Integers in Base

Consider the Gaussian integer $i - 1$ . A base $i - 1$ representation of a Gaussian integer $a + b i$ is a finite sequence of digits $d_{n - 1} d_{n - 2} \dots d_{1} d_{0}$ such that:

$a + b i = d_{n - 1} (i - 1)^{n - 1} + d_{n - 2} (i - 1)^{n - 2} + \dots + d_{1} (i - 1) + d_{0}$
Each $d_{k}$ is in ${0, 1}$
There are no leading zeroes, i.e. $d_{n - 1} \neq 0$ , unless $a + b i$ is itself $0$

Here are base $i - 1$ representations of a few Gaussian integers:

$11 + 24 i \to 111010110001101$
$24 - 11 i \to 110010110011$
$8 + 0 i \to 111000000$
$- 5 + 0 i \to 11001101$
$0 + 0 i \to 0$

Remarkably, every Gaussian integer has a unique base $i - 1$ representation!

Define $f (a + b i)$ as the number of $1$ s in the unique base $i - 1$ representation of $a + b i$ . For example, $f (11 + 24 i) = 9$ and $f (24 - 11 i) = 7$ .

Define $B (L)$ as the sum of $f (a + b i)$ for all integers $a, b$ such that $| a | \leq L$ and $| b | \leq L$ . For example, $B (500) = 10795060$ .

Find $B (10^{15}) mod 1 000 000 007$ .

Write-up coming later

The complete problem is available here. An approach, code, and answer will be added later.