LMFDB - Newform orbit 2160.2.o.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,2,Mod(2159,2160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2160, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))

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

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

chi := DirichletCharacter("2160.2159");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$2160 = 2^{4} \cdot 3^{3} \cdot 5$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2160.o (of order $2$ , degree $1$ , 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:	$17.2476868366$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - x^{3} - 2x^{2} - 3x + 9$ x^4 - x^3 - 2x^2 - 3x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$2$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$ -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 basis $1,\beta_1,\beta_2,\beta_3$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + ( - \beta_{3} - \beta_{2} - 1) q^{5} + (\beta_{3} - \beta_1 + 2) q^{7} + (\beta_{3} - \beta_1 + 2) q^{11} - 2 \beta_{2} q^{13} + ( - \beta_{3} + \beta_1 + 1) q^{17} + (3 \beta_{3} + \beta_{2} + 3 \beta_1) q^{19}+ \cdots + ( - 3 \beta_{3} - 6 \beta_{2} - 3 \beta_1) q^{97}+O(q^{100})$ q + (-b3 - b2 - 1) * q^5 + (b3 - b1 + 2) * q^7 + (b3 - b1 + 2) * q^11 - 2b2 q^13 + (-b3 + b1 + 1) * q^17 + (3b3 + b2 + 3b1) * q^19 + (-3b3 - b2 - 3b1) * q^23 + (-b2 - 3b1 + 1) q^25 + (-2b3 + 2b2 - 2b1) q^29 - b2 * q^31 + (b2 + 3b1 - 6) q^35 - 4b2 q^37 + (2b3 - 2b2 + 2b1) q^41 - 6 * q^43 - 2b2 q^47 + (3b3 - 3b1 + 5) * q^49 - 3 * q^53 + (b2 + 3b1 - 6) q^55 + (-2b3 + 2b1 + 2) * q^59 + (3b3 - 3b1 + 7) * q^61 + (-2b3 - 4b1 - 4) * q^65 + (2b3 - 2b1 - 8) * q^67 + (4b3 - 4b1 + 2) * q^71 + (-9b3 - 4b2 - 9b1) q^73 + (3b3 - 3b1 + 12) * q^77 + (-3b3 - 3b2 - 3b1) q^79 + (6b3 + 5b2 + 6b1) q^83 + (-3b3 - 4b2 - 3b1 + 3) q^85 + (4b3 + 2b2 + 4b1) q^89 + (6b3 + 6b1) * q^91 + (-5b3 + 3b2 - b1 + 2) * q^95 + (-3b3 - 6b2 - 3b1) q^97
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 3 q^{5} + 6 q^{7} + 6 q^{11} + 6 q^{17} + q^{25} - 21 q^{35} - 24 q^{43} + 14 q^{49} - 12 q^{53} - 21 q^{55} + 12 q^{59} + 22 q^{61} - 18 q^{65} - 36 q^{67} + 42 q^{77} + 12 q^{85} + 12 q^{95}+O(q^{100})$ 4 * q - 3 * q^5 + 6 * q^7 + 6 * q^11 + 6 * q^17 + q^25 - 21 * q^35 - 24 * q^43 + 14 * q^49 - 12 * q^53 - 21 * q^55 + 12 * q^59 + 22 * q^61 - 18 * q^65 - 36 * q^67 + 42 * q^77 + 12 * q^85 + 12 * q^95

Basis of coefficient ring in terms of a root $\nu$ of $x^{4} - x^{3} - 2x^{2} - 3x + 9$ x^4 - x^3 - 2*x^2 - 3*x + 9 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( \nu^{3} + 2\nu^{2} - 2\nu - 6 ) / 3$ (v^3 + 2v^2 - 2v - 6) / 3
$\beta_{3}$	$=$	$( \nu^{3} - \nu^{2} - 2\nu - 3 ) / 3$ (v^3 - v^2 - 2*v - 3) / 3

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$-\beta_{3} + \beta_{2} + 1$ -b3 + b2 + 1
$\nu^{3}$	$=$	$2\beta_{3} + \beta_{2} + 2\beta _1 + 4$ 2b3 + b2 + 2b1 + 4

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times$ .

