# Properties

 Label 7056.2.a.co Level $7056$ Weight $2$ Character orbit 7056.a Self dual yes Analytic conductor $56.342$ Analytic rank $1$ Dimension $2$ CM discriminant -7 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7056,2,Mod(1,7056)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7056, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("7056.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7056 = 2^{4} \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7056.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: $$56.3424436662$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{7})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 7$$ x^2 - 7 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 441) Fricke sign: $$1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-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 $$\beta = 2\sqrt{7}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q+O(q^{10})$$ q $$q + \beta q^{11} - \beta q^{23} - 5 q^{25} - 2 \beta q^{29} + 6 q^{37} - 12 q^{43} - 2 \beta q^{53} - 4 q^{67} + \beta q^{71} - 8 q^{79} +O(q^{100})$$ q + b * q^11 - b * q^23 - 5 * q^25 - 2*b * q^29 + 6 * q^37 - 12 * q^43 - 2*b * q^53 - 4 * q^67 + b * q^71 - 8 * q^79 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q - 10 q^{25} + 12 q^{37} - 24 q^{43} - 8 q^{67} - 16 q^{79}+O(q^{100})$$ 2 * q - 10 * q^25 + 12 * q^37 - 24 * q^43 - 8 * q^67 - 16 * q^79

## 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
 −2.64575 2.64575
0 0 0 0 0 0 0 0 0
1.2 0 0 0 0 0 0 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$7$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
3.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7056.2.a.co 2
3.b odd 2 1 inner 7056.2.a.co 2
4.b odd 2 1 441.2.a.h 2
7.b odd 2 1 CM 7056.2.a.co 2
12.b even 2 1 441.2.a.h 2
21.c even 2 1 inner 7056.2.a.co 2
28.d even 2 1 441.2.a.h 2
28.f even 6 2 441.2.e.h 4
28.g odd 6 2 441.2.e.h 4
84.h odd 2 1 441.2.a.h 2
84.j odd 6 2 441.2.e.h 4
84.n even 6 2 441.2.e.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.a.h 2 4.b odd 2 1
441.2.a.h 2 12.b even 2 1
441.2.a.h 2 28.d even 2 1
441.2.a.h 2 84.h odd 2 1
441.2.e.h 4 28.f even 6 2
441.2.e.h 4 28.g odd 6 2
441.2.e.h 4 84.j odd 6 2
441.2.e.h 4 84.n even 6 2
7056.2.a.co 2 1.a even 1 1 trivial
7056.2.a.co 2 3.b odd 2 1 inner
7056.2.a.co 2 7.b odd 2 1 CM
7056.2.a.co 2 21.c even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7056))$$:

 $$T_{5}$$ T5 $$T_{11}^{2} - 28$$ T11^2 - 28 $$T_{13}$$ T13 $$T_{17}$$ T17 $$T_{23}^{2} - 28$$ T23^2 - 28

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 28$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2} - 28$$
$29$ $$T^{2} - 112$$
$31$ $$T^{2}$$
$37$ $$(T - 6)^{2}$$
$41$ $$T^{2}$$
$43$ $$(T + 12)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} - 112$$
$59$ $$T^{2}$$
$61$ $$T^{2}$$
$67$ $$(T + 4)^{2}$$
$71$ $$T^{2} - 28$$
$73$ $$T^{2}$$
$79$ $$(T + 8)^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$