Buggy python submission (verdict: Time Limit Exceeded):
#!/usr/bin/env python3
# -*- coding: utf-8 -*-
from functools import lru_cache

PRIME = 10 ** 9 + 7


@lru_cache(maxsize=None)
def digitmod13(y):
    if y == 0:
        return 1
    elif y == 1:
        return 10 % 13
    elif y % 2 == 0:
        return digitmod13(y // 2) ** 2 % 13
    else:
        return digitmod13(y // 2) ** 2 * 10 % 13


def main():
    S = input()
    N = len(S)
    dp = [[0 for _ in range(13)] for _ in range(N)]
    S = S[::-1]
    
    # init
    first = S[0]
    if first == '?':
        for k in range(10):
            dp[0][k] = 1
    else:
        first = int(first)
        dp[0][first] = 1
    
    # dp
    for i in range(1, N):
        c = S[i]
        if c == '?':
            for d in range(13):
                for k in range(10):
                    dp[i][(d + k * digitmod13(i)) % 13] += dp[i - 1][d]
                    dp[i][(d + k * digitmod13(i)) % 13] %= PRIME
        else:
            c = int(c)
            for d in range(13):
                dp[i][(d + c * digitmod13(i)) % 13] += dp[i - 1][d]
                dp[i][(d + c * digitmod13(i)) % 13] %= PRIME
    
    print(dp[N - 1][5] % PRIME)


if __name__ == "__main__":
    main()


Fixed submission:
#!/usr/bin/env python3
# -*- coding: utf-8 -*-
from functools import lru_cache

PRIME = 10 ** 9 + 7


@lru_cache(maxsize=None)
def digitmod13(y):
    if y == 0:
        return 1
    elif y == 1:
        return 10 % 13
    elif y % 2 == 0:
        return digitmod13(y // 2) ** 2 % 13
    else:
        return digitmod13(y // 2) ** 2 * 10 % 13


def main():
    S = input()
    N = len(S)
    dp = [[0 for _ in range(13)] for _ in range(N)]
    S = S[::-1]
    
    # init
    first = S[0]
    if first == '?':
        for k in range(10):
            dp[0][k] = 1
    else:
        first = int(first)
        dp[0][first] = 1
    
    # prepare table
    table = [[[(i + (j * k)) % 13 for k in range(13)] for j in range(10)] for i in range(13)]
    
    # dp
    for i in range(1, N):
        c = S[i]
        if c == '?':
            for d in range(13):
                for k in range(10):
                    mod = table[d][k][digitmod13(i)]
                    dp[i][mod] += dp[i - 1][d]
                    dp[i][mod] %= PRIME
        else:
            c = int(c)
            for d in range(13):
                mod = table[d][c][digitmod13(i)]
                dp[i][mod] += dp[i - 1][d]
                dp[i][mod] %= PRIME
    
    print(dp[N - 1][5] % PRIME)


if __name__ == "__main__":
    main()

Buggy python submission (verdict: Time Limit Exceeded):
#!/usr/bin/env python3
# -*- coding: utf-8 -*-
from functools import lru_cache

PRIME = 10 ** 9 + 7


@lru_cache(maxsize=None)
def digitmod13(n):
    return (10 ** n) % 13


def main():
    S = input()
    N = len(S)
    dp = [[0 for _ in range(13)] for _ in range(N)]
    S = S[::-1]
    # init
    first = S[0]
    if first == '?':
        for k in range(10):
            dp[0][k] = 1
    else:
        first = int(first)
        dp[0][first] = 1
    for i in range(1, N):
        c = S[i]
        if c == '?':
            for d in range(13):
                for k in range(10):
                    dp[i][(d + k * digitmod13(i)) % 13] += dp[i - 1][d]
        else:
            c = int(c)
            for d in range(13):
                dp[i][(d + c * digitmod13(i)) % 13] += dp[i - 1][d]
        for d in range(13):
            dp[i][d] %= PRIME
    
    print(dp[N - 1][5] % PRIME)


if __name__ == "__main__":
    main()


