# Properties

 Label 6840.2.a.bg Level $6840$ Weight $2$ Character orbit 6840.a Self dual yes Analytic conductor $54.618$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6840,2,Mod(1,6840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6840, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6840.a (trivial)

## 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: yes Analytic conductor: $$54.6176749826$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.229.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 4x - 1$$ x^3 - 4*x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 760) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

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

 $$f(q)$$ $$=$$ $$q - q^{5} + \beta_1 q^{7}+O(q^{10})$$ q - q^5 + b1 * q^7 $$q - q^{5} + \beta_1 q^{7} + (\beta_{2} + 2) q^{13} + ( - \beta_1 - 2) q^{17} + q^{19} + ( - 2 \beta_{2} + \beta_1) q^{23} + q^{25} + ( - \beta_1 - 6) q^{29} - 2 \beta_1 q^{31} - \beta_1 q^{35} + ( - \beta_{2} - \beta_1 + 2) q^{37} + (2 \beta_{2} - 2 \beta_1 - 2) q^{41} + ( - 2 \beta_{2} - 4) q^{43} + (2 \beta_{2} + 2 \beta_1) q^{47} + ( - 2 \beta_{2} + \beta_1 + 1) q^{49} + (\beta_{2} - 2 \beta_1 - 2) q^{53} + (4 \beta_{2} - \beta_1 - 4) q^{59} + ( - 2 \beta_{2} + 2) q^{61} + ( - \beta_{2} - 2) q^{65} + ( - 5 \beta_{2} + 2 \beta_1) q^{67} - 4 \beta_1 q^{71} + (4 \beta_{2} - \beta_1 + 2) q^{73} - 2 \beta_{2} q^{79} + (2 \beta_1 + 4) q^{83} + (\beta_1 + 2) q^{85} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{89} + ( - 2 \beta_{2} + \beta_1) q^{91} - q^{95} + (\beta_{2} - 3 \beta_1 - 2) q^{97}+O(q^{100})$$ q - q^5 + b1 * q^7 + (b2 + 2) * q^13 + (-b1 - 2) * q^17 + q^19 + (-2*b2 + b1) * q^23 + q^25 + (-b1 - 6) * q^29 - 2*b1 * q^31 - b1 * q^35 + (-b2 - b1 + 2) * q^37 + (2*b2 - 2*b1 - 2) * q^41 + (-2*b2 - 4) * q^43 + (2*b2 + 2*b1) * q^47 + (-2*b2 + b1 + 1) * q^49 + (b2 - 2*b1 - 2) * q^53 + (4*b2 - b1 - 4) * q^59 + (-2*b2 + 2) * q^61 + (-b2 - 2) * q^65 + (-5*b2 + 2*b1) * q^67 - 4*b1 * q^71 + (4*b2 - b1 + 2) * q^73 - 2*b2 * q^79 + (2*b1 + 4) * q^83 + (b1 + 2) * q^85 + (-2*b2 + 2*b1 - 2) * q^89 + (-2*b2 + b1) * q^91 - q^95 + (b2 - 3*b1 - 2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{5} - q^{7}+O(q^{10})$$ 3 * q - 3 * q^5 - q^7 $$3 q - 3 q^{5} - q^{7} + 5 q^{13} - 5 q^{17} + 3 q^{19} + q^{23} + 3 q^{25} - 17 q^{29} + 2 q^{31} + q^{35} + 8 q^{37} - 6 q^{41} - 10 q^{43} - 4 q^{47} + 4 q^{49} - 5 q^{53} - 15 q^{59} + 8 q^{61} - 5 q^{65} + 3 q^{67} + 4 q^{71} + 3 q^{73} + 2 q^{79} + 10 q^{83} + 5 q^{85} - 6 q^{89} + q^{91} - 3 q^{95} - 4 q^{97}+O(q^{100})$$ 3 * q - 3 * q^5 - q^7 + 5 * q^13 - 5 * q^17 + 3 * q^19 + q^23 + 3 * q^25 - 17 * q^29 + 2 * q^31 + q^35 + 8 * q^37 - 6 * q^41 - 10 * q^43 - 4 * q^47 + 4 * q^49 - 5 * q^53 - 15 * q^59 + 8 * q^61 - 5 * q^65 + 3 * q^67 + 4 * q^71 + 3 * q^73 + 2 * q^79 + 10 * q^83 + 5 * q^85 - 6 * q^89 + q^91 - 3 * q^95 - 4 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 4x - 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2} + \nu - 3$$ v^2 + v - 3 $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ v^2 - v - 3
 $$\nu$$ $$=$$ $$( -\beta_{2} + \beta_1 ) / 2$$ (-b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{2} + \beta _1 + 6 ) / 2$$ (b2 + b1 + 6) / 2

