Sparse Inverse Fourier Transforms
The FFT is faster than the naive DFT through matrix-vector products because it can reuse many intermediate results.
However, for inputs this sparse, there's really not much that can be re-used, even in the best case.
So, the most efficient way here is probably to work straight forward:

Allocate the space ($N=7408800$ values) for your output (initialize with zeros) $o$.
Go through your input $i$. Nested loops:

For every nonzero input index $n$:

For every output index $k$: add $i_ne^{j2\pi \frac{nk}N}$ to $o_k$.





Notes:

since with sparse input, your output is non-sparse, it's usually faster to traverse the input only once, but the output many times, instead of the other way around (that is because memory is much faster when addressed linearly)
to calculate the complex exponential, you can use SIMD-accelerated calculations, because the values are in consecutive addresses and have proportional phase. LibVOLK has a number of implementations that might be helpful for fast calculation here.