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==As a function on lattices==
A modular form can be thought of as a function ''F'' from the set of [[latticeperiod (group)lattice|lattice]]s Λ in '''C''' to the set of complex numbers which satisfies certain conditions. In order to visualise Λ, consider that any lattice has a ''lattice basis'': two complex numbers ''z''<sub>1</sub> and ''z''<sub>2</sub> such that Λ is the set of all sums
:''mz''<sub>1</sub> + ''nz''<sub>2</sub>
where ''m'' and ''n'' are integers. Here ''z''<sub>1</sub> and ''z''<sub>2</sub> are any complex numbers, such that they are [[linearly independent]] over the real numbers. This means that the lines
:''Rz''<sub>1</sub>
and
:''Rz''<sub>2</sub>
are different; so that in other words ''z''<sub>1</sub> and ''z''<sub>2</sub> point in different directions (which are not at 180°).
There is a distinction, though, between a function ''F'' applying to such pairs
:(''z''<sub>1</sub>, ''z''<sub>2</sub>)
and a function of lattices; a function of lattices must not depend on the lattice basis we choose. In fact we consider only functions ''F'' that are [[well-defined]] with respect to the [[equivalence relation]] 'generates the same lattice as'. More explicitly, pairs
:(''z''<sub>1</sub>, ''z''<sub>2</sub>)
and
:(''w''<sub>1</sub>, ''w''<sub>2</sub>)
are considered equivalent if each ''z''<sub>''i''</sub> is a [[linear combination]] with ''integer coefficients'' of the ''w''<sub>''j''</sub>, and ''vice versa''. (NB that ''real'' coefficients is automatic; those coefficients are uniquely determined, and we require them to be integers.)
Given that preliminary description, the conditions applied to a modular form are these:
:(1) If we consider the lattice Λ = <α, ''z''> generated by a constant α and a variable ''z'', then ''F''(Λ) is an analytic function of ''z''.
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