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −0.254102 −1.86081 2.11491
0 0 0 −1.00000 0 −3.18953 0 0 0
1.2 0 0 0 −1.00000 0 −1.39821 0 0 0
1.3 0 0 0 −1.00000 0 3.58774 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$+1$$
$$3$$ $$-1$$
$$5$$ $$+1$$
$$19$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6840.2.a.bg 3
3.b odd 2 1 760.2.a.j 3
12.b even 2 1 1520.2.a.s 3
15.d odd 2 1 3800.2.a.x 3
15.e even 4 2 3800.2.d.l 6
24.f even 2 1 6080.2.a.bq 3
24.h odd 2 1 6080.2.a.bv 3
60.h even 2 1 7600.2.a.bq 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.a.j 3 3.b odd 2 1
1520.2.a.s 3 12.b even 2 1
3800.2.a.x 3 15.d odd 2 1
3800.2.d.l 6 15.e even 4 2
6080.2.a.bq 3 24.f even 2 1
6080.2.a.bv 3 24.h odd 2 1
6840.2.a.bg 3 1.a even 1 1 trivial
7600.2.a.bq 3 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6840))$$:

 $$T_{7}^{3} + T_{7}^{2} - 12T_{7} - 16$$ T7^3 + T7^2 - 12*T7 - 16 $$T_{11}$$ T11 $$T_{13}^{3} - 5T_{13}^{2} + 2T_{13} + 4$$ T13^3 - 5*T13^2 + 2*T13 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3}$$
$5$ $$(T + 1)^{3}$$
$7$ $$T^{3} + T^{2} + \cdots - 16$$
$11$ $$T^{3}$$
$13$ $$T^{3} - 5 T^{2} + \cdots + 4$$
$17$ $$T^{3} + 5 T^{2} + \cdots - 4$$
$19$ $$(T - 1)^{3}$$
$23$ $$T^{3} - T^{2} + \cdots + 64$$
$29$ $$T^{3} + 17 T^{2} + \cdots + 124$$
$31$ $$T^{3} - 2 T^{2} + \cdots + 128$$
$37$ $$T^{3} - 8T^{2} + 8$$
$41$ $$T^{3} + 6 T^{2} + \cdots - 56$$
$43$ $$T^{3} + 10 T^{2} + \cdots - 32$$
$47$ $$T^{3} + 4 T^{2} + \cdots + 128$$
$53$ $$T^{3} + 5 T^{2} + \cdots + 52$$
$59$ $$T^{3} + 15 T^{2} + \cdots - 784$$
$61$ $$T^{3} - 8 T^{2} + \cdots + 64$$
$67$ $$T^{3} - 3 T^{2} + \cdots + 1052$$
$71$ $$T^{3} - 4 T^{2} + \cdots + 1024$$
$73$ $$T^{3} - 3 T^{2} + \cdots - 292$$
$79$ $$T^{3} - 2 T^{2} + \cdots + 32$$
$83$ $$T^{3} - 10 T^{2} + \cdots + 32$$
$89$ $$T^{3} + 6 T^{2} + \cdots - 184$$
$97$ $$T^{3} + 4 T^{2} + \cdots + 296$$