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Not an answer, but a comment needing more room than allowed in a comment :

sage: %cpaste
Pasting code; enter '--' alone on the line to stop or use Ctrl-D.
:# Raw cut and paste :
:
:f(x) = (3/174465461165747500*pi*(-1750000*I*pi*x^3 - 31250000*(224*pi + 45*sqrt(448*pi + 2025) + 2025)*x^2
:                                + 17500000000000000*I*pi*x)
:        *sqrt(92821652156334811582567480952850314403/10*pi^2/(224*pi + 45*sqrt(448*pi + 2025) + 2025)
:              + 98489794142024498175862287197250000*pi*sqrt(448*pi + 2025)
:              /(224*pi + 45*sqrt(448*pi + 2025) + 2025) + 7713517620898636162808584411766250000*pi
:              /(224*pi + 45*sqrt(448*pi + 2025) + 2025) + 659225266976959904108326638192187500*sqrt(448*pi + 2025)
:              /(224*pi + 45*sqrt(448*pi + 2025) + 2025) + 29665137013963195684874698718648437500
:              /(224*pi + 45*sqrt(448*pi + 2025) + 2025))
:        /(63*pi^2*x^4 - (504000*I*pi^2 + 67500*I*pi*(sqrt(448*pi + 2025) + 45))*x^3
:          - 3000000*(560224*pi^2 + 45*pi*(sqrt(448*pi + 2025) + 45))*x^2 + 8400000000000000000000*pi^2
:          - (-5040000000000000*I*pi^2 - 675000000000000*I*pi*(sqrt(448*pi + 2025) + 45))*x))
:
:# deleting blanks and newlines (adding a space before "+" and "-") : 
:
:g(x) = (3/174465461165747500*pi*(-1750000*I*pi*x^3 - 31250000*(224*pi + 45*sqrt(448*pi + 2025) + 2025)*x^2 + 17500000000000000*I*pi*x)*sqrt(92821652156334811582567480952850314403/10*pi^2/(224*pi + 45*sqrt(448*pi + 2025) + 2025) + 98489794142024498175862287197250000*pi*sqrt(448*pi + 2025)/(224*pi + 45*sqrt(448*pi + 2025) + 2025) + 7713517620898636162808584411766250000*pi/(224*pi + 45*sqrt(448*pi + 2025) + 2025) + 659225266976959904108326638192187500*sqrt(448*pi + 2025)/(224*pi + 45*sqrt(448*pi + 2025) + 2025) + 29665137013963195684874698718648437500/(224*pi + 45*sqrt(448*pi + 2025) + 2025))/(63*pi^2*x^4 - (504000*I*pi^2 + 67500*I*pi*(sqrt(448*pi + 2025) + 45))*x^3 - 3000000*(560224*pi^2 + 45*pi*(sqrt(448*pi + 2025) + 45))*x^2 + 8400000000000000000000*pi^2 - (-5040000000000000*I*pi^2 - 675000000000000*I*pi*(sqrt(448*pi + 2025) + 45))*x))
:
:--
sage: f(1).n()
418409.917305474 + 1.28757494213663e11*I
sage: g(1).n()
1.39111866114058e-12 + 6.95559330500736e-6*I

Something is amiss in your code entry... or Sage's interpretation.

And the problem you noted is still present :

sage: F = f.real()^2 + f.imag()^2
sage: Ff = (f.real()^2 + f.imag()^2).factor()
sage: G = g.real()^2 + g.imag()^2
sage: Gf = (g.real()^2 + g.imag()^2).factor()
sage: F(1).n()
1.65784923163565e22
sage: Ff(1).n()
4.77205703148314e29
sage: G(1).n()
4.83802782246652e-11
sage: Gf(1).n()
0.00139260822082924

This smells of a bug, or rather two bugs :d

• Code interpretation on entry

• Factoring

with a possible common origin (pynac comes to mind...)

I'll report that on sage support, if only to get advice on how to fole useful tickets...

