
// Make newform 5625.2.a.o in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5625_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_5625_2_a_o();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_5625_2_a_o();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        poly := [-9, -15, 31, 8, -11, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [-4, 0, 1, 0, 0, 0], [-6, 13, 5, -8, -1, 1], [3, 3, -6, -1, 1, 0], [24, -43, -11, 14, 1, -1]];
    Rf_basisdens := [1, 1, 1, 6, 2, 6];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_5625_a();"
function MakeCharacter_5625_a()
    N := 5625;
    order := 1;
    char_gens := [4376, 1252];
    v := [1, 1];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

function MakeCharacter_5625_a_Hecke(Kf)
    return MakeCharacter_5625_a();
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..(prec+1)]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 2;
    raw_aps := [[0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, -1, 1, 0, 0], [0, 0, 0, -1, 0, -1], [-1, 2, -1, -1, 0, 0], [0, -2, 0, 1, 0, 1], [1, 0, 1, 1, -2, 0], [-1, 2, -1, -1, 2, -1], [-5, 0, 1, 0, 0, 0], [0, 0, 1, 0, -2, 1], [-5, 2, 1, 0, 2, 1], [-5, -2, 1, 1, 0, 1], [0, -2, 0, -1, -2, 0], [2, 0, 2, 2, -2, 2], [-6, 0, 0, -4, 2, 2], [2, -2, 2, 1, 0, 1], [4, 2, -1, -2, 2, -1], [0, 0, -1, 2, 2, 1], [1, -2, 1, -2, 0, -2], [-6, 2, -1, -1, 0, -2], [3, 2, -2, -1, 0, 0], [2, 2, 2, -1, 0, -1], [-4, 4, 2, -1, 2, -1], [1, 0, 0, 4, 2, 3], [-3, 2, 3, -5, -2, 1], [3, -4, 3, 7, -2, 2], [-7, 4, -1, -4, 0, 2], [0, -4, 0, -3, -4, 2], [-2, 2, -2, 3, 4, -3], [-4, -4, 1, -4, -2, 1], [4, -2, -2, 4, 0, -2], [-5, 0, 1, -1, 0, -1], [4, 0, -1, 1, -4, -4], [1, -4, 1, 1, 0, 1], [-3, 6, -2, 2, 6, -3], [-1, -2, 1, -6, -4, 0], [3, -2, -1, 4, -4, -2], [-5, -2, 1, -6, -4, 0], [-3, 2, 3, 3, -2, 3], [-8, -2, -2, 5, -2, -1], [3, 2, -3, 4, -2, -3], [-5, 4, 1, -2, 6, -2], [4, -4, 0, 9, -2, 3], [-9, 4, -3, 0, 6, 0], [-6, 0, 1, -7, 0, 0], [9, 0, -3, 7, 6, 0], [-12, 2, 0, -7, 2, 0], [-4, 6, 2, 5, 0, -1], [9, -2, 3, 6, 2, -1], [-8, 2, -2, 4, 0, -2], [-7, 0, -1, 5, -2, -1], [10, -4, 4, 4, -2, 3], [3, 6, -3, -9, 4, -3], [15, -2, 3, 0, -2, 0], [11, 2, -1, -3, 0, -3], [4, -2, -2, 11, 2, -1], [7, -4, 1, 12, -4, -1], [12, -6, 5, -1, -4, -2], [-4, -2, 2, -6, 2, 0], [-10, 0, -2, 0, 4, 0], [3, 0, -3, -7, 0, -1], [4, -2, -2, 6, 0, 1], [-4, -2, -4, -6, 2, 0], [6, -2, 5, -5, -4, 0], [3, -2, -3, 