LMFDB - Newform orbit 2548.2.a.m

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2548,2,Mod(1,2548)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2548, 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("2548.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$2548 = 2^{2} \cdot 7^{2} \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2548.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:	$20.3458824350$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - 6$ x^2 - 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 364)
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 $\beta = \sqrt{6}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + \beta q^{3} + (\beta + 1) q^{5} + 3 q^{9} + (\beta + 4) q^{11} + q^{13} + (\beta + 6) q^{15} - \beta q^{17} + (\beta - 3) q^{19} + (2 \beta - 1) q^{23} + (2 \beta + 2) q^{25} + ( - 2 \beta - 1) q^{29} + \cdots + (3 \beta + 12) q^{99} +O(q^{100})$ q + b * q^3 + (b + 1) * q^5 + 3 * q^9 + (b + 4) * q^11 + q^13 + (b + 6) * q^15 - b * q^17 + (b - 3) * q^19 + (2b - 1) q^23 + (2b + 2) q^25 + (-2b - 1) q^29 + (-b - 1) * q^31 + (4b + 6) q^33 + (-b - 6) * q^37 + b * q^39 + (-2b - 6) q^41 + q^43 + (3b + 3) q^45 + (-b - 1) * q^47 - 6 * q^51 + (2b - 3) q^53 + (5b + 10) q^55 + (-3b + 6) q^57 - 14 * q^59 + 2 * q^61 + (b + 1) * q^65 + (-2b - 2) q^67 + (-b + 12) * q^69 + (b + 10) * q^71 + (-5b + 1) q^73 + (2b + 12) q^75 + (2b - 7) q^79 - 9 * q^81 + (3b - 3) q^83 + (-b - 6) * q^85 + (-b - 12) * q^87 + (b + 9) * q^89 + (-b - 6) * q^93 + (-2b + 3) q^95 + (-b + 13) * q^97 + (3b + 12) q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 2 q^{5} + 6 q^{9} + 8 q^{11} + 2 q^{13} + 12 q^{15} - 6 q^{19} - 2 q^{23} + 4 q^{25} - 2 q^{29} - 2 q^{31} + 12 q^{33} - 12 q^{37} - 12 q^{41} + 2 q^{43} + 6 q^{45} - 2 q^{47} - 12 q^{51} - 6 q^{53}+ \cdots + 24 q^{99}+O(q^{100})$ 2 * q + 2 * q^5 + 6 * q^9 + 8 * q^11 + 2 * q^13 + 12 * q^15 - 6 * q^19 - 2 * q^23 + 4 * q^25 - 2 * q^29 - 2 * q^31 + 12 * q^33 - 12 * q^37 - 12 * q^41 + 2 * q^43 + 6 * q^45 - 2 * q^47 - 12 * q^51 - 6 * q^53 + 20 * q^55 + 12 * q^57 - 28 * q^59 + 4 * q^61 + 2 * q^65 - 4 * q^67 + 24 * q^69 + 20 * q^71 + 2 * q^73 + 24 * q^75 - 14 * q^79 - 18 * q^81 - 6 * q^83 - 12 * q^85 - 24 * q^87 + 18 * q^89 - 12 * q^93 + 6 * q^95 + 26 * q^97 + 24 * q^99

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

−2.44949
2.44949

−2.44949

−1.44949

3.00000

1.2

2.44949

3.44949

3.00000

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$7$	$-1$
$13$	$-1$

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	2548.2.a.m		2
7.b	odd	2	1		364.2.a.c	✓	2
7.c	even	3	2		2548.2.j.l		4
7.d	odd	6	2		2548.2.j.m		4
21.c	even	2	1		3276.2.a.q		2
28.d	even	2	1		1456.2.a.p		2
35.c	odd	2	1		9100.2.a.v		2
56.e	even	2	1		5824.2.a.bn		2
56.h	odd	2	1		5824.2.a.bm		2
91.b	odd	2	1		4732.2.a.i		2
91.i	even	4	2		4732.2.g.f		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
364.2.a.c	✓	2	7.b	odd	2	1
1456.2.a.p		2	28.d	even	2	1
2548.2.a.m		2	1.a	even	1	1	trivial
2548.2.j.l		4	7.c	even	3	2
2548.2.j.m		4	7.d	odd	6	2
3276.2.a.q		2	21.c	even	2	1
4732.2.a.i		2	91.b	odd	2	1
4732.2.g.f		4	91.i	even	4	2
5824.2.a.bm		2	56.h	odd	2	1
5824.2.a.bn		2	56.e	even	2	1
9100.2.a.v		2	35.c	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(2548))$ :

T_{3}^{2} - 6

T_{5}^{2} - 2T_{5} - 5

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} - 6$ T^2 - 6
$5$	$T^{2} - 2T - 5$ T^2 - 2*T - 5
$7$	$T^{2}$ T^2
$11$	$T^{2} - 8T + 10$ T^2 - 8*T + 10
$13$	$(T - 1)^{2}$ (T - 1)^2
$17$	$T^{2} - 6$ T^2 - 6
$19$	$T^{2} + 6T + 3$ T^2 + 6*T + 3
$23$	$T^{2} + 2T - 23$ T^2 + 2*T - 23
$29$	$T^{2} + 2T - 23$ T^2 + 2*T - 23
$31$	$T^{2} + 2T - 5$ T^2 + 2*T - 5
$37$	$T^{2} + 12T + 30$ T^2 + 12*T + 30
$41$	$T^{2} + 12T + 12$ T^2 + 12*T + 12
$43$	$(T - 1)^{2}$ (T - 1)^2
$47$	$T^{2} + 2T - 5$ T^2 + 2*T - 5
$53$	$T^{2} + 6T - 15$ T^2 + 6*T - 15
$59$	$(T + 14)^{2}$ (T + 14)^2
$61$	$(T - 2)^{2}$ (T - 2)^2
$67$	$T^{2} + 4T - 20$ T^2 + 4*T - 20
$71$	$T^{2} - 20T + 94$ T^2 - 20*T + 94
$73$	$T^{2} - 2T - 149$ T^2 - 2*T - 149
$79$	$T^{2} + 14T + 25$ T^2 + 14*T + 25
$83$	$T^{2} + 6T - 45$ T^2 + 6*T - 45
$89$	$T^{2} - 18T + 75$ T^2 - 18*T + 75
$97$	$T^{2} - 26T + 163$ T^2 - 26*T + 163
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