Problem Description
We consider a sequence A with p float-point numbers denoted by a0,a1,...,ap−1 where p is a prime number. To simplify our problem, we guarantee that p must be 13, 103 or 100003.
To make a decomposition for this sequence, we define the kernal functions
r(h,k)=2sin3(2πhkp)
Therefore we can get a new sequence B = {b0,b1,...,bp−1} tranformed from the original sequence A where
bk=∑h=0p−1ah∗r(h,k)
Your mission is to calculate the new sequence B.
Input
The first line is the number of test cases. Each test case contains two lines. The first line contains an integer p. The second line contains p float-point numbers corresponding to the sequence A.
Output
For each test case, output p float-point numbers rounded to three decimal places in one line corresponding to the sequence B.
Sample Input
13
7 0 0 0 0 0 0 0 0 0 0 0 0
13
1 2 3 4 5 6 7 8 9 10 11 12 13
13
11 7 7 7 7 7 7 7 7 7 7 7 7
Sample Output
7.000 7.000 7.000 7.000 7.000 7.000 7.000
7.000 7.000 7.000 7.000 7.000 7.000
91.000 85.477 92.015 93.543 91.049 99.763
98.551 98.517 97.304 106.018 103.525
105.053 111.590
95.000 102.032 102.032 102.032 102.032
102.032 102.032 102.032 102.032 102.032
102.032 102.032 102.032