0, -4, 0], [-4, -6, -4, 5, 2, -2], [0, 6, -1, 6, 2, -1], [9, 0, 3, 2, 0, -4], [7, -6, 1, 6, 2, 5], [-11, 2, 1, -16, 2, -4], [2, 8, -4, -3, 2, -3], [-3, 0, -3, 7, -6, -4], [2, -8, 4, 8, -4, 8], [-9, -6, 4, -5, 4, 2], [10, -2, -2, 14, 0, 2], [-12, 2, 0, -4, -6, -4], [9, 6, -3, 7, 8, 2], [3, -6, 3, -10, -2, 2], [-10, 6, -3, -1, 0, 0], [15, -4, 3, 7, -4, 1], [4, -2, -1, 1, 0, 2], [-10, -2, -4, 1, 0, 1], [-10, 2, -5, -10, -2, 1], [-8, -6, 4, 9, -2, 8], [-14, 2, 4, -6, 4, 0], [5, 8, -1, -5, 6, -5], [-11, -2, 7, -6, 6, 1], [10, 0, 4, 9, 0, 3], [-12, 6, -6, 8, 0, -3], [-18, 0, 0, -15, 4, -3], [5, 0, 5, 5, 0, 5], [-17, 6, -3, -13, 2, -1], [5, -10, -1, 13, -2, 1], [2, -8, 0, -9, -2, -3], [-17, 0, 1, -7, 10, -1], [-4, -4, -4, -1, 6, -1], [3, -6, 9, 2, -8, 2], [-5, -6, 0, -2, -2, -3], [12, 0, -2, 10, 0, -2], [-1, 0, -1, 13, -2, 2], [2, 8, -4, 3, 6, -3], [-2, -8, -2, -1, -4, 5], [-7, 2, -7, 8, -2, -4], [8, -4, 3, 2, 2, -3], [16, 0, -2, 4, 2, -1], [-8, 4, -8, -8, 6, -8], [-7, 6, -7, -14, -2, -2], [5, 0, -1, 21, 2, 3], [-2, 0, 2, 5, 6, -1], [-3, 2, 2, 5, 4, 4], [4, 2, -3, 2, 6, 1], [8, -6, 8, 6, 0, 6], [6, 2, 0, 0, 0, -7], [-7, -8, -1, 10, 0, -3], [-9, -4, -3, 10, 0, 4], [-5, 10, -6, -17, 0, 0], [3, 0, -3, 12, 4, 0], [10, -2, -8, 1, -6, 1], [-6, -18, 6, 13, -6, 7], [-6, 6, 1, -6, 2, 7], [-8, 8, 0, -7, -2, -1], [4, 0, -2, 5, -4, -1], [-2, 0, -2, -1, 2, -1], [3, -14, 4, 6, -10, 1], [-20, 8, -2, 0, 8, 0], [7, -2, -1, -1, 6, -1], [6, 6, -6, -4, 10, 2], [-13, 2, 10, -4, 6, 1], [7, -2, 3, -5, 2, 7], [-1, -4, -7, 20, -4, 1], [-24, -6, 0, -5, 0, 1], [25, 4, 1, 11, -2, -2], [-1, 4, -8, -5, 0, 0], [22, -6, 4, 11, 4, -1], [2, 10, -3, 12, 6, 1], [0, -10, 0, -18, 0, 0], [-19, 6, 1, -8, 0, -2], [2, -4, 2, 7, 0, 7], [-23, 4, 1, -19, -8, -7], [12, 0, 4, 6, -8, 6], [4, 6, -8, 11, 0, -7], [-4, -2, 2, -8, 0, -1], [13, -8, -5, 5, -6, 5], [-5, 6, -5, 11, 4, -2], [9, -10, -3, 0, -6, 0], [-15, 4, 2, 0, 2, 5], [6, -2, 0, 8, 0, 2], [-2, -12, -1, 3, 0, 4], [0, 14, -6, -3, -2, -9], [-4, 4, -2, 8, 8, 8], [11, -6, 11, 9, -16, 3], [-2, 6, -2, 7, -2, 2], [1, 14, -11, 2, 10, 2], [13, -4, 1, 21, -10, -2], [-26, 8, -2, -25, -8, 5], [-18, 4, -5, 8, 6, -3], [-2, 6, 4, -16, 8, -4], [5, -6, 5, -3, 0, 4], [-21, -4, 3, -8, 12, 4], [26, 0, -4, -3, 2, -3], [12, -6, -6, -15, -4, -3], [0, -8, -1, 11, 4, -2], [14, 0, -10, 6, -2, 0], [18, 6, -6, 3, 8, -3], [9, -10, -3, 3, -4, -3], [-15, 0, 3, -4, 0, 1], [7, -2, 