ABSTRACT
To understand the representation numbers of a positive definite quadratic form, one can study the associated theta series. This theta series is a modular form whose Fourier coefficients are the representation numbers of the quadratic form. It is well known that the space of modular forms has a basis consisting of eigenforms for the Hecke algebra T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203747018/bef25d74-ef02-4914-ac4f-27ec89fc30ac/content/eq2560.tif"/> , and the Fourier coefficients of these eigenforms satisfy the arithmetic relations satisfied by the operators of T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203747018/bef25d74-ef02-4914-ac4f-27ec89fc30ac/content/eq2561.tif"/> . It is also known that the subspace of modular forms spanned by theta series is invariant under a subalgebra T L https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203747018/bef25d74-ef02-4914-ac4f-27ec89fc30ac/content/eq2562.tif"/> of T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203747018/bef25d74-ef02-4914-ac4f-27ec89fc30ac/content/eq2563.tif"/> ; the Fourier coefficents of eigenforms for T L https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203747018/bef25d74-ef02-4914-ac4f-27ec89fc30ac/content/eq2564.tif"/> satisfy only some of the relations satisfied by T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203747018/bef25d74-ef02-4914-ac4f-27ec89fc30ac/content/eq2565.tif"/> . To use our knowledge of eigenforms for the full Hecke algebra to help us understand the representation numbers of quadratic forms, we need to understand how to write a T L https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203747018/bef25d74-ef02-4914-ac4f-27ec89fc30ac/content/eq2566.tif"/> -eigenform as a linear combination of T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203747018/bef25d74-ef02-4914-ac4f-27ec89fc30ac/content/eq2567.tif"/> -eigenforms.