Not an answer, but a comment needing more room than allowed in a comment :

sage: %cpaste
Pasting code; enter '--' alone on the line to stop or use Ctrl-D.
:# Raw cut and paste :
:
:f(x) = (3/174465461165747500*pi*(-1750000*I*pi*x^3 - 31250000*(224*pi + 45*sqrt(448*pi + 2025) + 2025)*x^2
:                                + 17500000000000000*I*pi*x)
:        *sqrt(92821652156334811582567480952850314403/10*pi^2/(224*pi + 45*sqrt(448*pi + 2025) + 2025)
:              + 98489794142024498175862287197250000*pi*sqrt(448*pi + 2025)
:              /(224*pi + 45*sqrt(448*pi + 2025) + 2025) + 7713517620898636162808584411766250000*pi
:              /(224*pi + 45*sqrt(448*pi + 2025) + 2025) + 659225266976959904108326638192187500*sqrt(448*pi + 2025)
:              /(224*pi + 45*sqrt(448*pi + 2025) + 2025) + 29665137013963195684874698718648437500
:              /(224*pi + 45*sqrt(448*pi + 2025) + 2025))
:        /(63*pi^2*x^4 - (504000*I*pi^2 + 67500*I*pi*(sqrt(448*pi + 2025) + 45))*x^3
:          - 3000000*(560224*pi^2 + 45*pi*(sqrt(448*pi + 2025) + 45))*x^2 + 8400000000000000000000*pi^2
:          - (-5040000000000000*I*pi^2 - 675000000000000*I*pi*(sqrt(448*pi + 2025) + 45))*x))
:
:# deleting blanks and newlines (adding a space before "+" and "-") : 
:
:g(x) = (3/174465461165747500*pi*(-1750000*I*pi*x^3 - 31250000*(224*pi + 45*sqrt(448*pi + 2025) + 2025)*x^2 + 17500000000000000*I*pi*x)*sqrt(92821652156334811582567480952850314403/10*pi^2/(224*pi + 45*sqrt(448*pi + 2025) + 2025) + 98489794142024498175862287197250000*pi*sqrt(448*pi + 2025)/(224*pi + 45*sqrt(448*pi + 2025) + 2025) + 7713517620898636162808584411766250000*pi/(224*pi + 45*sqrt(448*pi + 2025) + 2025) + 659225266976959904108326638192187500*sqrt(448*pi + 2025)/(224*pi + 45*sqrt(448*pi + 2025) + 2025) + 29665137013963195684874698718648437500/(224*pi + 45*sqrt(448*pi + 2025) + 2025))/(63*pi^2*x^4 - (504000*I*pi^2 + 67500*I*pi*(sqrt(448*pi + 2025) + 45))*x^3 - 3000000*(560224*pi^2 + 45*pi*(sqrt(448*pi + 2025) + 45))*x^2 + 8400000000000000000000*pi^2 - (-5040000000000000*I*pi^2 - 675000000000000*I*pi*(sqrt(448*pi + 2025) + 45))*x))
:
:--
sage: f(1).n()
418409.917305474 + 1.28757494213663e11*I
sage: g(1).n()
1.39111866114058e-12 + 6.95559330500736e-6*I

Something is amiss in your code entry... or Sage's interpretation.

And the problem you noted is still present :

sage: F = f.real()^2 + f.imag()^2
sage: Ff = (f.real()^2 + f.imag()^2).factor()
sage: G = g.real()^2 + g.imag()^2
sage: Gf = (g.real()^2 + g.imag()^2).factor()
sage: F(1).n()
1.65784923163565e22
sage: Ff(1).n()
4.77205703148314e29
sage: G(1).n()
4.83802782246652e-11
sage: Gf(1).n()
0.00139260822082924

This smells of a bug, or rather two bugs :d

• Code interpretation on entry

• Factoring

with a possible common origin (pynac comes to mind...)

I'll report that on sage support, if only to get for confirmation and advice on how to fole file useful tickets...