7, 23, -8, 6], [12, 2, -5, 17, 0, -6], [-19, 4, -1, -17, 6, 1], [22, -8, 4, 5, -4, 5], [-19, -2, -3, -11, -2, 1], [-6, 8, 0, -5, -8, 1], [-1, 10, -1, -7, 4, -6], [-13, 14, -6, -6, -2, -5], [22, -6, 10, -10, -10, 2], [-8, -6, -8, 13, -6, 0], [10, -2, 4, 17, 2, 11], [-19, 2, 10, 1, -4, 0], [4, -4, -4, 3, 2, -3], [-8, -8, 3, -20, -6, 3], [9, 0, 3, -2, -8, -2], [-25, 0, 4, -13, 8, 4], [11, -2, -7, 16, -4, -8], [2, -4, -4, -17, -14, 1], [-12, -2, 0, -11, 2, -11], [-14, -4, -2, -5, -4, 2], [1, 10, -6, -15, 0, -4], [-19, 0, 6, -10, 6, 3], [9, 18, -3, -5, 2, -5], [4, -12, 10, -4, -14, -3], [-3, -2, 9, 0, -6, 6], [19, 10, -7, 15, 2, 3], [-34, 12, -10, 13, 8, -5], [24, -14, 6, 0, -2, 6], [2, -18, 8, -1, -10, 11], [-6, -4, 6, -29, -4, -6], [-6, 0, 2, -25, -2, 5], [-1, -2, 5, 7, 6, 1], [-38, 6, -8, -7, 6, 5], [-8, -18, 4, 9, -6, 15], [-8, -2, -2, -2, -8, 9], [-6, 2, -7, 0, 6, 5], [15, 8, -2, 13, -8, -4], [-27, 12, -3, -6, 0, 0], [19, -6, 1, 21, -4, -3], [-5, -4, 7, -19, -6, -2], [-19, 10, -1, -5, 12, 1], [2, 12, 2, -11, 4, 1], [-29, 4, 1, 6, 4, -1], [-1, -10, -5, 15, 2, -9], [-30, 6, -6, -5, 4, 1], [-1, -4, -1, -9, 2, 4], [-13, 6, -1, -9, -12, -3], [19, -2, 1, -18, -6, 0], [-9, 2, -3, -28, 2, -5], [4, -10, 4, 17, -14, 0], [37, 14, 1, 0, 2, -6], [-39, -2, -3, -2, 0, 4], [-4, 4, -4, -1, 18, 5], [28, 0, -2, 16, -2, -9], [-10, 20, -12, 3, 6, -3], [-29, 2, 7, -22, 16, 2], [29, 0, -7, 23, -6, 6], [-13, 0, 5, -26, 2, 4], [-2, 10, -7, 8, -6, -3], [-24, -2, -6, 2, -8, 2], [28, 8, -8, 14, 2, 2], [-30, 2, -12, -10, 12, -3], [34, -2, -2, 29, 0, -1], [-2, 2, -3, -32, 2, -3], [6, 2, 0, 4, 8, -3], [0, 2, 0, 5, 2, -2], [-12, 10, 0, -5, 20, 1], [14, -8, -5, 29, 0, -2], [12, -6, 0, 4, -10, -2], [-15, -12, -2, -10, 6, 3], [7, 12, 7, -19, 4, -1], [-2, 4, -2, -1, 14, -1], [-12, 10, 0, -31, -2, -7], [-31, -2, -1, 3, 2, 3], [-36, 0, 7, -28, 6, -9], [-14, -8, 12, -9, -14, -3], [23, 18, -7, 4, 2, 3], [-2, -6, 4, -11, 0, 1], [-7, 8, -13, -17, 0, 1], [-19, 20, -1, -20, 4, 5], [16, 14, -8, -9, 8, -3], [-12, 14, 6, -18, 0, -1], [12, -10, 12, -9, -10, 9], [16, -16, 6, 14, 0, 2], [6, 10, -12, -9, -10, -9], [5, 4, -1, 28, -4, -2], [9, -10, 15, 28, -10, 9], [-9, 2, 3, -11, 14, 1], [25, 4, 1, -1, 6, -13], [-4, 12, -4, 19, 16, 0], [-27, 2, 8, -23, 4, 6], [-31, -10, 5, 8, -8, 2], [-16, -18, 7, 1, -12, 6], [-3, 0, -3, 1, -2, 8], [-3, -18, -3, 14, 6, -10], [-5, 2, 7, -29, -4, -6], [19, -24, 