LMFDB - Newform orbit 2548.2.l.h

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2548,2,Mod(373,2548)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2548, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 4])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2548.373"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$2548 = 2^{2} \cdot 7^{2} \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2548.l (of order $3$ , degree $2$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,6,0,-2,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$20.3458824350$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-3})$
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:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 52)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$ -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 primitive root of unity $\zeta_{6}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + 3 q^{3} + (2 \zeta_{6} - 2) q^{5} + 6 q^{9} - 5 q^{11} + (4 \zeta_{6} - 3) q^{13} + (6 \zeta_{6} - 6) q^{15} + (3 \zeta_{6} - 3) q^{17} - 3 q^{19} + \zeta_{6} q^{23} + \zeta_{6} q^{25} + 9 q^{27} + ( - \zeta_{6} + 1) q^{29}+ \cdots - 30 q^{99}+O(q^{100})$ q + 3 * q^3 + (2z - 2) q^5 + 6 * q^9 - 5 * q^11 + (4z - 3) q^13 + (6z - 6) q^15 + (3z - 3) q^17 - 3 * q^19 + z * q^23 + z * q^25 + 9 * q^27 + (-z + 1) * q^29 + 8z q^31 - 15 * q^33 - 3z q^37 + (12z - 9) q^39 + (3z - 3) q^41 - z * q^43 + (12z - 12) q^45 + (4z - 4) q^47 + (9z - 9) q^51 + 6z q^53 + (-10z + 10) q^55 - 9 * q^57 + (5z - 5) q^59 - 5 * q^61 + (-6z - 2) q^65 + 7 * q^67 + 3z q^69 + 11z q^71 - 14z q^73 + 3z q^75 + (-4z + 4) q^79 + 9 * q^81 + 12 * q^83 - 6z q^85 + (-3z + 3) q^87 + 9z q^89 + 24z q^93 + (-6z + 6) q^95 + z * q^97 - 30 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 6 q^{3} - 2 q^{5} + 12 q^{9} - 10 q^{11} - 2 q^{13} - 6 q^{15} - 3 q^{17} - 6 q^{19} + q^{23} + q^{25} + 18 q^{27} + q^{29} + 8 q^{31} - 30 q^{33} - 3 q^{37} - 6 q^{39} - 3 q^{41} - q^{43} - 12 q^{45}+ \cdots - 60 q^{99}+O(q^{100})$ 2 * q + 6 * q^3 - 2 * q^5 + 12 * q^9 - 10 * q^11 - 2 * q^13 - 6 * q^15 - 3 * q^17 - 6 * q^19 + q^23 + q^25 + 18 * q^27 + q^29 + 8 * q^31 - 30 * q^33 - 3 * q^37 - 6 * q^39 - 3 * q^41 - q^43 - 12 * q^45 - 4 * q^47 - 9 * q^51 + 6 * q^53 + 10 * q^55 - 18 * q^57 - 5 * q^59 - 10 * q^61 - 10 * q^65 + 14 * q^67 + 3 * q^69 + 11 * q^71 - 14 * q^73 + 3 * q^75 + 4 * q^79 + 18 * q^81 + 24 * q^83 - 6 * q^85 + 3 * q^87 + 9 * q^89 + 24 * q^93 + 6 * q^95 + q^97 - 60 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2548\mathbb{Z}\right)^\times$ .

