Problem:
Score : 600 points

Problem Statement

Given is an integer N. Snuke will choose integers s_1, s_2, n_1, n_2, u_1, u_2, k_1, k_2, e_1, and e_2 so that all of the following six conditions will be satisfied:

0 \leq s_1 < s_2

0 \leq n_1 < n_2

0 \leq u_1 < u_2

0 \leq k_1 < k_2

0 \leq e_1 < e_2

s_1 + s_2 + n_1 + n_2 + u_1 + u_2 + k_1 + k_2 + e_1 + e_2 \leq N

For every possible choice (s_1,s_2,n_1,n_2,u_1,u_2,k_1,k_2,e_1,e_2), compute (s_2 − s_1)(n_2 − n_1)(u_2 − u_1)(k_2 - k_1)(e_2 - e_1), and find the sum of all computed values, modulo (10^{9} +7).

Solve this problem for each of the T test cases given.

Constraints

All values in input are integers.

1 \leq T \leq 100

1 \leq N \leq 10^{9}

Input

Input is given from Standard Input in the following format:

T
\mathrm{case}_1
\vdots
\mathrm{case}_T

Each case is given in the following format:

N

Output

Print T lines. The i-th line should contain the answer to the i-th test case.

Sample Input 1

4
4
6
10
1000000000

Sample Output 1

0
11
4598
257255556

When N=4, there is no possible choice (s_1,s_2,n_1,n_2,u_1,u_2,k_1,k_2,e_1,e_2). Thus, the answer is 0.

When N=6, there are six possible choices (s_1,s_2,n_1,n_2,u_1,u_2,k_1,k_2,e_1,e_2) as follows:

(0,1,0,1,0,1,0,1,0,1)

(0,2,0,1,0,1,0,1,0,1)

(0,1,0,2,0,1,0,1,0,1)

(0,1,0,1,0,2,0,1,0,1)

(0,1,0,1,0,1,0,2,0,1)

(0,1,0,1,0,1,0,1,0,2)

We have one choice where (s_2 − s_1)(n_2 − n_1)(u_2 − u_1)(k_2 - k_1)(e_2 - e_1) is 1 and five choices where (s_2 − s_1)(n_2 − n_1)(u_2 − u_1)(k_2 - k_1)(e_2 - e_1) is 2, so the answer is 11.

Be sure to find the sum modulo (10^{9} +7).

Solution in Python (3.8.2) (judge verdict: Accepted)

import numpy as np

MOD = 10 ** 9 + 7

l_matrix = np.zeros((32, 32), np.uint64)
for i in range(16):
    l_matrix[i+16][i] = 1
    l_matrix[i][i+16] = 1
for i in range(15):
    l_matrix[i+1][i] = 1
for i in range(5, 15):
    l_matrix[i+1][i+16] = 1
    
doubling = [l_matrix] #doubling[i] = (l_matrix) ** (2 ** i) % MOD
for _ in range(30):
    doubling.append(doubling[-1] @ doubling[-1] % MOD)

T = int(input())
for _ in range(T):
    N = int(input())
    N += 10
    dp = np.zeros(32, np.uint64)
    dp[0] = 1
    for i, c in enumerate(reversed(format(N, 'b'))):
        if c == '1':
            dp = doubling[i] @ dp
            dp %= MOD
    print(int((dp[15] + dp[31]) % MOD))
    #print(dp)