LMFDB - Newform orbit 936.2.q.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [936,2,Mod(313,936)]

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, 4, 0]))

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

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

chi := DirichletCharacter("936.313");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$936 = 2^{3} \cdot 3^{2} \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	936.q (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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	yes
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.

$f(q)$	$=$	$q + ( - \zeta_{6} - 1) q^{3} + 4 \zeta_{6} q^{5} + (2 \zeta_{6} - 2) q^{7} + 3 \zeta_{6} q^{9} + (\zeta_{6} - 1) q^{11} + \zeta_{6} q^{13} + ( - 8 \zeta_{6} + 4) q^{15} - 3 q^{17} - 5 q^{19} + ( - 2 \zeta_{6} + 4) q^{21}+ \cdots - 3 q^{99}+O(q^{100})$ q + (-z - 1) * q^3 + 4z q^5 + (2z - 2) q^7 + 3z q^9 + (z - 1) * q^11 + z * q^13 + (-8z + 4) q^15 - 3 * q^17 - 5 * q^19 + (-2z + 4) q^21 + 2z q^23 + (11z - 11) q^25 + (-6z + 3) q^27 + (-6z + 6) q^29 - 8z q^31 + (-z + 2) * q^33 - 8 * q^35 - 4 * q^37 + (-2z + 1) q^39 - 5z q^41 + (-z + 1) * q^43 + (12z - 12) q^45 + (6z - 6) q^47 + 3z q^49 + (3z + 3) q^51 - 4 * q^55 + (5z + 5) q^57 + 9z q^59 + (12z - 12) q^61 - 6 * q^63 + (4z - 4) q^65 + 15z q^67 + (-4z + 2) q^69 - 8 * q^71 + 7 * q^73 + (-11z + 22) q^75 - 2z q^77 + (9z - 9) q^81 + (-4z + 4) q^83 - 12z q^85 + (6z - 12) q^87 - 10 * q^89 - 2 * q^91 + (16z - 8) q^93 - 20z q^95 + (17z - 17) q^97 - 3 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 3 q^{3} + 4 q^{5} - 2 q^{7} + 3 q^{9} - q^{11} + q^{13} - 6 q^{17} - 10 q^{19} + 6 q^{21} + 2 q^{23} - 11 q^{25} + 6 q^{29} - 8 q^{31} + 3 q^{33} - 16 q^{35} - 8 q^{37} - 5 q^{41} + q^{43} - 12 q^{45}+ \cdots - 6 q^{99}+O(q^{100})$ 2 * q - 3 * q^3 + 4 * q^5 - 2 * q^7 + 3 * q^9 - q^11 + q^13 - 6 * q^17 - 10 * q^19 + 6 * q^21 + 2 * q^23 - 11 * q^25 + 6 * q^29 - 8 * q^31 + 3 * q^33 - 16 * q^35 - 8 * q^37 - 5 * q^41 + q^43 - 12 * q^45 - 6 * q^47 + 3 * q^49 + 9 * q^51 - 8 * q^55 + 15 * q^57 + 9 * q^59 - 12 * q^61 - 12 * q^63 - 4 * q^65 + 15 * q^67 - 16 * q^71 + 14 * q^73 + 33 * q^75 - 2 * q^77 - 9 * q^81 + 4 * q^83 - 12 * q^85 - 18 * q^87 - 20 * q^89 - 4 * q^91 - 20 * q^95 - 17 * q^97 - 6 * q^99

Character values

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

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

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}

313.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−1.50000

−

0.866025i

2.00000

3.46410i

−1.00000

1.73205i

1.50000

2.59808i

625.1

−1.50000

0.866025i

2.00000

−

3.46410i

−1.00000

−

1.73205i

1.50000

−

2.59808i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.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.q.a	✓	2
3.b	odd	2	1		2808.2.q.a		2
9.c	even	3	1	inner	936.2.q.a	✓	2
9.c	even	3	1		8424.2.a.b		1
9.d	odd	6	1		2808.2.q.a		2
9.d	odd	6	1		8424.2.a.h		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
936.2.q.a	✓	2	1.a	even	1	1	trivial
936.2.q.a	✓	2	9.c	even	3	1	inner
2808.2.q.a		2	3.b	odd	2	1
2808.2.q.a		2	9.d	odd	6	1
8424.2.a.b		1	9.c	even	3	1
8424.2.a.h		1	9.d	odd	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}^{2} - 4T_{5} + 16

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

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} + 3T + 3$ T^2 + 3*T + 3
$5$	$T^{2} - 4T + 16$ T^2 - 4*T + 16
$7$	$T^{2} + 2T + 4$ T^2 + 2*T + 4
$11$	$T^{2} + T + 1$ T^2 + T + 1
$13$	$T^{2} - T + 1$ T^2 - T + 1
$17$	$(T + 3)^{2}$ (T + 3)^2
$19$	$(T + 5)^{2}$ (T + 5)^2
$23$	$T^{2} - 2T + 4$ T^2 - 2*T + 4
$29$	$T^{2} - 6T + 36$ T^2 - 6*T + 36
$31$	$T^{2} + 8T + 64$ T^2 + 8*T + 64
$37$	$(T + 4)^{2}$ (T + 4)^2
$41$	$T^{2} + 5T + 25$ T^2 + 5*T + 25
$43$	$T^{2} - T + 1$ T^2 - T + 1
$47$	$T^{2} + 6T + 36$ T^2 + 6*T + 36
$53$	$T^{2}$ T^2
$59$	$T^{2} - 9T + 81$ T^2 - 9*T + 81
$61$	$T^{2} + 12T + 144$ T^2 + 12*T + 144
$67$	$T^{2} - 15T + 225$ T^2 - 15*T + 225
$71$	$(T + 8)^{2}$ (T + 8)^2
$73$	$(T - 7)^{2}$ (T - 7)^2
$79$	$T^{2}$ T^2
$83$	$T^{2} - 4T + 16$ T^2 - 4*T + 16
$89$	$(T + 10)^{2}$ (T + 10)^2
$97$	$T^{2} + 17T + 289$ T^2 + 17*T + 289
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