LMFDB - Newform orbit 1456.2.s.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1456,2,Mod(113,1456)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1456, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 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("1456.113"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$1456 = 2^{4} \cdot 7 \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1456.s (of order $3$ , degree $2$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$11.6262185343$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 364)
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.

$f(q)$	$=$	$q - q^{5} - \zeta_{6} q^{7} + 3 \zeta_{6} q^{9} + (2 \zeta_{6} - 2) q^{11} + (\zeta_{6} + 3) q^{13} + 3 \zeta_{6} q^{17} - 6 \zeta_{6} q^{19} + (4 \zeta_{6} - 4) q^{23} - 4 q^{25} + ( - 7 \zeta_{6} + 7) q^{29} + \cdots - 6 q^{99} +O(q^{100})$ q - q^5 - z * q^7 + 3z q^9 + (2z - 2) q^11 + (z + 3) * q^13 + 3z q^17 - 6z q^19 + (4z - 4) q^23 - 4 * q^25 + (-7z + 7) q^29 - 4 * q^31 + z * q^35 + (9z - 9) q^37 + (9z - 9) q^41 + 10z q^43 - 3z q^45 + 2 * q^47 + (z - 1) * q^49 + 9 * q^53 + (-2z + 2) q^55 + 14z q^59 + 5z q^61 + (-3z + 3) q^63 + (-z - 3) * q^65 + (8z - 8) q^67 + 10z q^71 - 7 * q^73 + 2 * q^77 - 2 * q^79 + (9z - 9) q^81 + 6 * q^83 - 3z q^85 + (6z - 6) q^89 + (-4z + 1) q^91 + 6z q^95 - 2z q^97 - 6 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 2 q^{5} - q^{7} + 3 q^{9} - 2 q^{11} + 7 q^{13} + 3 q^{17} - 6 q^{19} - 4 q^{23} - 8 q^{25} + 7 q^{29} - 8 q^{31} + q^{35} - 9 q^{37} - 9 q^{41} + 10 q^{43} - 3 q^{45} + 4 q^{47} - q^{49} + 18 q^{53}+ \cdots - 12 q^{99}+O(q^{100})$ 2 * q - 2 * q^5 - q^7 + 3 * q^9 - 2 * q^11 + 7 * q^13 + 3 * q^17 - 6 * q^19 - 4 * q^23 - 8 * q^25 + 7 * q^29 - 8 * q^31 + q^35 - 9 * q^37 - 9 * q^41 + 10 * q^43 - 3 * q^45 + 4 * q^47 - q^49 + 18 * q^53 + 2 * q^55 + 14 * q^59 + 5 * q^61 + 3 * q^63 - 7 * q^65 - 8 * q^67 + 10 * q^71 - 14 * q^73 + 4 * q^77 - 4 * q^79 - 9 * q^81 + 12 * q^83 - 3 * q^85 - 6 * q^89 - 2 * q^91 + 6 * q^95 - 2 * q^97 - 12 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1456\mathbb{Z}\right)^\times$ .

$n$	$561$	$911$	$1093$	$1249$
$\chi(n)$	$-\zeta_{6}$	$1$	$1$	$1$

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}

113.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−1.00000

−0.500000

−

0.866025i

1.50000

2.59808i

1121.1

−1.00000

−0.500000

0.866025i

1.50000

−

2.59808i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1456.2.s.c		2
4.b	odd	2	1		364.2.k.a	✓	2
12.b	even	2	1		3276.2.z.c		2
13.c	even	3	1	inner	1456.2.s.c		2
28.d	even	2	1		2548.2.k.c		2
28.f	even	6	1		2548.2.i.d		2
28.f	even	6	1		2548.2.l.d		2
28.g	odd	6	1		2548.2.i.e		2
28.g	odd	6	1		2548.2.l.e		2
52.i	odd	6	1		4732.2.a.d		1
52.j	odd	6	1		364.2.k.a	✓	2
52.j	odd	6	1		4732.2.a.c		1
52.l	even	12	2		4732.2.g.b		2
156.p	even	6	1		3276.2.z.c		2
364.q	odd	6	1		2548.2.i.e		2
364.v	even	6	1		2548.2.k.c		2
364.ba	even	6	1		2548.2.l.d		2
364.bi	odd	6	1		2548.2.l.e		2
364.br	even	6	1		2548.2.i.d		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
364.2.k.a	✓	2	4.b	odd	2	1
364.2.k.a	✓	2	52.j	odd	6	1
1456.2.s.c		2	1.a	even	1	1	trivial
1456.2.s.c		2	13.c	even	3	1	inner
2548.2.i.d		2	28.f	even	6	1
2548.2.i.d		2	364.br	even	6	1
2548.2.i.e		2	28.g	odd	6	1
2548.2.i.e		2	364.q	odd	6	1
2548.2.k.c		2	28.d	even	2	1
2548.2.k.c		2	364.v	even	6	1
2548.2.l.d		2	28.f	even	6	1
2548.2.l.d		2	364.ba	even	6	1
2548.2.l.e		2	28.g	odd	6	1
2548.2.l.e		2	364.bi	odd	6	1
3276.2.z.c		2	12.b	even	2	1
3276.2.z.c		2	156.p	even	6	1
4732.2.a.c		1	52.j	odd	6	1
4732.2.a.d		1	52.i	odd	6	1
4732.2.g.b		2	52.l	even	12	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1456, [\chi])$ :

T_{3}

T3

T_{5} + 1

T5 + 1

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2}$ T^2
$5$	$(T + 1)^{2}$ (T + 1)^2
$7$	$T^{2} + T + 1$ T^2 + T + 1
$11$	$T^{2} + 2T + 4$ T^2 + 2*T + 4
$13$	$T^{2} - 7T + 13$ T^2 - 7*T + 13
$17$	$T^{2} - 3T + 9$ T^2 - 3*T + 9
$19$	$T^{2} + 6T + 36$ T^2 + 6*T + 36
$23$	$T^{2} + 4T + 16$ T^2 + 4*T + 16
$29$	$T^{2} - 7T + 49$ T^2 - 7*T + 49
$31$	$(T + 4)^{2}$ (T + 4)^2
$37$	$T^{2} + 9T + 81$ T^2 + 9*T + 81
$41$	$T^{2} + 9T + 81$ T^2 + 9*T + 81
$43$	$T^{2} - 10T + 100$ T^2 - 10*T + 100
$47$	$(T - 2)^{2}$ (T - 2)^2
$53$	$(T - 9)^{2}$ (T - 9)^2
$59$	$T^{2} - 14T + 196$ T^2 - 14*T + 196
$61$	$T^{2} - 5T + 25$ T^2 - 5*T + 25
$67$	$T^{2} + 8T + 64$ T^2 + 8*T + 64
$71$	$T^{2} - 10T + 100$ T^2 - 10*T + 100
$73$	$(T + 7)^{2}$ (T + 7)^2
$79$	$(T + 2)^{2}$ (T + 2)^2
$83$	$(T - 6)^{2}$ (T - 6)^2
$89$	$T^{2} + 6T + 36$ T^2 + 6*T + 36
$97$	$T^{2} + 2T + 4$ T^2 + 2*T + 4
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