$n$	$271$	$1297$	$1621$	$2081$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

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}

2159.1

−1.18614	−	1.26217i
−1.18614	+	1.26217i
1.68614	+	0.396143i
1.68614	−	0.396143i

−2.18614

−

0.469882i

4.37228

2159.2

−2.18614

0.469882i

4.37228

2159.3

0.686141

−

2.12819i

−1.37228

2159.4

0.686141

2.12819i

−1.37228

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
60.h	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2160.2.o.b	yes	4
3.b	odd	2	1		2160.2.o.d	yes	4
4.b	odd	2	1		2160.2.o.a	✓	4
5.b	even	2	1		2160.2.o.c	yes	4
12.b	even	2	1		2160.2.o.c	yes	4
15.d	odd	2	1		2160.2.o.a	✓	4
20.d	odd	2	1		2160.2.o.d	yes	4
60.h	even	2	1	inner	2160.2.o.b	yes	4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2160.2.o.a	✓	4	4.b	odd	2	1
2160.2.o.a	✓	4	15.d	odd	2	1
2160.2.o.b	yes	4	1.a	even	1	1	trivial
2160.2.o.b	yes	4	60.h	even	2	1	inner
2160.2.o.c	yes	4	5.b	even	2	1
2160.2.o.c	yes	4	12.b	even	2	1
2160.2.o.d	yes	4	3.b	odd	2	1
2160.2.o.d	yes	4	20.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(2160, [\chi])$ :

T_{7}^{2} - 3T_{7} - 6

T_{11}^{2} - 3T_{11} - 6

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{4}$ T^4
$3$	$T^{4}$ T^4
$5$	$T^{4} + 3 T^{3} + \cdots + 25$ T^4 + 3T^3 + 4T^2 + 15*T + 25
$7$	$(T^{2} - 3 T - 6)^{2}$ (T^2 - 3*T - 6)^2
$11$	$(T^{2} - 3 T - 6)^{2}$ (T^2 - 3*T - 6)^2
$13$	$(T^{2} + 12)^{2}$ (T^2 + 12)^2
$17$	$(T^{2} - 3 T - 6)^{2}$ (T^2 - 3*T - 6)^2
$19$	$T^{4} + 51T^{2} + 576$ T^4 + 51*T^2 + 576
$23$	$T^{4} + 51T^{2} + 576$ T^4 + 51*T^2 + 576
$29$	$T^{4} + 76T^{2} + 256$ T^4 + 76*T^2 + 256
$31$	$(T^{2} + 3)^{2}$ (T^2 + 3)^2
$37$	$(T^{2} + 48)^{2}$ (T^2 + 48)^2
$41$	$T^{4} + 76T^{2} + 256$ T^4 + 76*T^2 + 256
$43$	$(T + 6)^{4}$ (T + 6)^4
$47$	$(T^{2} + 12)^{2}$ (T^2 + 12)^2
$53$	$(T + 3)^{4}$ (T + 3)^4
$59$	$(T^{2} - 6 T - 24)^{2}$ (T^2 - 6*T - 24)^2
$61$	$(T^{2} - 11 T - 44)^{2}$ (T^2 - 11*T - 44)^2
$67$	$(T^{2} + 18 T + 48)^{2}$ (T^2 + 18*T + 48)^2
$71$	$(T^{2} - 132)^{2}$ (T^2 - 132)^2
$73$	$T^{4} + 447 T^{2} + 49284$ T^4 + 447*T^2 + 49284
$79$	$T^{4} + 63T^{2} + 324$ T^4 + 63*T^2 + 324
$83$	$T^{4} + 222T^{2} + 7569$ T^4 + 222*T^2 + 7569
$89$	$(T^{2} + 44)^{2}$ (T^2 + 44)^2
$97$	$T^{4} + 171T^{2} + 1296$ T^4 + 171*T^2 + 1296
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