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The first Fourier coefficients of E4, E6, Delta and j

Example

With $q = e^{2 π i τ}$ , $E_{4} = 1 + 240 q + 2160 q^{2} + 6720 q^{3} + \dots, E_{6} = 1 - 504 q - 16632 q^{2} - 122976 q^{3} + \dots,$ $Δ = q - 24 q^{2} + 252 q^{3} - 1472 q^{4} + \dots, j = q^{- 1} + 744 + 196884 q + 21493760 q^{2} + \dots .$ The coefficients are read from $E_{k} = 1 - \frac{2 k}{B_{k}} \sum σ_{k - 1} (n) q^{n}$ : $240 = - 8 / B_{4}$ , $- 504 = - 12 / B_{6}$ , and the higher coefficients from $σ_{3} (n), σ_{5} (n)$ and the binomial expansions.

Facts & Assumptions

Given: The expansion $E_{k} = 1 - \frac{2 k}{B_{k}} \sum_{n \geq 1} σ_{k - 1} (n) q^{n}$ of Eisenstein series are modular forms; their Fourier coefficients with $B_{4} = - 1 / 30$ , $B_{6} = 1 / 42$ (The Bernoulli numbers are defined by the generating series $t / (e^{t} - 1)$ ), the divisor sums of The divisor power sums $σ_{k}$ , and $Δ = q \prod (1 - q^{n})^{24} = (E_{4}^{3} - E_{6}^{2}) / 1728$ , $j = E_{4}^{3} / Δ$ (The Jacobi product formula for the discriminant, The discriminant is a nonvanishing cusp form of weight 12, The modular discriminant and the j-invariant).

[F1]

$σ_{1} (1) = 1$ and for prime powers $σ_{k} (p^{e}) = 1 + p^{k} + \dots + p^{k e}$ ; in particular $σ_{3} (1) = 1$ , $σ_{3} (2) = 9$ , $σ_{3} (3) = 28$ , $σ_{3} (4) = 73$ and $σ_{5} (1) = 1$ , $σ_{5} (2) = 33$ , $σ_{5} (3) = 244$ (The divisor power sums $σ_{k}$ ).

[F2]

Multiplication of modular forms adds weights and multiplies $q$ -expansions as absolutely convergent Cauchy products; the product formula for $Δ$ is available (Eisenstein series are modular forms; their Fourier coefficients, The Jacobi product formula for the discriminant).

Verification

1.1F1givenalgebra

By [F1] and the Bernoulli values, $240 = - \frac{8}{- 1 / 30}$ and $- 504 = - \frac{12}{1 / 42}$ , so $E_{4} = 1 + 240 q + 240 \cdot 9 q^{2} + 240 \cdot 28 q^{3} + \dots = 1 + 240 q + 2160 q^{2} + 6720 q^{3} + \dots$ and $E_{6} = 1 - 504 q - 504 \cdot 33 q^{2} - 504 \cdot 244 q^{3} + \dots = 1 - 504 q - 16632 q^{2} - 122976 q^{3} + \dots$ .

2.1F2step 1.1givenalgebra

Squaring and cubing these expansions by the binomial theorem [F2], $E_{4}^{3} = 1 + 720 q + 179280 q^{2} + 16954560 q^{3} + \dots$ and $E_{6}^{2} = 1 - 1008 q + 220752 q^{2} + 16519104 q^{3} + \dots$ ; hence $Δ = (E_{4}^{3} - E_{6}^{2}) / 1728 = q - 24 q^{2} + 252 q^{3} + O (q^{4})$ . The product formula gives $Δ = q \prod (1 - q^{n})^{24} = q (1 - 24 q + 252 q^{2} - 1472 q^{3} + \dots) = q - 24 q^{2} + 252 q^{3} - 1472 q^{4} + \dots$ , in agreement through $q^{3}$ and supplying the fourth coefficient.

3.1F2step 2.1givenalgebra∎

Dividing $E_{4}^{3}$ by $Δ = q P$ with $P = 1 - 24 q + 252 q^{2} - 1472 q^{3} + O (q^{4})$ , whose inverse begins $P^{- 1} = 1 + 24 q + 324 q^{2} + 3200 q^{3} + O (q^{4})$ , gives $q j = E_{4}^{3} P^{- 1} = 1 + 744 q + 196884 q^{2} + 21493760 q^{3} + O (q^{4})$ , that is $j = q^{- 1} + 744 + 196884 q + 21493760 q^{2} + \dots$ .

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Sources

D. Zagier, Elliptic Modular Forms and Their Applications, in The 1-2-3 of Modular Forms (Universitext, Springer, 2008) (standard reference, not scraped)
J. S. Milne, Modular Functions and Modular Forms (v1.31, 2017) (standard reference, not scraped)