Buggy python submission (verdict: Time Limit Exceeded):
S = list(input())

DIVIDER = 13

mod_digit = [i % DIVIDER for i in range(10)]
mod_level = []
tmp = 1
for _ in range(len(S)):
    mod_level.append(tmp % DIVIDER)
    tmp *= 10
mod_level = mod_level[::-1]

memo = [[0] * DIVIDER for _ in range(len(S) + 1)]
# def dp(pos, mod_tmp):
for idx in range(len(S) + 1):
    # if pos < 0:
    #     if mod_tmp == 5:
    #         return 1
    #     else:
    #         return 0
    # elif memo[pos][mod_tmp] != None: return memo[pos][mod_tmp]
    if idx == 0:
        for mod_tmp in range(DIVIDER):
            if mod_tmp == 5:
                memo[0][mod_tmp] = 1
            else:
                memo[0][mod_tmp] = 0
    else:
        pos = idx - 1
        for mod_tmp in range(DIVIDER):
            ret = 0
            if S[pos] == '?':
                for d in range(10):
                    mod_nxt = (mod_tmp + mod_digit[d] * mod_level[pos]) % DIVIDER
                    ret += memo[idx - 1][mod_nxt]
            else:
                d = int(S[pos])
                mod_nxt = (mod_tmp + mod_digit[d] * mod_level[pos]) % DIVIDER
                ret += memo[idx - 1][mod_nxt]
            #
            memo[idx][mod_tmp] = ret % 1000000007

print(memo[len(S)][0])

Fixed submission:
inf = 10 ** 18
P = 10 ** 9 + 7

S = input()
N = len(S)

tmp = 1
modrow = [1] * 6
for i in range(5):
    tmp = (tmp * 10) % 13
    modrow[1 + i] = tmp

modtab = [[None] * 10 for i in range(6)]
for i in range(6):
    for j in range(10):
        modtab[i][j] = (modrow[i] * j) % 13

# for modcol in modtab:
#     print("#", sorted(modcol))

# X + question === 5 (mod 13)
X = 0
q_idx = []
for i, Si in enumerate(S[::-1]):
    if Si == '?':
        q_idx.append(i)
    else:
        X = (X + modtab[i % 6][int(Si)]) % 13


M = len(q_idx)
ans_tab = [[0] * 13 for _ in range(M + 1)]

ans_tab[0][0] = 1
for i, idx in enumerate(q_idx):
    for src in range(13):
        for coeff in range(10):
            dst = (src + modtab[idx % 6][coeff]) % 13
            ans_tab[1 + i][dst] = (ans_tab[1 + i][dst] + ans_tab[i][src]) % P
        
# print("#", ans_tab[-1])
idx_ans = (5 - X) % 13
print(ans_tab[-1][idx_ans])

Buggy python submission (verdict: Time Limit Exceeded):
S = list(input())

DIVIDER = 13

mod_digit = [i % DIVIDER for i in range(10)]
mod_level = []
tmp = 1
for _ in range(len(S)):
    mod_level.append(tmp % DIVIDER)
    tmp *= 10
mod_level = mod_level[::-1]

memo = [[0] * DIVIDER for _ in range(len(S) + 1)]
# def dp(pos, mod_tmp):
for idx in range(len(S) + 1):
    # if pos < 0:
    #     if mod_tmp == 5:
    #         return 1
    #     else:
    #         return 0
    # elif memo[pos][mod_tmp] != None: return memo[pos][mod_tmp]
    if idx == 0:
        for mod_tmp in range(DIVIDER):
            if mod_tmp == 5:
                memo[0][mod_tmp] = 1
            else:
                memo[0][mod_tmp] = 0
    else:
        pos = idx - 1
        for mod_tmp in range(DIVIDER):
            ret = 0
            if S[pos] == '?':
                for d in range(10):
                    mod_nxt = (mod_tmp + mod_digit[d] * mod_level[pos]) % DIVIDER
                    ret += memo[idx - 1][mod_nxt]
            else:
                d = int(S[pos])
                mod_nxt = (mod_tmp + mod_digit[d] * mod_level[pos]) % DIVIDER
                ret += memo[idx - 1][mod_nxt]
            #
            memo[idx][mod_tmp] = ret % 1000000007

print(memo[len(S)][0])
            

Fixed submission:
inf = 10 ** 18
P = 10 ** 9 + 7

S = input()
N = len(S)

tmp = 1
modrow = [1] * 6
for i in range(5):
    tmp = (tmp * 10) % 13
    modrow[1 + i] = tmp

modtab = [[None] * 10 for i in range(6)]
for i in range(6):
    for j in range(10):
        modtab[i][j] = (modrow[i] * j) % 13

# for modcol in modtab:
#     print("#", sorted(modcol))

# X + question === 5 (mod 13)
X = 0
q_idx = []
for i, Si in enumerate(S[::-1]):
    if Si == '?':
        q_idx.append(i)
    else:
        X = (X + modtab[i % 6][int(Si)]) % 13


M = len(q_idx)
ans_tab = [[0] * 13 for _ in range(M + 1)]

ans_tab[0][0] = 1
for i, idx in enumerate(q_idx):
    for src in range(13):
        for coeff in range(10):
            dst = (src + modtab[idx % 6][coeff]) % 13
            ans_tab[1 + i][dst] = (ans_tab[1 + i][dst] + ans_tab[i][src]) % P
        
# print("#", ans_tab[-1])
idx_ans = (5 - X) % 13
print(ans_tab[-1][idx_ans])