$	$T^{3} + 15 T^{2} + \cdots + 111$ T^3 + 15T^2 + 72T + 111
$97$	$T^{3} + 45 T^{2} + \cdots + 3329$ T^3 + 45T^2 + 672T + 3329
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