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### Modular forms

Recall the formal product in the variable q we considered above. If we replace q with exp(2πiz), where z is a complex number, we obtain a function of z, which is called the Ramanujan Δ Function or discriminant function:
Of course, in order for this product and this sum to converge, we must have Im(z)>0. In other words, z is an element of the complex upper half plane consisting of all complex numbers with positive imaginary part. Obviously Δ has the property that &Delta(z+1)=&Delta(z). One can show that Δ satisfies the additional transformation property
The number 12 in the exponent is called the weight. In general, if k is a positive integer, and if f(z) is a holomorphic function on the complex upper half plane that satisfies
and that has a Fourier expansion of the form
then f is called a modular form of weight k. The numbers an are called the Fourier coefficients of f, and if the constant term a0 is zero, then f is called a cusp form.
 k dim Mk dim Sk 2 0 0 4 1 0 6 1 0 8 1 0 10 1 0 12 2 1 14 1 0 16 2 1 18 2 1 20 2 1 22 2 1 24 3 2
Hence Δ is a cusp form of weight 12, and the numbers τ(n) are its Fourier coefficients. A fundamental result says that the vector space Mk of cusp forms of weight k is finite-dimensional. Of course, the same is then true for the subspace Sk of cusp forms. If the Fourier coefficients of a cusp form f have the property that aman=amn whenever gcd(m,n)=1, then f is called an eigenform. It can be proved that Sk has a vector space basis consisting of eigenforms. The Sato-Tate conjecture is actually a statement for eigen-cuspforms, in the following sense. Let an be the Fourier coefficients of a cusp form of weight k that is also an eigenform. By Deligne's theorem, we have
Therefore, there exists a Frobenius angle φp between 0 and π for which
The Sato-Tate conjecture then asserts that these angles are distributed according to the same function as before,
The display below demonstrates the Sato-Tate conjecture for the two eigenforms of weight 24.
First Sato-Tate Graph for weight 24 (1000 Parameters, Step 1)
Second Sato-Tate Graph for weight 24 (1000 Parameters, Step 1)
More to come - go to page 3.