$n$	$197$	$885$	$1275$
$\chi(n)$	$-\zeta_{6}$	$-1 + \zeta_{6}$	$1$

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}

373.1

0.500000	+	0.866025i
0.500000	−	0.866025i

3.00000

−1.00000

1.73205i

6.00000

1537.1

3.00000

−1.00000

−

1.73205i

6.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
91.g	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2548.2.l.h		2
7.b	odd	2	1		2548.2.l.a		2
7.c	even	3	1		52.2.e.a	✓	2
7.c	even	3	1		2548.2.i.a		2
7.d	odd	6	1		2548.2.i.h		2
7.d	odd	6	1		2548.2.k.d		2
13.c	even	3	1		2548.2.i.a		2
21.h	odd	6	1		468.2.l.a		2
28.g	odd	6	1		208.2.i.d		2
35.j	even	6	1		1300.2.i.f		2
35.l	odd	12	2		1300.2.bb.f		4
56.k	odd	6	1		832.2.i.a		2
56.p	even	6	1		832.2.i.j		2
84.n	even	6	1		1872.2.t.f		2
91.g	even	3	1		676.2.a.e		1
91.g	even	3	1	inner	2548.2.l.h		2
91.h	even	3	1		52.2.e.a	✓	2
91.k	even	6	1		676.2.e.a		2
91.m	odd	6	1		2548.2.l.a		2
91.n	odd	6	1		2548.2.i.h		2
91.r	even	6	1		676.2.e.a		2
91.u	even	6	1		676.2.a.d		1
91.v	odd	6	1		2548.2.k.d		2
91.x	odd	12	2		676.2.h.b		4
91.z	odd	12	2		676.2.h.b		4
91.bd	odd	12	2		676.2.d.d		2
273.s	odd	6	1		468.2.l.a		2
273.x	odd	6	1		6084.2.a.k		1
273.bm	odd	6	1		6084.2.a.f		1
273.bw	even	12	2		6084.2.b.l		2
364.q	odd	6	1		2704.2.a.b		1
364.s	odd	6	1		2704.2.a.a		1
364.bi	odd	6	1		208.2.i.d		2
364.bt	even	12	2		2704.2.f.a		2
455.ba	even	6	1		1300.2.i.f		2
455.cq	odd	12	2		1300.2.bb.f		4
728.bx	odd	6	1		832.2.i.a		2
728.cw	even	6	1		832.2.i.j		2
1092.ck	even	6	1		1872.2.t.f		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
52.2.e.a	✓	2	7.c	even	3	1
52.2.e.a	✓	2	91.h	even	3	1
208.2.i.d		2	28.g	odd	6	1
208.2.i.d		2	364.bi	odd	6	1
468.2.l.a		2	21.h	odd	6	1
468.2.l.a		2	273.s	odd	6	1
676.2.a.d		1	91.u	even	6	1
676.2.a.e		1	91.g	even	3	1
676.2.d.d		2	91.bd	odd	12	2
676.2.e.a		2	91.k	even	6	1
676.2.e.a		2	91.r	even	6	1
676.2.h.b		4	91.x	odd	12	2
676.2.h.b		4	91.z	odd	12	2
832.2.i.a		2	56.k	odd	6	1
832.2.i.a		2	728.bx	odd	6	1
832.2.i.j		2	56.p	even	6	1
832.2.i.j		2	728.cw	even	6	1
1300.2.i.f		2	35.j	even	6	1
1300.2.i.f		2	455.ba	even	6	1
1300.2.bb.f		4	35.l	odd	12	2
1300.2.bb.f		4	455.cq	odd	12	2
1872.2.t.f		2	84.n	even	6	1
1872.2.t.f		2	1092.ck	even	6	1
2548.2.i.a		2	7.c	even	3	1
2548.2.i.a		2	13.c	even	3	1
2548.2.i.h		2	7.d	odd	6	1
2548.2.i.h		2	91.n	odd	6	1
2548.2.k.d		2	7.d	odd	6	1
2548.2.k.d		2	91.v	odd	6	1
2548.2.l.a		2	7.b	odd	2	1
2548.2.l.a		2	91.m	odd	6	1
2548.2.l.h		2	1.a	even	1	1	trivial
2548.2.l.h		2	91.g	even	3	1	inner
2704.2.a.a		1	364.s	odd	6	1
2704.2.a.b		1	364.q	odd	6	1
2704.2.f.a		2	364.bt	even	12	2
6084.2.a.f		1	273.bm	odd	6	1
6084.2.a.k		1	273.x	odd	6	1
6084.2.b.l		2	273.bw	even	12	2

Hecke kernels

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

T_{3} - 3

T3 - 3

T_{5}^{2} + 2T_{5} + 4

T5^2 + 2*T5 + 4

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$(T - 3)^{2}$ (T - 3)^2
$5$	$T^{2} + 2T + 4$ T^2 + 2*T + 4
$7$	$T^{2}$ T^2
$11$	$(T + 5)^{2}$ (T + 5)^2
$13$	$T^{2} + 2T + 13$ T^2 + 2*T + 13
$17$	$T^{2} + 3T + 9$ T^2 + 3*T + 9
$19$	$(T + 3)^{2}$ (T + 3)^2
$23$	$T^{2} - T + 1$ T^2 - T + 1
$29$	$T^{2} - T + 1$ T^2 - T + 1
$31$	$T^{2} - 8T + 64$ T^2 - 8*T + 64
$37$	$T^{2} + 3T + 9$ T^2 + 3*T + 9
$41$	$T^{2} + 3T + 9$ T^2 + 3*T + 9
$43$	$T^{2} + T + 1$ T^2 + T + 1
$47$	$T^{2} + 4T + 16$ T^2 + 4*T + 16
$53$	$T^{2} - 6T + 36$ T^2 - 6*T + 36
$59$	$T^{2} + 5T + 25$ T^2 + 5*T + 25
$61$	$(T + 5)^{2}$ (T + 5)^2
$67$	$(T - 7)^{2}$ (T - 7)^2
$71$	$T^{2} - 11T + 121$ T^2 - 11*T + 121
$73$	$T^{2} + 14T + 196$ T^2 + 14*T + 196
$79$	$T^{2} - 4T + 16$ T^2 - 4*T + 16
$83$	$(T - 12)^{2}$ (T - 12)^2
$89$	$T^{2} - 9T + 81$ T^2 - 9*T + 81
$97$	$T^{2} - T + 1$ T^2 - T + 1
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