# IntegerModRing representation

How can I generate the elements of IntegerModring(n) in symmetric representation? For example Integers(6) = {0, 1,-1,2, -2, 3}

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Well, if you want to change the representation of the objects, so when you type Integers(6)(5) it returns -1, I think that it is not implemented in sage.

However, if computing the symmetric representant of an element in Integers(n) can be done with the following code:

I use this function in some multimodular algorithms.

def lift_pm(p):
"""
Compute a representative of the element p mod n in [-n/2 , n/2]

INPUT:

- p an integer mod n

OUTPUT:

- An integer r such that  -n/2 < r <= n/2

EXAMPLES::

sage: p = Mod(2,4)
sage: lift_pm(p)
2
sage: p = Mod(3,4)
sage: lift_pm(p)
-1
"""
r = p.lift()
if r*2 > p.modulus():
r -= p.modulus()
return r

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The true solution would be like in Maple : mod:=mods

Surprisingly in IntegerModrings Sage uses positive representations, but in polynomialrings over these it uses symmetric representation. So we can make the computations with the residues in the polynomialring.

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See the file sage/rings/polynomial/multi_polynomial_libsingular.pyx, in particular the comment at the beginning that Singular is used for multipolynomial ring over GF(p) when p is prime, and the _repr_ method at

You could use Python's list comprehension to make a list of the numbers you want:

sage: N = 5
sage: [(-1)^(n+1)*floor((n+1)/2) for n in range(N)]
[0, 1, -1, 2, -2]
sage: N = 6
sage: [(-1)^(n+1)*floor((n+1)/2) for n in range(N)]
[0, 1, -1, 2, -2, 3]


Is this the kind of thing you were looking for, or something else?

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Thank you. It's a good idea. I believed that the IntegerModRing have a special option.

But it is not teh solition. I think, My question was'nt clear. The positive representation for mod 6 is {1,2,3,4,5,6}. The symmetric representation is {1,2,3,-2,-1}. The problem is'nt that, how can generate this remainder set, but the Integers(6) ring how can use it permanently.