LMFDB - Newform orbit 1125.2.a.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1125,2,Mod(1,1125)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1125, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1125.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1125 = 3^{2} \cdot 5^{3}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1125.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:	$8.98317022739$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x - 1$ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 125)
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 $\beta = \frac{1}{2}(1 + \sqrt{5})$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + \beta q^{2} + (\beta - 1) q^{4} - 3 q^{7} + ( - 2 \beta + 1) q^{8} + 3 q^{11} - 3 \beta q^{13} - 3 \beta q^{14} - 3 \beta q^{16} + ( - 2 \beta - 1) q^{17} + ( - \beta - 2) q^{19} + 3 \beta q^{22} + (2 \beta - 2) q^{23} + \cdots + 2 \beta q^{98} +O(q^{100})$ q + b * q^2 + (b - 1) * q^4 - 3 * q^7 + (-2b + 1) q^8 + 3 * q^11 - 3b q^13 - 3b q^14 - 3b q^16 + (-2b - 1) q^17 + (-b - 2) * q^19 + 3b q^22 + (2b - 2) q^23 + (-3b - 3) q^26 + (-3b + 3) q^28 + (-6b + 3) q^29 + (5b - 3) q^31 + (b - 5) * q^32 + (-3b - 2) q^34 + (6b - 6) q^37 + (-3b - 1) q^38 + 3 * q^41 - 9 * q^43 + (3b - 3) q^44 + 2 * q^46 + (7b - 3) q^47 + 2 * q^49 - 3 * q^52 + (-b - 3) * q^53 + (6b - 3) q^56 + (-3b - 6) q^58 + (3b - 9) q^59 + (-5b + 2) q^61 + (2b + 5) q^62 + (2b + 1) q^64 + (-3b - 9) q^67 + (-b - 1) * q^68 + 3 * q^71 + (3b - 3) q^73 + 6 * q^74 + (-2b + 1) q^76 - 9 * q^77 + (-4b + 7) q^79 + 3b q^82 + (-4b + 6) q^83 - 9b q^86 + (-6b + 3) q^88 + (12b - 6) q^89 + 9b q^91 + (-2b + 4) q^92 + (4b + 7) q^94 + (3b + 3) q^97 + 2b q^98
$\operatorname{Tr}(f)(q)$	$=$	$2 q + q^{2} - q^{4} - 6 q^{7} + 6 q^{11} - 3 q^{13} - 3 q^{14} - 3 q^{16} - 4 q^{17} - 5 q^{19} + 3 q^{22} - 2 q^{23} - 9 q^{26} + 3 q^{28} - q^{31} - 9 q^{32} - 7 q^{34} - 6 q^{37} - 5 q^{38} + 6 q^{41}+ \cdots + 2 q^{98}+O(q^{100})$ 2 * q + q^2 - q^4 - 6 * q^7 + 6 * q^11 - 3 * q^13 - 3 * q^14 - 3 * q^16 - 4 * q^17 - 5 * q^19 + 3 * q^22 - 2 * q^23 - 9 * q^26 + 3 * q^28 - q^31 - 9 * q^32 - 7 * q^34 - 6 * q^37 - 5 * q^38 + 6 * q^41 - 18 * q^43 - 3 * q^44 + 4 * q^46 + q^47 + 4 * q^49 - 6 * q^52 - 7 * q^53 - 15 * q^58 - 15 * q^59 - q^61 + 12 * q^62 + 4 * q^64 - 21 * q^67 - 3 * q^68 + 6 * q^71 - 3 * q^73 + 12 * q^74 - 18 * q^77 + 10 * q^79 + 3 * q^82 + 8 * q^83 - 9 * q^86 + 9 * q^91 + 6 * q^92 + 18 * q^94 + 9 * q^97 + 2 * q^98

