LMFDB - Newform orbit 392.2.i.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [392,2,Mod(177,392)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(392, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("392.177");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$392 = 2^{3} \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	392.i (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:	$3.13013575923$
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:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 56)
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.

$f(q)$	$=$	$q + 2 \zeta_{6} q^{5} + 3 \zeta_{6} q^{9} + ( - 4 \zeta_{6} + 4) q^{11} - 2 q^{13} + (6 \zeta_{6} - 6) q^{17} + 8 \zeta_{6} q^{19} + ( - \zeta_{6} + 1) q^{25} + 6 q^{29} + ( - 8 \zeta_{6} + 8) q^{31} + 2 \zeta_{6} q^{37}+ \cdots + 12 q^{99}+O(q^{100})$ q + 2z q^5 + 3z q^9 + (-4z + 4) q^11 - 2 * q^13 + (6z - 6) q^17 + 8z q^19 + (-z + 1) * q^25 + 6 * q^29 + (-8z + 8) q^31 + 2z q^37 - 2 * q^41 - 4 * q^43 + (6z - 6) q^45 - 8z q^47 + (6z - 6) q^53 + 8 * q^55 - 6z q^61 - 4z q^65 + (-4z + 4) q^67 - 8 * q^71 + (-10z + 10) q^73 - 16z q^79 + (9z - 9) q^81 - 8 * q^83 - 12 * q^85 - 6z q^89 + (16z - 16) q^95 + 6 * q^97 + 12 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 2 q^{5} + 3 q^{9} + 4 q^{11} - 4 q^{13} - 6 q^{17} + 8 q^{19} + q^{25} + 12 q^{29} + 8 q^{31} + 2 q^{37} - 4 q^{41} - 8 q^{43} - 6 q^{45} - 8 q^{47} - 6 q^{53} + 16 q^{55} - 6 q^{61} - 4 q^{65} + 4 q^{67}+ \cdots + 24 q^{99}+O(q^{100})$ 2 * q + 2 * q^5 + 3 * q^9 + 4 * q^11 - 4 * q^13 - 6 * q^17 + 8 * q^19 + q^25 + 12 * q^29 + 8 * q^31 + 2 * q^37 - 4 * q^41 - 8 * q^43 - 6 * q^45 - 8 * q^47 - 6 * q^53 + 16 * q^55 - 6 * q^61 - 4 * q^65 + 4 * q^67 - 16 * q^71 + 10 * q^73 - 16 * q^79 - 9 * q^81 - 16 * q^83 - 24 * q^85 - 6 * q^89 - 16 * q^95 + 12 * q^97 + 24 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/392\mathbb{Z}\right)^\times$ .

