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On the Fourier coefficients of modular forms of half-integral weight

On the Fourier coefficients of modular forms of half-integral weight 2025

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The document presents a mathematical study on the Fourier coefficients of modular forms of half-integral weight, specifically proving a formula that connects these coefficients to special values of L-functions. The work involves an explicit theta lift from the multiplicative group of an indefinite quaternion algebra over Q and discusses applications such as demonstrating the nonvanishing of this lift and providing an explicit version of the Rallis inner product formula. The article is structured into sections detailing the introduction, methods, computations, and final results related to these mathematical concepts.

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Begin by reviewing the abstract section, which outlines the main findings regarding the Fourier coefficients and their relation to L-functions. This will provide context for filling out any relevant fields.
Proceed to the introduction section. Here, you may need to summarize your understanding of Waldspurger's proportionality relation as it pertains to your specific application.
In sections discussing Fourier coefficients and toric integrals, ensure that you accurately input any mathematical expressions or values that are required based on your analysis.
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What is the coefficient rule for integrals?

The coefficients ck are called the (kth) Fourier (series) coefficients of (the signal) r (t). These are, in general, complex numbers. is called the fundamental frequency of the signal r (t). The kth multiple of the fundamental frequency (for positive ks) is called the kth harmonic.

What are the Fourier coefficients?

The Fourier transform of a function f(x) is defined as. (1) F ( u ) = f ( x ) e 2 i u x d x. F(u) is in turn related to f(x) by the inverse Fourier transform: (2) f ( x ) = F ( u ) e 2 i u x d u.

What is the fourier integral formula?

From the constant coefficient rule each coefficient can be written in front of the integral sign. Note: We do not repeat the c for each term, we use one c for the entire integral. Rewrite the integrand to simplify the exponents. Then, to evaluate the integral, integrate each of these terms separately.

What is the effect of half-wave symmetry on the Fourier coefficients of a signal?

Conclusion: For the half-wave symmetry, only odd harmonics are present and all the remaining Fourier coefficients will be zero.

What are the Fourier coefficients of integrals?

If a 2-periodic function f:RR is Lebesgue integrable in [,], and the series a02+n=1[ancosnx+bnsinnx], where (an),(bn) are some real sequences, is convergent to f uniformly or in Lp norm or pointwise, then it is known that an, bn are Fourier coefficients of f.

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