Buggy python submission (verdict: Time Limit Exceeded):
S = input()
S = S[::-1]
N = len(S)

mod = 10**9 + 7
R = []
digit = 1
while True:
  r = [i*digit % 13 for i in range(10)]
  if r in R:
    break
  else:
    R.append(r)
  
  digit = digit*10 % 13


nr = len(R)
dp = [[1] + [0]*12] + [[0]*13 for i in range(N)]
for i, s in enumerate(S):
  if s == '?':
    _R = R[i%nr]
    for j in range(13):
      for r in _R:
        dp[i+1][(j+r)%13] = (dp[i+1][(j+r)%13] + dp[i][j]) % mod
      
  else:
    s = int(s)
    r = s*10**i % 13
    dp[i+1] = dp[i][-r:] + dp[i][:-r]

print(dp[-1][5])

Fixed submission:
S = input()
S = S[::-1]
N = len(S)

mod = 10**9 + 7

dp = [[1] + [0]*12] + [[0]*13 for i in range(N)]

digit = 1
for i, s in enumerate(S):
  if s == '?':
    R = [j*digit % 13 for j in range(10)]
    for j in range(13):
      for r in R:
        dp[i+1][(j+r)%13] = (dp[i+1][(j+r)%13] + dp[i][j]) % mod
      
  else:
    s = int(s)
    r = s*digit % 13
    dp[i+1] = dp[i][-r:] + dp[i][:-r]
  
  digit *= 10
  digit %= 13

print(dp[-1][5])

Buggy python submission (verdict: Time Limit Exceeded):
S = input()
S = S[::-1]
N = len(S)

mod = 10**9 + 7

dp = [[1] + [0]*12] + [[0]*13 for i in range(N)]

for i, s in enumerate(S):
  if s == '?':
    R = [j*10**i % 13 for j in range(10)]
    for j in range(13):
      for r in R:
        dp[i+1][(j+r)%13] = (dp[i+1][(j+r)%13] + dp[i][j]) % mod
      
  else:
    s = int(s)
    r = s*10**i % 13
    dp[i+1] = dp[i][-r:] + dp[i][:-r]

print(dp[-1][5])

Fixed submission:
S = input()
S = S[::-1]
N = len(S)

mod = 10**9 + 7

dp = [[1] + [0]*12] + [[0]*13 for i in range(N)]

digit = 1
for i, s in enumerate(S):
  if s == '?':
    R = [j*digit % 13 for j in range(10)]
    for j in range(13):
      for r in R:
        dp[i+1][(j+r)%13] = (dp[i+1][(j+r)%13] + dp[i][j]) % mod
      
  else:
    s = int(s)
    r = s*digit % 13
    dp[i+1] = dp[i][-r:] + dp[i][:-r]
  
  digit *= 10
  digit %= 13

print(dp[-1][5])

Buggy python submission (verdict: Time Limit Exceeded):
S = input()
S = S[::-1]
N = len(S)

mod = 10**9 + 7

dp = [[1] + [0]*12] + [[0]*13 for i in range(N)]

for i, s in enumerate(S):
  if s == '?':
    R = [j*10**i % 13 for j in range(10)]
    for j in range(13):
      for r in R:
        dp[i+1][(j+r)%13] = (dp[i+1][(j+r)%13] + dp[i][j]) % mod
      
  else:
    s = int(s)
    r = s*10**i % 13
    dp[i+1] = dp[i][-r:] + dp[i][:-r]

print(dp[-1][5])

Fixed submission:
S = input()
S = S[::-1]
N = len(S)

mod = 10**9 + 7

dp = [[1] + [0]*12] + [[0]*13 for i in range(N)]

digit = 1
for i, s in enumerate(S):
  if s == '?':
    R = [j*digit % 13 for j in range(10)]
    for j in range(13):
      for r in R:
        dp[i+1][(j+r)%13] = (dp[i+1][(j+r)%13] + dp[i][j]) % mod
      
  else:
    s = int(s)
    r = s*digit % 13
    dp[i+1] = dp[i][-r:] + dp[i][:-r]
  
  digit *= 10
  digit %= 13

print(dp[-1][5])

Buggy python submission (verdict: Time Limit Exceeded):
S = input()
S = S[::-1]

mod = 10**9 + 7

dp = [1] + [0]*12

for i, s in enumerate(S):
  if s == '?':
    R = [j*10**i % 13 for j in range(1, 10)]
    _dp = dp.copy()
    for r in R:
      dp = [(a1+a2)%mod for a1, a2 in zip(dp, _dp[-r:] + _dp[:-r])]
      
  else:
    s = int(s)
    r = s*10**i % 13
    dp = dp[-r:] + dp[:-r]

print(dp[5])

Fixed submission:
S = input()
S = S[::-1]
N = len(S)

mod = 10**9 + 7

dp = [[1] + [0]*12] + [[0]*13 for i in range(N)]

digit = 1
for i, s in enumerate(S):
  if s == '?':
    R = [j*digit % 13 for j in range(10)]
    for j in range(13):
      for r in R:
        dp[i+1][(j+r)%13] = (dp[i+1][(j+r)%13] + dp[i][j]) % mod
      
  else:
    s = int(s)
    r = s*digit % 13
    dp[i+1] = dp[i][-r:] + dp[i][:-r]
  
  digit *= 10
  digit %= 13

print(dp[-1][5])

Buggy python submission (verdict: Time Limit Exceeded):
import numpy as np

S = list(input())[::-1]

dp = np.array([1] + [0]*12)

nd = len(S)
for i in range(nd):
  if S[i] != '?':
    slide = (int(S[i])*(10**i)) % 13
    dp = np.r_[dp[-slide:], dp[:-slide]]
  
  else:
    new_dp = dp.copy()
    for j in range(1, 10):
      slide = int(j*(10**i)) % 13
      new_dp += np.r_[dp[-slide:], dp[:-slide]]
    
    dp = new_dp % (10**9 + 7)

print(dp[5])

Fixed submission:
S = input()
S = S[::-1]
N = len(S)

mod = 10**9 + 7

dp = [[1] + [0]*12] + [[0]*13 for i in range(N)]

digit = 1
for i, s in enumerate(S):
  if s == '?':
    R = [j*digit % 13 for j in range(10)]
    for j in range(13):
      for r in R:
        dp[i+1][(j+r)%13] = (dp[i+1][(j+r)%13] + dp[i][j]) % mod
      
  else:
    s = int(s)
    r = s*digit % 13
    dp[i+1] = dp[i][-r:] + dp[i][:-r]
  
  digit *= 10
  digit %= 13

print(dp[-1][5])