$n$	$197$	$295$	$297$
$\chi(n)$	$1$	$1$	$-\zeta_{6}$

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}

177.1

0.500000	−	0.866025i
0.500000	+	0.866025i

1.00000

−

1.73205i

1.50000

−

2.59808i

361.1

1.00000

1.73205i

1.50000

2.59808i

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	392.2.i.d		2
3.b	odd	2	1		3528.2.s.e		2
4.b	odd	2	1		784.2.i.g		2
7.b	odd	2	1		392.2.i.c		2
7.c	even	3	1		392.2.a.d		1
7.c	even	3	1	inner	392.2.i.d		2
7.d	odd	6	1		56.2.a.a	✓	1
7.d	odd	6	1		392.2.i.c		2
21.c	even	2	1		3528.2.s.t		2
21.g	even	6	1		504.2.a.c		1
21.g	even	6	1		3528.2.s.t		2
21.h	odd	6	1		3528.2.a.x		1
21.h	odd	6	1		3528.2.s.e		2
28.d	even	2	1		784.2.i.e		2
28.f	even	6	1		112.2.a.b		1
28.f	even	6	1		784.2.i.e		2
28.g	odd	6	1		784.2.a.e		1
28.g	odd	6	1		784.2.i.g		2
35.i	odd	6	1		1400.2.a.g		1
35.j	even	6	1		9800.2.a.u		1
35.k	even	12	2		1400.2.g.g		2
56.j	odd	6	1		448.2.a.d		1
56.k	odd	6	1		3136.2.a.p		1
56.m	even	6	1		448.2.a.e		1
56.p	even	6	1		3136.2.a.q		1
77.i	even	6	1		6776.2.a.g		1
84.j	odd	6	1		1008.2.a.d		1
84.n	even	6	1		7056.2.a.bo		1
91.s	odd	6	1		9464.2.a.c		1
112.v	even	12	2		1792.2.b.d		2
112.x	odd	12	2		1792.2.b.i		2
140.s	even	6	1		2800.2.a.p		1
140.x	odd	12	2		2800.2.g.p		2
168.ba	even	6	1		4032.2.a.bb		1
168.be	odd	6	1		4032.2.a.bk		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
56.2.a.a	✓	1	7.d	odd	6	1
112.2.a.b		1	28.f	even	6	1
392.2.a.d		1	7.c	even	3	1
392.2.i.c		2	7.b	odd	2	1
392.2.i.c		2	7.d	odd	6	1
392.2.i.d		2	1.a	even	1	1	trivial
392.2.i.d		2	7.c	even	3	1	inner
448.2.a.d		1	56.j	odd	6	1
448.2.a.e		1	56.m	even	6	1
504.2.a.c		1	21.g	even	6	1
784.2.a.e		1	28.g	odd	6	1
784.2.i.e		2	28.d	even	2	1
784.2.i.e		2	28.f	even	6	1
784.2.i.g		2	4.b	odd	2	1
784.2.i.g		2	28.g	odd	6	1
1008.2.a.d		1	84.j	odd	6	1
1400.2.a.g		1	35.i	odd	6	1
1400.2.g.g		2	35.k	even	12	2
1792.2.b.d		2	112.v	even	12	2
1792.2.b.i		2	112.x	odd	12	2
2800.2.a.p		1	140.s	even	6	1
2800.2.g.p		2	140.x	odd	12	2
3136.2.a.p		1	56.k	odd	6	1
3136.2.a.q		1	56.p	even	6	1
3528.2.a.x		1	21.h	odd	6	1
3528.2.s.e		2	3.b	odd	2	1
3528.2.s.e		2	21.h	odd	6	1
3528.2.s.t		2	21.c	even	2	1
3528.2.s.t		2	21.g	even	6	1
4032.2.a.bb		1	168.ba	even	6	1
4032.2.a.bk		1	168.be	odd	6	1
6776.2.a.g		1	77.i	even	6	1
7056.2.a.bo		1	84.n	even	6	1
9464.2.a.c		1	91.s	odd	6	1
9800.2.a.u		1	35.j	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(392, [\chi])$ :

T_{3}

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

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2}$ T^2
$5$	$T^{2} - 2T + 4$ T^2 - 2*T + 4
$7$	$T^{2}$ T^2
$11$	$T^{2} - 4T + 16$ T^2 - 4*T + 16
$13$	$(T + 2)^{2}$ (T + 2)^2
$17$	$T^{2} + 6T + 36$ T^2 + 6*T + 36
$19$	$T^{2} - 8T + 64$ T^2 - 8*T + 64
$23$	$T^{2}$ T^2
$29$	$(T - 6)^{2}$ (T - 6)^2
$31$	$T^{2} - 8T + 64$ T^2 - 8*T + 64
$37$	$T^{2} - 2T + 4$ T^2 - 2*T + 4
$41$	$(T + 2)^{2}$ (T + 2)^2
$43$	$(T + 4)^{2}$ (T + 4)^2
$47$	$T^{2} + 8T + 64$ T^2 + 8*T + 64
$53$	$T^{2} + 6T + 36$ T^2 + 6*T + 36
$59$	$T^{2}$ T^2
$61$	$T^{2} + 6T + 36$ T^2 + 6*T + 36
$67$	$T^{2} - 4T + 16$ T^2 - 4*T + 16
$71$	$(T + 8)^{2}$ (T + 8)^2
$73$	$T^{2} - 10T + 100$ T^2 - 10*T + 100
$79$	$T^{2} + 16T + 256$ T^2 + 16*T + 256
$83$	$(T + 8)^{2}$ (T + 8)^2
$89$	$T^{2} + 6T + 36$ T^2 + 6*T + 36
$97$	$(T - 6)^{2}$ (T - 6)^2
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