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COMPUTATION OF MODULAR FORMS OF WEIGHT 3 2 Abstract In - math clemson

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The document presents a computational investigation into the basis problem for modular forms of weight 3/2, utilizing SAGE software. The authors, Gehret, Kottmeyer, and Salter, explore whether a basis can be constructed from lifts and twists of forms derived from ternary quadratic forms. They find that this approach does not yield a basis and further investigate the addition of forms created by multiplying weight 1/2 forms with weight 1 forms. Despite various attempts, including applying Hecke operators, they conclude that these methods also fail to generate a basis. The paper includes conjectures on linear dependence among certain classes of forms and provides an appendix detailing dimensions of subspaces for levels up to 600.

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What is the formula for modular form?

A modular form f : H C of weight k for SL2(Z) is a function on H with the following properties: 1. f : H C is a holomorphic function (meromorphic function in case of a modular function). anqn, q = e2iz.

What is an example of a modular form?

This writeup gives first examples of modular forms: Eisenstein series, the dis- criminant, and the j-function. The modular group is the group of 2-by-2 matrices with integer entries and determinant 1, SL2(Z) = a b c d : a, b, c, d Z, ad bc = 1 . c d () = a + b c + d , bC.

What is the simplest modular form?

Any modular form is going to look very complicated. Some of the simplest which are used as building blocks for other modular forms are called Eisenstein series. You can think of an Eisenstein series as an infinite sum of functions.

What is an example of a modular math?

A familiar example of modular arithmetic is the hour hand on a 12-hour clock. If the hour hand points to 7 now, then 8 hours later it will point to 3. Ordinary addition would result in 7 + 8 = 15, but 15 reads as 3 on the clock face.

What is the modular form of weight K?

Definition. A modular form for G of weight k is a function on H satisfying the above functional equation for all matrices in G, that is holomorphic on H and at all cusps of G. Again, modular forms that vanish at all cusps are called cusp forms for G.

What is a modular form in math?

Introduction. A modular form is a holomorphic function on the upper half-plane. h = {x + iy : x R,y 0} = { C : Im 0} that transforms in a certain way under a discrete matrix group and has a nice behavior at infinity.