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The coefficients of certain integral modular forms

Published online by Cambridge University Press:  24 October 2008

R. A. Rankin  and
J. M. Rushforth
R. A. Rankin
Affiliation:
The UniversityBirmingham 15
J. M. Rushforth
Affiliation:
The UniversityBirmingham 15
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Extract

The notation which we use is that of a recent paper by one of us, and we quote results from that paper as they are required. It is known (see R, Theorem 1, for example) that the vector space k of all cusp-forms f(z) of even negative dimension – k (k ≥ 12), belonging to the full modular group Γ(1), possesses a finite basis of forms

where k is defined by (2·10) of R and the coefficients possess the following properties:

for a prime p, where p is a positive integer.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 50 , Issue 2 , April 1954 , pp. 305 - 308
Copyright
Copyright © Cambridge Philosophical Society 1954

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References

REFERENCES

(1)Hecke, E.Analytische Arithmetik der positiven quadratischen Formen. Math.-Fys. Medd. XVII, 12 (1940), 1–134.Google Scholar
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(3)Petersson, H.Konstruktion der sämtlichen Lösungen einer Riemannschen Funktional-gleichung durch Dirichlet-Reihen mit Eulerscher Produktentwicklung I. Math. Ann., 116 (1939), 401–12.CrossRefGoogle Scholar
(4)Rankin, R. A.The scalar product of modular forms. Proc. Lond. math. Soc. (3), 2 (1962), 198–217.Google Scholar