# Properties

 Label 7056.2.a.bf Level $7056$ Weight $2$ Character orbit 7056.a Self dual yes Analytic conductor $56.342$ Analytic rank $1$ Dimension $1$ CM discriminant -3 Inner twists $2$

# 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 63) 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)

 $$f(q)$$ $$=$$ $$q+O(q^{10})$$ q $$q + 7 q^{13} - 7 q^{19} - 5 q^{25} - 7 q^{31} - q^{37} - 5 q^{43} - 14 q^{61} - 11 q^{67} + 7 q^{73} + 13 q^{79} - 14 q^{97}+O(q^{100})$$ q + 7 * q^13 - 7 * q^19 - 5 * q^25 - 7 * q^31 - q^37 - 5 * q^43 - 14 * q^61 - 11 * q^67 + 7 * q^73 + 13 * q^79 - 14 * q^97

## 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
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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7056.2.a.bf 1
3.b odd 2 1 CM 7056.2.a.bf 1
4.b odd 2 1 441.2.a.e 1
7.b odd 2 1 7056.2.a.y 1
7.d odd 6 2 1008.2.s.j 2
12.b even 2 1 441.2.a.e 1
21.c even 2 1 7056.2.a.y 1
21.g even 6 2 1008.2.s.j 2
28.d even 2 1 441.2.a.d 1
28.f even 6 2 63.2.e.a 2
28.g odd 6 2 441.2.e.c 2
84.h odd 2 1 441.2.a.d 1
84.j odd 6 2 63.2.e.a 2
84.n even 6 2 441.2.e.c 2
252.n even 6 2 567.2.g.d 2
252.r odd 6 2 567.2.h.c 2
252.bj even 6 2 567.2.h.c 2
252.bn odd 6 2 567.2.g.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.e.a 2 28.f even 6 2
63.2.e.a 2 84.j odd 6 2
441.2.a.d 1 28.d even 2 1
441.2.a.d 1 84.h odd 2 1
441.2.a.e 1 4.b odd 2 1
441.2.a.e 1 12.b even 2 1
441.2.e.c 2 28.g odd 6 2
441.2.e.c 2 84.n even 6 2
567.2.g.d 2 252.n even 6 2
567.2.g.d 2 252.bn odd 6 2
567.2.h.c 2 252.r odd 6 2
567.2.h.c 2 252.bj even 6 2
1008.2.s.j 2 7.d odd 6 2
1008.2.s.j 2 21.g even 6 2
7056.2.a.y 1 7.b odd 2 1
7056.2.a.y 1 21.c even 2 1
7056.2.a.bf 1 1.a even 1 1 trivial
7056.2.a.bf 1 3.b odd 2 1 CM

## 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 $$T_{11}$$ T11 $$T_{13} - 7$$ T13 - 7 $$T_{17}$$ T17 $$T_{23}$$ T23

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T - 7$$
$17$ $$T$$
$19$ $$T + 7$$
$23$ $$T$$
$29$ $$T$$
$31$ $$T + 7$$
$37$ $$T + 1$$
$41$ $$T$$
$43$ $$T + 5$$
$47$ $$T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T + 14$$
$67$ $$T + 11$$
$71$ $$T$$
$73$ $$T - 7$$
$79$ $$T - 13$$
$83$ $$T$$
$89$ $$T$$
$97$ $$T + 14$$