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Let us see what is exactly CC. For this, we compare:

sage: CC(0.13816890584139213)
sage: CC
Complex Field with 53 bits of precision
sage: CC(0.1234567890123456789012345678901234567890)

The first input corresponds to initializing the real part of the wanted point. Instead of 0.13816890584139213 we have a printed version going only up to 0.138168905841392. Sometimes the printed version is such a rough information. So what is CC. It is an object collecting inexact information, only $53$ bits are collected. So from the next test number we have only 0.123456789012346. If we try to print more... print(a.n(200)) runs into a TypeError: cannot approximate to a precision of 200 bits, use at most 53 bits...

So let us try from the start with a higher precision:

C = ComplexField(150)
print(f"C is {C}")

PD = HyperbolicPlane().PD()
p = PD.get_point(C(0.138168905841392130000000000) + C(0.4878012008585488000000000)*i)    # our C instead of CC


And we obtain:

C is Complex Field with 150 bits of precision
Point in PD 0.13816890584139213000000000000000000000000000 + 0.48780120085854880000000000000000000000000000*I

We have the decimals we want...