Fixed submission:
#!/usr/bin/env python3
# -*- coding: utf-8 -*-
from functools import lru_cache

PRIME = 10 ** 9 + 7


@lru_cache(maxsize=None)
def digitmod13(y):
    if y == 0:
        return 1
    elif y == 1:
        return 10 % 13
    elif y % 2 == 0:
        return digitmod13(y // 2) ** 2 % 13
    else:
        return digitmod13(y // 2) ** 2 * 10 % 13


def main():
    S = input()
    N = len(S)
    dp = [[0 for _ in range(13)] for _ in range(N)]
    S = S[::-1]
    
    # init
    first = S[0]
    if first == '?':
        for k in range(10):
            dp[0][k] = 1
    else:
        first = int(first)
        dp[0][first] = 1
    
    # prepare table
    table = [[[(i + (j * k)) % 13 for k in range(13)] for j in range(10)] for i in range(13)]
    
    # dp
    for i in range(1, N):
        c = S[i]
        if c == '?':
            for d in range(13):
                for k in range(10):
                    mod = table[d][k][digitmod13(i)]
                    dp[i][mod] += dp[i - 1][d]
                    dp[i][mod] %= PRIME
        else:
            c = int(c)
            for d in range(13):
                mod = table[d][c][digitmod13(i)]
                dp[i][mod] += dp[i - 1][d]
                dp[i][mod] %= PRIME
    
    print(dp[N - 1][5] % PRIME)


if __name__ == "__main__":
    main()

Buggy python submission (verdict: Time Limit Exceeded):
#!/usr/bin/env python3
# -*- coding: utf-8 -*-

PRIME = 10 ** 9 + 7


def digitmod13(k, n):
    return (k * (10 ** n)) % 13


def main():
    S = input()
    N = len(S)
    dp = [[0 for _ in range(13)] for _ in range(N)]
    S = S[::-1]
    # init
    first = S[0]
    if first == '?':
        for k in range(10):
            dp[0][k] = 1
    else:
        first = int(first)
        dp[0][first] = 1
    for i in range(1, N):
        c = S[i]
        if c == '?':
            for d in range(13):
                for k in range(10):
                    dp[i][(d + digitmod13(k, i)) % 13] += dp[i - 1][d]
        else:
            c = int(c)
            for d in range(13):
                dp[i][(d + digitmod13(c, i)) % 13] += dp[i - 1][d]
        for d in range(13):
            dp[i][d] %= PRIME
    
    print(dp[N - 1][5] % PRIME)


if __name__ == "__main__":
    main()


Fixed submission:
#!/usr/bin/env python3
# -*- coding: utf-8 -*-
from functools import lru_cache

PRIME = 10 ** 9 + 7


@lru_cache(maxsize=None)
def digitmod13(y):
    if y == 0:
        return 1
    elif y == 1:
        return 10 % 13
    elif y % 2 == 0:
        return digitmod13(y // 2) ** 2 % 13
    else:
        return digitmod13(y // 2) ** 2 * 10 % 13


def main():
    S = input()
    N = len(S)
    dp = [[0 for _ in range(13)] for _ in range(N)]
    S = S[::-1]
    
    # init
    first = S[0]
    if first == '?':
        for k in range(10):
            dp[0][k] = 1
    else:
        first = int(first)
        dp[0][first] = 1
    
    # prepare table
    table = [[[(i + (j * k)) % 13 for k in range(13)] for j in range(10)] for i in range(13)]
    
    # dp
    for i in range(1, N):
        c = S[i]
        if c == '?':
            for d in range(13):
                for k in range(10):
                    mod = table[d][k][digitmod13(i)]
                    dp[i][mod] += dp[i - 1][d]
                    dp[i][mod] %= PRIME
        else:
            c = int(c)
            for d in range(13):
                mod = table[d][c][digitmod13(i)]
                dp[i][mod] += dp[i - 1][d]
                dp[i][mod] %= PRIME
    
    print(dp[N - 1][5] % PRIME)


if __name__ == "__main__":
    main()