# Properties

 Label 2736.1.bk.a Level $2736$ Weight $1$ Character orbit 2736.bk Analytic conductor $1.365$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ CM discriminant -3 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,1,Mod(847,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 0, 4]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("2736.847");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2736 = 2^{4} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2736.bk (of order $$6$$, 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: $$1.36544187456$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ 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: yes Projective image: $$D_{6}$$ Projective field: Galois closure of 6.2.225194688.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6}^{2} - \zeta_{6}) q^{7}+O(q^{10})$$ q + (-z^2 - z) * q^7 $$q + ( - \zeta_{6}^{2} - \zeta_{6}) q^{7} - \zeta_{6} q^{13} - q^{19} + \zeta_{6} q^{25} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{31} - q^{37} + ( - \zeta_{6} - 1) q^{43} + (\zeta_{6}^{2} - \zeta_{6} - 1) q^{49} + \zeta_{6} q^{61} + ( - \zeta_{6}^{2} + 1) q^{67} + \zeta_{6}^{2} q^{73} + (\zeta_{6} + 1) q^{79} + (\zeta_{6}^{2} - 1) q^{91} - 2 \zeta_{6}^{2} q^{97} +O(q^{100})$$ q + (-z^2 - z) * q^7 - z * q^13 - q^19 + z * q^25 + (-z^2 - z) * q^31 - q^37 + (-z - 1) * q^43 + (z^2 - z - 1) * q^49 + z * q^61 + (-z^2 + 1) * q^67 + z^2 * q^73 + (z + 1) * q^79 + (z^2 - 1) * q^91 - 2*z^2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q - q^{13} - 2 q^{19} + q^{25} - 2 q^{37} - 3 q^{43} - 4 q^{49} + q^{61} + 3 q^{67} - q^{73} + 3 q^{79} - 3 q^{91} + 2 q^{97}+O(q^{100})$$ 2 * q - q^13 - 2 * q^19 + q^25 - 2 * q^37 - 3 * q^43 - 4 * q^49 + q^61 + 3 * q^67 - q^73 + 3 * q^79 - 3 * q^91 + 2 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$$.

 $$n$$ $$1009$$ $$1217$$ $$1711$$ $$2053$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
847.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 0 0 1.73205i 0 0 0
2287.1 0 0 0 0 0 1.73205i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
76.g odd 6 1 inner
228.m even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.1.bk.a 2
3.b odd 2 1 CM 2736.1.bk.a 2
4.b odd 2 1 2736.1.bk.b yes 2
12.b even 2 1 2736.1.bk.b yes 2
19.c even 3 1 2736.1.bk.b yes 2
57.h odd 6 1 2736.1.bk.b yes 2
76.g odd 6 1 inner 2736.1.bk.a 2
228.m even 6 1 inner 2736.1.bk.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2736.1.bk.a 2 1.a even 1 1 trivial
2736.1.bk.a 2 3.b odd 2 1 CM
2736.1.bk.a 2 76.g odd 6 1 inner
2736.1.bk.a 2 228.m even 6 1 inner
2736.1.bk.b yes 2 4.b odd 2 1
2736.1.bk.b yes 2 12.b even 2 1
2736.1.bk.b yes 2 19.c even 3 1
2736.1.bk.b yes 2 57.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{43}^{2} + 3T_{43} + 3$$ acting on $$S_{1}^{\mathrm{new}}(2736, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 3$$
$11$ $$T^{2}$$
$13$ $$T^{2} + T + 1$$
$17$ $$T^{2}$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2} + 3$$
$37$ $$(T + 1)^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 3T + 3$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} - T + 1$$
$67$ $$T^{2} - 3T + 3$$
$71$ $$T^{2}$$
$73$ $$T^{2} + T + 1$$
$79$ $$T^{2} - 3T + 3$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2} - 2T + 4$$