Not an answer, but a comment needing more room than allowed in a comment :

sage: %cpaste
Pasting code; enter '--' alone on the line to stop or use Ctrl-D.
:# Raw cut and paste :
:
:f(x) = (3/174465461165747500*pi*(-1750000*I*pi*x^3 - 31250000*(224*pi + 45*sqrt(448*pi + 2025) + 2025)*x^2
:                                + 17500000000000000*I*pi*x)
:        *sqrt(92821652156334811582567480952850314403/10*pi^2/(224*pi + 45*sqrt(448*pi + 2025) + 2025)
:              + 98489794142024498175862287197250000*pi*sqrt(448*pi + 2025)
:              /(224*pi + 45*sqrt(448*pi + 2025) + 2025) + 7713517620898636162808584411766250000*pi
:              /(224*pi + 45*sqrt(448*pi + 2025) + 2025) + 659225266976959904108326638192187500*sqrt(448*pi + 2025)
:              /(224*pi + 45*sqrt(448*pi + 2025) + 2025) + 29665137013963195684874698718648437500
:              /(224*pi + 45*sqrt(448*pi + 2025) + 2025))
:        /(63*pi^2*x^4 - (504000*I*pi^2 + 67500*I*pi*(sqrt(448*pi + 2025) + 45))*x^3
:          - 3000000*(560224*pi^2 + 45*pi*(sqrt(448*pi + 2025) + 45))*x^2 + 8400000000000000000000*pi^2
:          - (-5040000000000000*I*pi^2 - 675000000000000*I*pi*(sqrt(448*pi + 2025) + 45))*x))
:
:# deleting blanks and newlines (adding a space before "+" and "-") : 
:
:g(x) = (3/174465461165747500*pi*(-1750000*I*pi*x^3 - 31250000*(224*pi + 45*sqrt(448*pi + 2025) + 2025)*x^2 + 17500000000000000*I*pi*x)*sqrt(92821652156334811582567480952850314403/10*pi^2/(224*pi + 45*sqrt(448*pi + 2025) + 2025) + 98489794142024498175862287197250000*pi*sqrt(448*pi + 2025)/(224*pi + 45*sqrt(448*pi + 2025) + 2025) + 7713517620898636162808584411766250000*pi/(224*pi + 45*sqrt(448*pi + 2025) + 2025) + 659225266976959904108326638192187500*sqrt(448*pi + 2025)/(224*pi + 45*sqrt(448*pi + 2025) + 2025) + 29665137013963195684874698718648437500/(224*pi + 45*sqrt(448*pi + 2025) + 2025))/(63*pi^2*x^4 - (504000*I*pi^2 + 67500*I*pi*(sqrt(448*pi + 2025) + 45))*x^3 - 3000000*(560224*pi^2 + 45*pi*(sqrt(448*pi + 2025) + 45))*x^2 + 8400000000000000000000*pi^2 - (-5040000000000000*I*pi^2 - 675000000000000*I*pi*(sqrt(448*pi + 2025) + 45))*x))
:
:--
sage: f(1).n()
418409.917305474 + 1.28757494213663e11*I
sage: g(1).n()
1.39111866114058e-12 + 6.95559330500736e-6*I

Something is amiss in your code entry... or Sage's interpretation.

And the problem you noted is still present :

sage: F = f.real()^2 + f.imag()^2
sage: Ff = (f.real()^2 + f.imag()^2).factor()
sage: G = g.real()^2 + g.imag()^2
sage: Gf = (g.real()^2 + g.imag()^2).factor()
sage: F(1).n()
1.65784923163565e22
sage: Ff(1).n()
4.77205703148314e29
sage: G(1).n()
4.83802782246652e-11
sage: Gf(1).n()
0.00139260822082924

This smells of a bug, or rather two bugs :d

• Code interpretation on entry

• Factoring

with a possible common origin (pynac comes to mind...)

I'll report that on sage support, for confirmation and advice on how to file useful tickets...

EDIT : Reported in sage support