First of all, make sure to add type signatures and use the appropriate number type. All your numbers are fine with Int, since they're bounded by
2 * (1000000006 * 1000000006)
Therefore, you can get rid of Integer throughout your program and simply use Int:
fib2 :: Int -> (Int, Int)
fib2 0 = (1, 1)
fib2 1 = (1, 2)
fib2 n
| even n = ((a*a + b*b)`rem` 1000000007, (c*c - a*a)`rem` 1000000007)
| otherwise = ((c*c - a*a)`rem` 1000000007, (b*b + c*c)`rem` 1000000007)
where (a,b) = fib2 (n `quot` 2 - 1)
c = a + b
Next, you want to use rem and quot, since they're usually faster and provide the same result as mod and div on positive numbers.
We also add a type signature to solve:
solve :: Int -> Int
solve n = (a*b)`rem`1000000007
where (a,b) = fib2((2*n-1)`rem`2000000016)
Last but not least, we now simplify your main:
main :: IO ()
main = C.getContents >>= putStrLn . unlines . map (show.solve. fst . fromJust . C.readInt) . tail . C.words
However, I don't think that the IO part is the bottle-neck, so your usual String methods should be fine for this challenge, e.g.
main = getContents >>= putStrLn . unlines . map (show . solve . read) . tail . words
Indeed, this version of your program gets accepted of SPOJ. But so did your one. I guess that they updated the GHC version* and used the appropriate optimization flags. But you can make sure that they use optimization via
{-# OPTIONS_GHC -O2 #-}
* I just rememered that there was a defect with even in GHC 7.10there was a defect with even in GHC 7.10. If they used GHC 7.10 (which is unlikely), then you were affected by that defect.
