LMFDB - Newform orbit 1764.4.k.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1764,4,Mod(361,1764)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [2,0,0,0,-14,0,0,0,0,0,4,0,-108,0,0,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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.

$f(q)$	$=$	$q - 14 \zeta_{6} q^{5} + ( - 4 \zeta_{6} + 4) q^{11} - 54 q^{13} + ( - 14 \zeta_{6} + 14) q^{17} + 92 \zeta_{6} q^{19} - 152 \zeta_{6} q^{23} + (71 \zeta_{6} - 71) q^{25} + 106 q^{29} + (144 \zeta_{6} - 144) q^{31} + \cdots + 1502 q^{97} +O(q^{100})$ q - 14z q^5 + (-4z + 4) q^11 - 54 * q^13 + (-14z + 14) q^17 + 92z q^19 - 152z q^23 + (71z - 71) q^25 + 106 * q^29 + (144z - 144) q^31 - 158z q^37 - 390 * q^41 - 508 * q^43 + 528z q^47 + (-606z + 606) q^53 - 56 * q^55 + (-364z + 364) q^59 + 678z q^61 + 756z q^65 + (844z - 844) q^67 + 8 * q^71 + (422z - 422) q^73 - 384z q^79 - 548 * q^83 - 196 * q^85 - 1194z q^89 + (-1288z + 1288) q^95 + 1502 * q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 14 q^{5} + 4 q^{11} - 108 q^{13} + 14 q^{17} + 92 q^{19} - 152 q^{23} - 71 q^{25} + 212 q^{29} - 144 q^{31} - 158 q^{37} - 780 q^{41} - 1016 q^{43} + 528 q^{47} + 606 q^{53} - 112 q^{55} + 364 q^{59}+ \cdots + 3004 q^{97}+O(q^{100})$ 2 * q - 14 * q^5 + 4 * q^11 - 108 * q^13 + 14 * q^17 + 92 * q^19 - 152 * q^23 - 71 * q^25 + 212 * q^29 - 144 * q^31 - 158 * q^37 - 780 * q^41 - 1016 * q^43 + 528 * q^47 + 606 * q^53 - 112 * q^55 + 364 * q^59 + 678 * q^61 + 756 * q^65 - 844 * q^67 + 16 * q^71 - 422 * q^73 - 384 * q^79 - 1096 * q^83 - 392 * q^85 - 1194 * q^89 + 1288 * q^95 + 3004 * 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

−7.00000

−

12.1244i

1549.1

−7.00000

12.1244i

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.c		2
3.b	odd	2	1		588.4.i.h		2
7.b	odd	2	1		1764.4.k.n		2
7.c	even	3	1		1764.4.a.l		1
7.c	even	3	1	inner	1764.4.k.c		2
7.d	odd	6	1		252.4.a.a		1
7.d	odd	6	1		1764.4.k.n		2
21.c	even	2	1		588.4.i.a		2
21.g	even	6	1		84.4.a.b	✓	1
21.g	even	6	1		588.4.i.a		2
21.h	odd	6	1		588.4.a.a		1
21.h	odd	6	1		588.4.i.h		2
28.f	even	6	1		1008.4.a.d		1
84.j	odd	6	1		336.4.a.e		1
84.n	even	6	1		2352.4.a.v		1
105.p	even	6	1		2100.4.a.g		1
105.w	odd	12	2		2100.4.k.g		2
168.ba	even	6	1		1344.4.a.b		1
168.be	odd	6	1		1344.4.a.p		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
84.4.a.b	✓	1	21.g	even	6	1
252.4.a.a		1	7.d	odd	6	1
336.4.a.e		1	84.j	odd	6	1
588.4.a.a		1	21.h	odd	6	1
588.4.i.a		2	21.c	even	2	1
588.4.i.a		2	21.g	even	6	1
588.4.i.h		2	3.b	odd	2	1
588.4.i.h		2	21.h	odd	6	1
1008.4.a.d		1	28.f	even	6	1
1344.4.a.b		1	168.ba	even	6	1
1344.4.a.p		1	168.be	odd	6	1
1764.4.a.l		1	7.c	even	3	1
1764.4.k.c		2	1.a	even	1	1	trivial
1764.4.k.c		2	7.c	even	3	1	inner
1764.4.k.n		2	7.b	odd	2	1
1764.4.k.n		2	7.d	odd	6	1
2100.4.a.g		1	105.p	even	6	1
2100.4.k.g		2	105.w	odd	12	2
2352.4.a.v		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} + 14T_{5} + 196

T5^2 + 14*T5 + 196

T_{11}^{2} - 4T_{11} + 16

T11^2 - 4*T11 + 16

T_{13} + 54

T13 + 54

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2}$ T^2
$5$	$T^{2} + 14T + 196$ T^2 + 14*T + 196
$7$	$T^{2}$ T^2
$11$	$T^{2} - 4T + 16$ T^2 - 4*T + 16
$13$	$(T + 54)^{2}$ (T + 54)^2
$17$	$T^{2} - 14T + 196$ T^2 - 14*T + 196
$19$	$T^{2} - 92T + 8464$ T^2 - 92*T + 8464
$23$	$T^{2} + 152T + 23104$ T^2 + 152*T + 23104
$29$	$(T - 106)^{2}$ (T - 106)^2
$31$	$T^{2} + 144T + 20736$ T^2 + 144*T + 20736
$37$	$T^{2} + 158T + 24964$ T^2 + 158*T + 24964
$41$	$(T + 390)^{2}$ (T + 390)^2
$43$	$(T + 508)^{2}$ (T + 508)^2
$47$	$T^{2} - 528T + 278784$ T^2 - 528*T + 278784
$53$	$T^{2} - 606T + 367236$ T^2 - 606*T + 367236
$59$	$T^{2} - 364T + 132496$ T^2 - 364*T + 132496
$61$	$T^{2} - 678T + 459684$ T^2 - 678*T + 459684
$67$	$T^{2} + 844T + 712336$ T^2 + 844*T + 712336
$71$	$(T - 8)^{2}$ (T - 8)^2
$73$	$T^{2} + 422T + 178084$ T^2 + 422*T + 178084
$79$	$T^{2} + 384T + 147456$ T^2 + 384*T + 147456
$83$	$(T + 548)^{2}$ (T + 548)^2
$89$	$T^{2} + 1194 T + 1425636$ T^2 + 1194*T + 1425636
$97$	$(T - 1502)^{2}$ (T - 1502)^2
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