0, -8, -6, -3], [23, -14, -1, 14, 0, -10], [-13, 16, -1, 7, -2, 0], [3, -12, 3, 9, -18, 9], [2, 18, 2, -1, 18, -7], [-15, 14, -4, -30, 2, -1], [-32, 8, 4, -7, -4, -1], [20, -6, 15, 26, -2, 3], [-22, -4, 1, -16, -14, 7], [13, -4, 6, -9, -4, 8], [-3, 0, -3, 7, 6, 0], [9, -10, 3, 24, 10, -5], [-2, -18, -8, 4, 4, -1], [-18, 14, 6, -24, -6, 0], [10, -6, 5, -6, -14, 7], [-13, -2, -15, 0, -4, 6], [-10, 0, -10, 0, 2, 6], [-39, 2, 3, -6, -6, 0], [1, 30, -4, 4, 2, -7], [-13, -6, -1, 6, -6, 0], [-1, -4, 0, 5, -16, -6], [44, -10, 14, 14, 0, -3], [2, 6, 14, -5, -14, 7], [-3, 4, -10, -3, 8, -10], [-5, 24, -11, 10, 20, -2], [11, 18, -9, 12, 16, -6], [18, 2, -6, 7, 6, 1], [-28, -18, 8, 7, -8, 7], [-25, 0, -7, -21, 0, -3], [0, -2, -12, 0, -8, 0], [10, -8, -2, -19, -2, -1], [-16, 2, 7, -17, -12, 0], [-11, 16, -5, -27, 12, -15], [-5, 2, -1, 16, 4, -2], [0, -8, 0, -10, -10, -4], [-8, 12, 10, -22, 12, 2], [-3, 4, -3, -19, 10, 0], [13, 2, -11, -5, 16, -10], [47, -4, 11, -2, -8, 4], [-3, 4, 15, -28, 4, 1], [-4, -8, 8, 23, -18, -1], [-3, -4, -3, 7, 6, -6], [-12, 18, 5, -23, 16, 0], [-30, 16, 12, -13, -2, -1], [-15, -6, 7, -6, -4, 12], [6, 0, 6, 5, 6, 5], [-29, 0, -6, -3, 4, -4], [2, 4, -4, -1, 4, -1], [-12, 10, -12, 6, 0, 0], [4, -4, 10, 7, 2, 13], [44, 4, 7, 22, -10, -3], [-5, -18, 1, 19, -6, 13], [4, 8, -8, 1, 12, -6], [-24, -14, 12, 16, -10, 4], [-22, 16, 2, -25, 2, -1], [-41, 2, 1, -50, 2, 5], [21, -24, 3, -11, -14, -5], [14, 0, 12, -3, -14, 15], [41, -14, 5, 11, 8, 6], [-6, -6, 0, -29, -4, 1], [-5, 4, -17, 3, 18, -2], [-6, 20, 0, -11, -2, -5], [-18, 4, -6, -21, -4, -4], [-18, 12, 7, -17, 12, -4], [-20, 16, 3, -12, 2, 11], [-10, 2, 8, -5, 16, 1], [-24, 12, -18, -20, 6, -2], [9, 0, -14, -16, -10, -3], [-1, 2, -1, 12, 4, -12], [-16, -16, 9, -27, -12, -2], [-28, 16, -4, -22, -4, -10], [27, -4, 15, 5, 2, 4], [-19, 14, 5, 1, -2, 7], [-10, -16, -4, 4, 6, -3], [46, 20, -8, 1, 4, 1], [-11, 8, 2, 10, 18, 11], [-21, -6, 2, -20, -10, -3], [-35, 2, 13, -39, 0, -8], [-16, 10, -22, 3, 2, -3], [2, 20, -4, 9, 20, 3], [3, -6, 4, 0, -6, 7], [-30, 4, -6, -22, -10, 2], [-2, -10, 16, 5, -2, -1], [10, -2, -1, 26, -14, -3], [24, -16, 17, -14, -18, -9], [-14, -2, 10, -36, 12, 12], [12, 2, 0, 3, -6, 9], [11, -2, 12, -16, -2, 3], [-6, -4, 12, -14, 6, 11], [-12, 4, -18, 31, 0, -5], [-42, 14, 0, 14, 8, 3], [39, -18, -3, 18, 14, 5], [-34, -4, -10, 3, 10, 3], [17, 4, -7, 29, 8, 5], [-40, -2, 2, -2, -16, -2], [5, 8, 