LMFDB - Newform orbit 936.2.t.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [936,2,Mod(217,936)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(936, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("936.217");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$936 = 2^{3} \cdot 3^{2} \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	936.t (of order $3$ , degree $2$ , 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:	$7.47399762919$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 312)
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.

$f(q)$	$=$	$q - 3 q^{5} + 4 \zeta_{6} q^{7} + ( - 4 \zeta_{6} + 4) q^{11} + ( - 3 \zeta_{6} - 1) q^{13} + 3 \zeta_{6} q^{17} + 4 \zeta_{6} q^{19} + (8 \zeta_{6} - 8) q^{23} + 4 q^{25} + (5 \zeta_{6} - 5) q^{29} - 8 q^{31}+ \cdots + 14 \zeta_{6} q^{97}+O(q^{100})$ q - 3 * q^5 + 4z q^7 + (-4z + 4) q^11 + (-3z - 1) q^13 + 3z q^17 + 4z q^19 + (8z - 8) q^23 + 4 * q^25 + (5z - 5) q^29 - 8 * q^31 - 12z q^35 + (7z - 7) q^37 + (9z - 9) q^41 - 8z q^43 + 4 * q^47 + (9z - 9) q^49 + 5 * q^53 + (12z - 12) q^55 + 4z q^59 + 5z q^61 + (9z + 3) q^65 + (8z - 8) q^67 - 4z q^71 + 11 * q^73 + 16 * q^77 - 4 * q^79 - 9z q^85 + (6z - 6) q^89 + (-16z + 12) q^91 - 12z q^95 + 14z q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 6 q^{5} + 4 q^{7} + 4 q^{11} - 5 q^{13} + 3 q^{17} + 4 q^{19} - 8 q^{23} + 8 q^{25} - 5 q^{29} - 16 q^{31} - 12 q^{35} - 7 q^{37} - 9 q^{41} - 8 q^{43} + 8 q^{47} - 9 q^{49} + 10 q^{53} - 12 q^{55}+ \cdots + 14 q^{97}+O(q^{100})$ 2 * q - 6 * q^5 + 4 * q^7 + 4 * q^11 - 5 * q^13 + 3 * q^17 + 4 * q^19 - 8 * q^23 + 8 * q^25 - 5 * q^29 - 16 * q^31 - 12 * q^35 - 7 * q^37 - 9 * q^41 - 8 * q^43 + 8 * q^47 - 9 * q^49 + 10 * q^53 - 12 * q^55 + 4 * q^59 + 5 * q^61 + 15 * q^65 - 8 * q^67 - 4 * q^71 + 22 * q^73 + 32 * q^77 - 8 * q^79 - 9 * q^85 - 6 * q^89 + 8 * q^91 - 12 * q^95 + 14 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/936\mathbb{Z}\right)^\times$ .

$n$	$145$	$209$	$469$	$703$
$\chi(n)$	$-\zeta_{6}$	$1$	$1$	$1$

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}

217.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−3.00000

2.00000

3.46410i

289.1

−3.00000

2.00000

−

3.46410i

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	936.2.t.b		2
3.b	odd	2	1		312.2.q.c	✓	2
4.b	odd	2	1		1872.2.t.a		2
12.b	even	2	1		624.2.q.e		2
13.c	even	3	1	inner	936.2.t.b		2
39.h	odd	6	1		4056.2.a.b		1
39.i	odd	6	1		312.2.q.c	✓	2
39.i	odd	6	1		4056.2.a.j		1
39.k	even	12	2		4056.2.c.b		2
52.j	odd	6	1		1872.2.t.a		2
156.p	even	6	1		624.2.q.e		2
156.p	even	6	1		8112.2.a.bh		1
156.r	even	6	1		8112.2.a.r		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
312.2.q.c	✓	2	3.b	odd	2	1
312.2.q.c	✓	2	39.i	odd	6	1
624.2.q.e		2	12.b	even	2	1
624.2.q.e		2	156.p	even	6	1
936.2.t.b		2	1.a	even	1	1	trivial
936.2.t.b		2	13.c	even	3	1	inner
1872.2.t.a		2	4.b	odd	2	1
1872.2.t.a		2	52.j	odd	6	1
4056.2.a.b		1	39.h	odd	6	1
4056.2.a.j		1	39.i	odd	6	1
4056.2.c.b		2	39.k	even	12	2
8112.2.a.r		1	156.r	even	6	1
8112.2.a.bh		1	156.p	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(936, [\chi])$ :

T_{5} + 3

T_{7}^{2} - 4T_{7} + 16

Hecke characteristic polynomials

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