J-Invariant Fourier Coefficient Calculator
Exact positive/constantstandardj(q) Fourier coefficient by formalintegerE4³/Δdivision, index0…12; index0=744not constantofnormalizedJ. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Description
The standard modular j function has integer Fourier coefficients and constant term 744.
The standard modular j function has integer Fourier coefficients and constant term 744.1
This tool computes a bounded nonnegative-index coefficient exactly rather than using the normalized J convention.
When to use J-Invariant Fourier Coefficient Calculator
This tool computes a bounded nonnegative-index coefficient exactly rather than using the normalized J convention.
How to use J-Invariant Fourier Coefficient Calculator and how it works
- Prepare nonnegative fourier index, using the input definitions and matching units below.
- Enter the required fields and run J-Invariant Fourier Coefficient Calculator.
- Read result.exactModularJFourierCoefficient using the units, signs and ordering explained below.
- Correct any rejected inputs and rerun; interpret the result within the stated model and its assumptions.
| Input field | Meaning | Value type | Requirement |
|---|---|---|---|
| actualNonnegativeFourierIndex | Power of q whose coefficient is requested in standard j; integer 0–12. | number | Required; finite; other constraints below |
Require nonnegative Fourier index integer 0–12. Build E4 coefficients with exact divisor-cube sums, cube the finite series, build Delta/q coefficients with exact tau arithmetic and divide formal series using BigInt. Return the exact q^index coefficient of standard j as a decimal string; index zero returns 744. The q^-1 coefficient is outside this input branch and normalized J is not used.
j(q)=E4(q)^3/Delta(q); output=[q^index]j(q) Apply the input ordering, units and branch conditions stated above.
| Output field | Meaning | Value type |
|---|---|---|
| result | Object containing the calculated or selected fields below. | object |
| result.exactModularJFourierCoefficient | Exact decimal-string coefficient of standard j at the requested nonnegative index. | string |
{
"input": {
"actualNonnegativeFourierIndex": 1
},
"output": {
"result": {
"exactModularJFourierCoefficient": "196884"
}
}
}Limitations and assumptions
The calculation uses the supplied observations and model inputs. It does not collect measurements, select a reference population or coefficient, or establish that the model applies to the actual case. The stated branch conditions and output units remain part of the result.
All fields require explicit values. Unexpected fields, wrong types, nonfinite numbers and sparse arrays are rejected. Alignment, bounds and supported-token rules are explained above; extremely large inputs or nonfinite results can also fail. Input errors differ from successful false comparisons, unavailable-domain flags and empty match lists.
Alternative or Complementary methods
Use Ramanujan Tau Approximation for a complementary operation: Ramanujan tau is the integer coefficient sequence of the modular discriminant's q-product.
Use J-Invariant Approximation for a complementary operation: A truncated standard modular j expansion begins with q to minus one and the constant 744.
References
Similar or alternative tools
- Ramanujan Tau Approximation
Ramanujan tau is the integer coefficient sequence of the modular discriminant's q-product.
- J-Invariant Approximation
A truncated standard modular j expansion begins with q to minus one and the constant 744.