LMFDB - Newform orbit 1764.4.k.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,4,Mod(361,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1764.361");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$1764 = 2^{2} \cdot 3^{2} \cdot 7^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1764.k (of order $3$ , degree $2$ , not minimal)

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:	no
Analytic conductor:	$104.079369250$
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:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 84)
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.

$f(q)$	$=$	$q - 6 \zeta_{6} q^{5} + ( - 36 \zeta_{6} + 36) q^{11} - 62 q^{13} + (114 \zeta_{6} - 114) q^{17} - 76 \zeta_{6} q^{19} - 24 \zeta_{6} q^{23} + ( - 89 \zeta_{6} + 89) q^{25} - 54 q^{29} + (112 \zeta_{6} - 112) q^{31} + \cdots - 1010 q^{97} +O(q^{100})$ q - 6z q^5 + (-36z + 36) q^11 - 62 * q^13 + (114z - 114) q^17 - 76z q^19 - 24z q^23 + (-89z + 89) q^25 - 54 * q^29 + (112z - 112) q^31 + 178z q^37 + 378 * q^41 - 172 * q^43 + 192z q^47 + (402z - 402) q^53 - 216 * q^55 + (396z - 396) q^59 + 254z q^61 + 372z q^65 + (-1012z + 1012) q^67 - 840 * q^71 + (-890z + 890) q^73 - 80z q^79 - 108 * q^83 + 684 * q^85 + 1638z q^89 + (456z - 456) q^95 - 1010 * q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 6 q^{5} + 36 q^{11} - 124 q^{13} - 114 q^{17} - 76 q^{19} - 24 q^{23} + 89 q^{25} - 108 q^{29} - 112 q^{31} + 178 q^{37} + 756 q^{41} - 344 q^{43} + 192 q^{47} - 402 q^{53} - 432 q^{55} - 396 q^{59}+ \cdots - 2020 q^{97}+O(q^{100})$ 2 * q - 6 * q^5 + 36 * q^11 - 124 * q^13 - 114 * q^17 - 76 * q^19 - 24 * q^23 + 89 * q^25 - 108 * q^29 - 112 * q^31 + 178 * q^37 + 756 * q^41 - 344 * q^43 + 192 * q^47 - 402 * q^53 - 432 * q^55 - 396 * q^59 + 254 * q^61 + 372 * q^65 + 1012 * q^67 - 1680 * q^71 + 890 * q^73 - 80 * q^79 - 216 * q^83 + 1368 * q^85 + 1638 * q^89 - 456 * q^95 - 2020 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$ .

$n$	$785$	$883$	$1081$
$\chi(n)$	$1$	$1$	$-\zeta_{6}$

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}

361.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−3.00000

−

5.19615i

1549.1

−3.00000

5.19615i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1764.4.k.f		2
3.b	odd	2	1		588.4.i.c		2
7.b	odd	2	1		1764.4.k.l		2
7.c	even	3	1		1764.4.a.j		1
7.c	even	3	1	inner	1764.4.k.f		2
7.d	odd	6	1		252.4.a.b		1
7.d	odd	6	1		1764.4.k.l		2
21.c	even	2	1		588.4.i.f		2
21.g	even	6	1		84.4.a.a	✓	1
21.g	even	6	1		588.4.i.f		2
21.h	odd	6	1		588.4.a.d		1
21.h	odd	6	1		588.4.i.c		2
28.f	even	6	1		1008.4.a.h		1
84.j	odd	6	1		336.4.a.k		1
84.n	even	6	1		2352.4.a.d		1
105.p	even	6	1		2100.4.a.l		1
105.w	odd	12	2		2100.4.k.j		2
168.ba	even	6	1		1344.4.a.q		1
168.be	odd	6	1		1344.4.a.d		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
84.4.a.a	✓	1	21.g	even	6	1
252.4.a.b		1	7.d	odd	6	1
336.4.a.k		1	84.j	odd	6	1
588.4.a.d		1	21.h	odd	6	1
588.4.i.c		2	3.b	odd	2	1
588.4.i.c		2	21.h	odd	6	1
588.4.i.f		2	21.c	even	2	1
588.4.i.f		2	21.g	even	6	1
1008.4.a.h		1	28.f	even	6	1
1344.4.a.d		1	168.be	odd	6	1
1344.4.a.q		1	168.ba	even	6	1
1764.4.a.j		1	7.c	even	3	1
1764.4.k.f		2	1.a	even	1	1	trivial
1764.4.k.f		2	7.c	even	3	1	inner
1764.4.k.l		2	7.b	odd	2	1
1764.4.k.l		2	7.d	odd	6	1
2100.4.a.l		1	105.p	even	6	1
2100.4.k.j		2	105.w	odd	12	2
2352.4.a.d		1	84.n	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(1764, [\chi])$ :

T_{5}^{2} + 6T_{5} + 36

T_{11}^{2} - 36T_{11} + 1296

T_{13} + 62

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2}$ T^2
$5$	$T^{2} + 6T + 36$ T^2 + 6*T + 36
$7$	$T^{2}$ T^2
$11$	$T^{2} - 36T + 1296$ T^2 - 36*T + 1296
$13$	$(T + 62)^{2}$ (T + 62)^2
$17$	$T^{2} + 114T + 12996$ T^2 + 114*T + 12996
$19$	$T^{2} + 76T + 5776$ T^2 + 76*T + 5776
$23$	$T^{2} + 24T + 576$ T^2 + 24*T + 576
$29$	$(T + 54)^{2}$ (T + 54)^2
$31$	$T^{2} + 112T + 12544$ T^2 + 112*T + 12544
$37$	$T^{2} - 178T + 31684$ T^2 - 178*T + 31684
$41$	$(T - 378)^{2}$ (T - 378)^2
$43$	$(T + 172)^{2}$ (T + 172)^2
$47$	$T^{2} - 192T + 36864$ T^2 - 192*T + 36864
$53$	$T^{2} + 402T + 161604$ T^2 + 402*T + 161604
$59$	$T^{2} + 396T + 156816$ T^2 + 396*T + 156816
$61$	$T^{2} - 254T + 64516$ T^2 - 254*T + 64516
$67$	$T^{2} - 1012 T + 1024144$ T^2 - 1012*T + 1024144
$71$	$(T + 840)^{2}$ (T + 840)^2
$73$	$T^{2} - 890T + 792100$ T^2 - 890*T + 792100
$79$	$T^{2} + 80T + 6400$ T^2 + 80*T + 6400
$83$	$(T + 108)^{2}$ (T + 108)^2
$89$	$T^{2} - 1638 T + 2683044$ T^2 - 1638*T + 2683044
$97$	$(T + 1010)^{2}$ (T + 1010)^2
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