Buggy python submission (verdict: Time Limit Exceeded):
import sys
sys.setrecursionlimit(100000000)
def input():
    return sys.stdin.readline()[:-1]
from bisect import *
from collections import *
from heapq import *

mod = 10**9+7
def rec(k, m):
    if k == N:
        return m == 5
    if memo[k][m] != -1:
        return memo[k][m]
    res = 0
    d = D[N-k-1]
    if S[k] != '?':
        res += rec(k+1, (m+int(S[k])*d)%13)
    else:
        for i in range(10):
            res += rec(k+1, (m+i*d)%13)
    res %= mod
    memo[k][m] = res
    return res

S = input()
N = len(S)
D = [None] * N
t = 1
for i in range(N):
    D[i] = t % 13
    t = t * 10 % 13
memo = [[-1] * 13 for i in range(N+1)]
print(rec(0, 0))


Fixed submission:
import sys
sys.setrecursionlimit(100000000)
def input():
    return sys.stdin.readline()[:-1]
from bisect import *
from collections import *
from heapq import *

mod = 10**9+7
S = input()
N = len(S)
D = [None] * N
t = 1
for i in range(N):
    D[i] = t % 13
    t = t * 10 % 13
dp = [[0] * 13 for i in range(N+1)]
if S[N-1] != '?':
    dp[N-1][int(S[N-1])] = 1
else:
    for i in range(10):
        dp[N-1][i] = 1
for k in range(N-1)[::-1]:
    d = D[N-k-1]
    for m in range(13):
        if S[k] != '?':
            dp[k][m] += dp[k+1][(m-d*int(S[k]))%13]
            dp[k][m] %= mod
        else:
            for i in range(10):
                dp[k][m] += dp[k+1][(m-i*d)%13]
                dp[k][m] %= mod
print(dp[0][5])

Buggy python submission (verdict: Time Limit Exceeded):
import sys
sys.setrecursionlimit(100000000)
def input():
    return sys.stdin.readline()[:-1]
from bisect import *
from collections import *
from heapq import *

mod = 10**9+7
def rec(k, m):
    if k == N:
        return m == 5
    if memo[k][m] != -1:
        return memo[k][m]
    res = 0
    d = D[N-k-1]
    if S[k] != '?':
        res += rec(k+1, (m+int(S[k])*d)%13)
    else:
        for i in range(10):
            res += rec(k+1, (m+i*d)%13)
    res %= mod
    memo[k][m] = res
    return res

S = input()
N = len(S)
D = [None] * N
t = 1
for i in range(N):
    D[i] = t % 13
    t = t * 10 % 13
memo = [[-1] * 13 for i in range(N+1)]
# for k in range(N):
#     if S[k] != '?':
#         dp[]
#         for  in range(0, 1):
print(rec(0, 0))


Fixed submission:
import sys
sys.setrecursionlimit(100000000)
def input():
    return sys.stdin.readline()[:-1]
from bisect import *
from collections import *
from heapq import *

mod = 10**9+7
S = input()
N = len(S)
D = [None] * N
t = 1
for i in range(N):
    D[i] = t % 13
    t = t * 10 % 13
dp = [[0] * 13 for i in range(N+1)]
if S[N-1] != '?':
    dp[N-1][int(S[N-1])] = 1
else:
    for i in range(10):
        dp[N-1][i] = 1
for k in range(N-1)[::-1]:
    d = D[N-k-1]
    for m in range(13):
        if S[k] != '?':
            dp[k][m] += dp[k+1][(m-d*int(S[k]))%13]
            dp[k][m] %= mod
        else:
            for i in range(10):
                dp[k][m] += dp[k+1][(m-i*d)%13]
                dp[k][m] %= mod
print(dp[0][5])

Buggy python submission (verdict: Time Limit Exceeded):
import sys
sys.setrecursionlimit(100000000)
def input():
    return sys.stdin.readline()[:-1]
from bisect import *
from collections import *
from heapq import *

mod = 10**9+7
def rec(k, ti, m):
    if k == N:
        return m == 5
    if memo[k][ti][m] != -1:
        return memo[k][ti][m]
    res = 0
    d = D[N-k-1]
    if S[k] != '?':
        res += rec(k+1, ti, (m+int(S[k])*d)%13)
    else:
        for i in range(10):
            res += rec(k+1, 0, (m+i*d)%13)
    res %= mod
    memo[k][ti][m] = res
    return res

S = input()
N = len(S)
D = [None] * N
t = 1
for i in range(N):
    D[i] = t % 13
    t = t * 10 % 13
memo = [[[-1] * 13 for j in range(2)] for i in range(N+1)]
print(rec(0, 1, 0))


Fixed submission:
import sys
sys.setrecursionlimit(100000000)
def input():
    return sys.stdin.readline()[:-1]
from bisect import *
from collections import *
from heapq import *

mod = 10**9+7
S = input()
N = len(S)
D = [None] * N
t = 1
for i in range(N):
    D[i] = t % 13
    t = t * 10 % 13
dp = [[0] * 13 for i in range(N+1)]
if S[N-1] != '?':
    dp[N-1][int(S[N-1])] = 1
else:
    for i in range(10):
        dp[N-1][i] = 1
for k in range(N-1)[::-1]:
    d = D[N-k-1]
    for m in range(13):
        if S[k] != '?':
            dp[k][m] += dp[k+1][(m-d*int(S[k]))%13]
            dp[k][m] %= mod
        else:
            for i in range(10):
                dp[k][m] += dp[k+1][(m-i*d)%13]
                dp[k][m] %= mod
print(dp[0][5])