Compute the prime factorization of $1007021035035021007001$.  (You should write your answer in the form $p_1^{e_1}p_2^{e_2}\ldots p_k^{e_k}$ where $p_1,\ldots,p_k$ are distinct prime numbers and $e_1,\ldots,e_k$ are positive integers.)
1. **Identify the pattern in the number:**
   The number given is \(1007021035035021007001\). Notice the sequence of digits: \(1, 0, 0, 7, 0, 2, 1, 0, 3, 5, 0, 3, 5, 0, 2, 1, 0, 7, 0, 0, 1\). This sequence resembles the coefficients of the binomial expansion of \((a + b)^7\) with zeros inserted between each pair of coefficients.

2. **Recognize the binomial coefficients:**
   The coefficients of \((a + b)^7\) are \(1, 7, 21, 35, 35, 21, 7, 1\). When zeros are inserted between each pair of coefficients, the sequence becomes \(1, 0, 0, 7, 0, 2, 1, 0, 3, 5, 0, 3, 5, 0, 2, 1, 0, 7, 0, 0, 1\), which matches the given number.

3. **Express the number in terms of a power:**
   Since the number \(1007021035035021007001\) matches the pattern of \(1001^7\), we can write:
   \[
   1007021035035021007001 = 1001^7
   \]

4. **Factorize 1001:**
   The number \(1001\) can be factorized as:
   \[
   1001 = 7 \times 11 \times 13
   \]

5. **Raise each factor to the 7th power:**
   Using the prime factorization of \(1001\), we can write:
   \[
   1001^7 = (7 \times 11 \times 13)^7 = 7^7 \times 11^7 \times 13^7
   \]

6. **Combine the results:**
   Therefore, the prime factorization of \(1007021035035021007001\) is:
   \[
   1007021035035021007001 = 7^7 \times 11^7 \times 13^7
   \]

The final answer is \(\boxed{7^7 \times 11^7 \times 13^7}\).