Petersson Inner Product Approximation

ActualPeterssonquadratureΣfconj(g) y^(k−2) dxdyontheindependentlyqualifiedfundamentaldomainnodes/Euclidean area weights, complex tail error not certified; not unweightedinnerproductor claimed modularity. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.

Description

A Petersson inner product integrates a weighted complex product over a specified fundamental domain.

A Petersson inner product integrates a weighted complex product over a specified fundamental domain.^[1]

This tool sums caller-qualified quadrature samples and Euclidean area weights without validating modularity or certifying quadrature tails.

When to use Petersson Inner Product Approximation

This tool sums caller-qualified quadrature samples and Euclidean area weights without validating modularity or certifying quadrature tails.

How to use Petersson Inner Product Approximation and how it works

Prepare quadrature freal, quadrature fimaginary, quadrature greal, quadrature gimaginary, quadrature y, qualified euclidean area weights, modular weight, using the input definitions and matching units below.
Enter the required fields and run Petersson Inner Product Approximation.
Read result.real, result.imaginary 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
actualQuadratureFReal	Actual Quadrature F Real. Use the units and matching context stated in the calculation below.	array of number	Required; other constraints below
actualQuadratureFImaginary	Actual Quadrature F Imaginary. Use the units and matching context stated in the calculation below.	array of number	Required; other constraints below
actualQuadratureGReal	Actual Quadrature G Real. Use the units and matching context stated in the calculation below.	array of number	Required; other constraints below
actualQuadratureGImaginary	Actual Quadrature G Imaginary. Use the units and matching context stated in the calculation below.	array of number	Required; other constraints below
actualQuadratureY	Positive imaginary coordinate y of each independently qualified fundamental-domain quadrature node.	array of number	Required; other constraints below
callerQualifiedEuclideanAreaWeights	Nonnegative dx*dy quadrature weights in the same node order; not hyperbolic-area weights.	array of number	Required; other constraints below
actualModularWeight	Integer modular weight under the operation-specific range and normalization below.	number	Required; integer 0–16; finite; other constraints below

Require aligned nonempty arrays of real/imaginary f and g samples, positive y coordinates and nonnegative Euclidean area weights. Require modular weight k integer 0–16. Multiply f by conjugate(g), weight each node by y^(k-2)*areaWeight and sum real and imaginary parts separately. All-zero weights are accepted and return zero. Fundamental-domain eligibility, modularity and quadrature error are external.

integralApprox=sum(f_i*conj(g_i)*y_i^(k-2)*areaWeight_i)

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.real	Real component of the named finite complex calculation.	number
result.imaginary	Imaginary component of the named finite complex calculation.	number

json

{
  "input": {
    "actualQuadratureFReal": [
      1,
      2
    ],
    "actualQuadratureFImaginary": [
      0,
      0
    ],
    "actualQuadratureGReal": [
      1,
      1
    ],
    "actualQuadratureGImaginary": [
      0,
      1
    ],
    "actualQuadratureY": [
      1,
      1
    ],
    "callerQualifiedEuclideanAreaWeights": [
      1,
      1
    ],
    "actualModularWeight": 2
  },
  "output": {
    "result": {
      "real": 3,
      "imaginary": -2
    }
  }
}

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 Rankin-Selberg Convolution Approximation for a complementary operation: A Rankin-Selberg coefficient calculation depends on normalization and the independently verified form context.

Use Hecke Operator Approximation for a complementary operation: A level-one Hecke operator acts on Fourier coefficients under a specified modular-weight convention.

References

Petersson Inner Product Approximation — tool.io

Similar or alternative tools

Rankin-Selberg Convolution Approximation
A Rankin-Selberg coefficient calculation depends on normalization and the independently verified form context.
Hecke Operator Approximation
A level-one Hecke operator acts on Fourier coefficients under a specified modular-weight convention.