The existing set of type Ia supernovae (SNe Ia) is now sufficient to detect oscillatory deviations from the canonical $Lambda$CDM cosmology. We determine that the Fourier spectrum of the Pantheon data set of spectroscopically well-observed SNe Ia is consistent with the predictions of $Lambda$CDM. We also develop and describe two complementary techniques for using SNe Ia to constrain those alternate cosmological models that predict deviations from $Lambda$CDM that are oscillatory in conformal time. The first technique uses the reduced $chi^2$ statistic to determine the likelihood that the observed data would result from a given model. The second technique uses bootstrap analysis to determine the likelihood that the Fourier spectrum of a proposed model could result from statistical fluctuations around $Lambda$CDM. We constrain three oscillatory alternate cosmological models: one in which the dark energy equation of state parameter oscillates around the canonical value of $w_{Lambda} = -1$, one in which the energy density of dark energy oscillates around its $Lambda$CDM value, and one in which gravity derives from a scalar field evolving under an oscillatory coupling. We further determine that any alternate cosmological model that produces distance modulus residuals with a Fourier amplitude of $simeq 36$ millimags is strongly ruled out, given the existing data, for frequencies between $simeq 0.08 textrm{Gyr}^ {-1} h_{100}$ and $simeq 80 textrm{Gyr}^ {-1} h_{100}$.
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