5, 11, 0, 5], [-41, -16, 1, -22, 4, 2], [-12, 0, -12, 11, 8, -12], [20, 8, 14, 17, -18, -1], [-20, -2, 4, 9, -14, -3], [-4, -8, -10, 13, -4, -11], [-24, 24, -7, -12, -2, 5], [-12, -20, 12, -1, -20, 0], [7, -20, -5, 18, -10, 0], [-2, 8, 6, -2, -16, -2], [-11, 2, 1, -7, 12, 10], [11, 14, -1, -8, 10, -2], [3, -14, 7, 24, 8, -6], [-16, -12, 14, 13, -8, 1], [-8, 0, -20, 6, -6, 0], [2, 0, 0, -15, -10, 3], [21, 16, -16, -20, 6, -9], [-48, -2, 6, -35, -2, -11], [6, 0, 0, 20, 8, 8], [34, 8, -6, 41, 6, 11], [-14, 0, -8, -2, 22, 4], [31, 0, 6, 6, -14, 5], [-15, 10, -3, -21, 10, 9], [22, -6, 4, 27, 0, 9], [-7, 18, -17, -3, 6, -3], [43, -16, 25, -15, -12, 15], [16, 0, 8, -10, -4, 2], [-1, 0, 17, 4, -2, 4], [-40, 20, 15, -8, 2, 11], [16, 14, 3, -18, 6, -13], [-5, 16, -5, 29, 14, 0], [17, -8, -1, 18, -26, -6], [-21, 2, 9, -4, 10, 1], [30, -2, 0, 34, 0, -8], [9, 2, 15, 12, -18, 0], [-14, 4, -3, 5, 12, -8], [-3, -12, 3, 11, 12, 11], [38, -6, 1, 8, 6, -11], [8, 4, -9, 35, -4, -6], [-21, 24, 9, -2, -2, 4], [-5, 6, -4, -12, 6, -5], [21, -18, -3, -14, -6, -14], [9, 0, -14, 19, 12, 8], [-34, 16, 14, -29, 6, 1], [-29, 22, 3, 10, 8, 4], [-4, -22, 14, -6, 2, 0], [-34, 12, -22, -28, 14, -10], [34, -4, -1, 12, -22, -11], [-12, 2, -1, 8, 6, 1], [-5, 10, -5, -36, 10, -6], [6, -2, 6, -1, 14, -6], [46, -12, 4, -10, -8, 8], [-25, -12, 10, 5, -20, 4], [12, -8, 12, 32, 12, 2], [10, -22, -8, 10, 12, -8], [42, -12, 1, 19, -16, -10], [19, 26, 1, -6, 2, 1], [25, 2, 7, 20, -14, 14], [51, -2, -3, 31, 4, -5], [-18, 8, -1, -2, 22, -9], [-18, 8, 12, -8, 4, 4], [4, -12, -2, 14, -6, -10], [36, 2, 6, 8, -14, 2], [-30, -8, 6, -25, 12, 0], [8, 6, 2, -13, -4, -1], [21, -10, -9, 30, 6, -18], [18, 24, 0, 34, -2, -7], [15, -16, 3, -13, 4, 11], [39, -10, -9, 19, -12, -17], [-31, -4, -7, 2, 10, 8], [25, -10, 2, 37, -8, 8], [-11, 2, -5, -8, -18, -2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5625_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5625_2_a_o();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_5625_2_a_o(:prec:=6)
    chi := MakeCharacter_5625_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_5625_2_a_o();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_5625_2_a_o( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5625_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-9, 15, 31, -8, -11, 1, 1]>,<7,R![-100, 160, 101, -42, -21, 2, 1]>],Snew);
    return Vf;
end function;