Modular Forms with Only Nonnegative Coefficients

We study modular forms for $\textrm{SL}_2(\mathbb{Z})$ with no negative Fourier coefficients. Let $A(k)$ be the positive integer where if the first $A(k)$ Fourier coefficients of a modular form of weight $k$ for $\textrm{SL}_2(\mathbb{Z})$ are nonnegative, then all of its Fourier coefficients are nonnegative, so that $A(k)$ can be interpreted as a ``nonnegativity Sturm bound’’. We give upper and lower bounds for $A(k)$, as well as an upper bound on the $n$th Fourier coefficient of any form with no negative Fourier coefficients.

💡 Research Summary

The paper “Modular Forms with Only Nonnegative Coefficients” by Paul Jenkins and Jeremy Rouse investigates a new type of Sturm bound for holomorphic modular forms on SL₂(ℤ) that guarantees non‑negativity of all Fourier coefficients once a finite initial segment is known to be non‑negative.

For a modular form f(z)=∑ₙa(n)qⁿ of weight k (k≥12, k≡0 mod 4) they define
 N(f)=the smallest n with a(n)<0 (or ∞ if no such n exists) and
 A(k)=max{N(f) : f∈M_k, N(f)<∞}.
Thus A(k) is the smallest integer such that if the first A(k) coefficients of any weight‑k form are non‑negative, then every coefficient is non‑negative. This “non‑negativity Sturm bound” is well‑defined (proved in §4) and is intimately linked to sign changes of cusp forms.

The main quantitative result (Theorem 1) gives both a lower and an upper bound for A(k):

• Lower bound: L(k)=⌈(k−1)²/(16π²)⌉. It comes from analyzing the first Poincaré series P₁(z), the smallest non‑trivial cusp form of weight k. The first negative coefficient of P₁(z) occurs after roughly (k−1)²/(16π²) terms, showing that no universal bound can be smaller.

• Upper bound: U(k)=⌊k⁴(log k+log log k)²/7316⌋. The proof uses the modular transformation f(−1/z)=zᵏf(z) to obtain the identity
   1+∑{n≥1}a(n) e^{−2πn/y−yk e^{−2πny}}=yᵏ,
  and selects y≈k²/(2π log k) so that the exponential factor is positive for all n. This yields an explicit inequality for a(n) involving the Eisenstein coefficients (−2kB_kσ{k−1}(n)) and a remainder term coming from the cusp part. The crucial tool is an explicit bound from