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

−0.618034
1.61803

−0.618034

−1.61803

−3.00000

2.23607

1.2

1.61803

0.618034

−3.00000

−2.23607

Atkin-Lehner signs

$p$	Sign
$3$	$-1$
$5$	$-1$

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	1125.2.a.d		2
3.b	odd	2	1		125.2.a.a	✓	2
5.b	even	2	1		1125.2.a.c		2
5.c	odd	4	2		1125.2.b.f		4
12.b	even	2	1		2000.2.a.l		2
15.d	odd	2	1		125.2.a.b	yes	2
15.e	even	4	2		125.2.b.b		4
21.c	even	2	1		6125.2.a.d		2
24.f	even	2	1		8000.2.a.c		2
24.h	odd	2	1		8000.2.a.v		2
60.h	even	2	1		2000.2.a.a		2
60.l	odd	4	2		2000.2.c.e		4
75.h	odd	10	2		625.2.d.a		4
75.h	odd	10	2		625.2.d.g		4
75.j	odd	10	2		625.2.d.d		4
75.j	odd	10	2		625.2.d.j		4
75.l	even	20	4		625.2.e.d		8
75.l	even	20	4		625.2.e.g		8
105.g	even	2	1		6125.2.a.g		2
120.i	odd	2	1		8000.2.a.d		2
120.m	even	2	1		8000.2.a.u		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
125.2.a.a	✓	2	3.b	odd	2	1
125.2.a.b	yes	2	15.d	odd	2	1
125.2.b.b		4	15.e	even	4	2
625.2.d.a		4	75.h	odd	10	2
625.2.d.d		4	75.j	odd	10	2
625.2.d.g		4	75.h	odd	10	2
625.2.d.j		4	75.j	odd	10	2
625.2.e.d		8	75.l	even	20	4
625.2.e.g		8	75.l	even	20	4
1125.2.a.c		2	5.b	even	2	1
1125.2.a.d		2	1.a	even	1	1	trivial
1125.2.b.f		4	5.c	odd	4	2
2000.2.a.a		2	60.h	even	2	1
2000.2.a.l		2	12.b	even	2	1
2000.2.c.e		4	60.l	odd	4	2
6125.2.a.d		2	21.c	even	2	1
6125.2.a.g		2	105.g	even	2	1
8000.2.a.c		2	24.f	even	2	1
8000.2.a.d		2	120.i	odd	2	1
8000.2.a.u		2	120.m	even	2	1
8000.2.a.v		2	24.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(1125))$ :

T_{2}^{2} - T_{2} - 1

T_{7} + 3

T_{11} - 3

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} - T - 1$ T^2 - T - 1
$3$	$T^{2}$ T^2
$5$	$T^{2}$ T^2
$7$	$(T + 3)^{2}$ (T + 3)^2
$11$	$(T - 3)^{2}$ (T - 3)^2
$13$	$T^{2} + 3T - 9$ T^2 + 3*T - 9
$17$	$T^{2} + 4T - 1$ T^2 + 4*T - 1
$19$	$T^{2} + 5T + 5$ T^2 + 5*T + 5
$23$	$T^{2} + 2T - 4$ T^2 + 2*T - 4
$29$	$T^{2} - 45$ T^2 - 45
$31$	$T^{2} + T - 31$ T^2 + T - 31
$37$	$T^{2} + 6T - 36$ T^2 + 6*T - 36
$41$	$(T - 3)^{2}$ (T - 3)^2
$43$	$(T + 9)^{2}$ (T + 9)^2
$47$	$T^{2} - T - 61$ T^2 - T - 61
$53$	$T^{2} + 7T + 11$ T^2 + 7*T + 11
$59$	$T^{2} + 15T + 45$ T^2 + 15*T + 45
$61$	$T^{2} + T - 31$ T^2 + T - 31
$67$	$T^{2} + 21T + 99$ T^2 + 21*T + 99
$71$	$(T - 3)^{2}$ (T - 3)^2
$73$	$T^{2} + 3T - 9$ T^2 + 3*T - 9
$79$	$T^{2} - 10T + 5$ T^2 - 10*T + 5
$83$	$T^{2} - 8T - 4$ T^2 - 8*T - 4
$89$	$T^{2} - 180$ T^2 - 180
$97$	$T^{2} - 9T + 9$ T^2 - 9*T + 9
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