
// Make newform 6025.2.a.g in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6025_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_6025_2_a_g();".
// 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_6025_2_a_g();".
// 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 := [-1, -5, 1, 31, 14, -62, -28, 43, 15, -11, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-5, 1, 31, 14, -62, -28, 43, 15, -11, -2, 1], [17, 26, -109, -118, 220, 155, -157, -71, 42, 9, -4], [-19, -27, 110, 118, -220, -155, 157, 71, -42, -9, 4], [-17, -37, 132, 221, -349, -315, 280, 148, -80, -19, 8], [30, 78, -174, -366, 390, 441, -303, -187, 88, 23, -9], [-32, -68, 242, 367, -611, -505, 477, 235, -133, -30, 13], [-39, -93, 261, 471, -630, -609, 492, 271, -140, -34, 14], [-53, -117, 364, 588, -849, -764, 649, 342, -182, -43, 18], [61, 151, -435, -780, 1074, 1023, -843, -460, 240, 58, -24]];
    Rf_basisdens := [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1];
    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_6025_a();"
function MakeCharacter_6025_a()
    N := 6025;
    order := 1;
    char_gens := [2652, 2176];
    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_6025_a_Hecke(Kf)
    return MakeCharacter_6025_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 := [[1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 1, -1, 0, 0, -1, 0, -1], [0, 1, 1, 1, 1, -1, 0, 0, 0, 0, 0], [1, 0, 0, 0, -1, 0, 1, -1, 1, 0, 1], [1, 0, 0, -1, 0, -1, 2, 0, 2, 1, 0], [-3, 1, 1, -1, -1, -1, 1, -1, 2, 0, -1], [3, 0, 0, 0, 1, -1, 0, -1, 0, 0, 0], [2, -1, 1, -1, 0, -2, 3, 0, 1, 1, 0], [3, -2, 3, 1, 0, -2, 0, -1, -1, 0, 1], [3, 1, 1, -1, 1, -3, 1, 0, 1, 0, 0], [2, -4, 1, -1, 0, 0, 1, -2, 1, 0, 0], [2, 2, 0, 0, 0, 2, 0, 2, -1, 1, 1], [3, 2, -1, -3, -1, 1, 0, -1, 0, 0, 1], [-3, 1, 0, 2, 0, -2, -1, 2, 0, -3, 0], [-1, -1, -2, 3, -1, 1, -1, 2, 0, 2, 1], [0, 2, -2, 0, -1, 1, -2, 2, -1, 1, 1], [5, -2, 1, 0, 1, -4, 2, -1, 3, -1, -2], [0, -3, 2, -1, 2, -2, -1, -1, 0, -3, 0], [-3, 2, -2, 0, 0, 2, -1, -1, 3, 2, 1], [-1, 1, -3, -1, 3, -1, 4, 0, -1, 2, -1], [2, 3, 1, 1, -1, 0, -3, -1, -2, -2, 2], [0, 4, 2, -3, -3, 1, 0, 1, 0, 1, -2], [3, -5, 4, -1, -1, 1, 0, -2, -3, 0, 0], [-1, -3, 6, 2, -3, -1, 0, -2, 0, -5, -1], [6, -2, 1, 1, 1, 1, -1, 0, -1, 1, -1], [3, -3, -1, 0, -5, 1, -2, -2, 2, 3, 6], [1, 0, 0, -2, 3, -4, 2, -2, 0, -2, -1], [-2, -1, 2, 2, 2, -5, -2, -1, 0, -4, -4], [4, -2, 1, -3, -1, 1, -2, 0, 2, -1, -2], [-4, -3, 2, 3, 1, -3, 0, 0, 4, 0, -1], [-1, 1, -1, 3, 2, 0, 0, 3, -2, 3, -3], [-11, 3, -3, -3, 1, 2, -1, 1, 3, -3, 0], [-5, 4, -2, 2, -2, -2, -4, -2, 2, -1, 0], [1, -1, 1, -1, 3, -5, 6, -2, 1, -1, -2], [4, -3, 2, 7, 2, 1, -2, -2, 0, -2, 3], [3, 1, 3, 5, 2, -4, 0, 3, -1, -4, -2], [10, -2, 3, 1, 0, -2, 3, -2, 3, 4, 0], [-4, 2, -3, -7, 1, 1, 3, 1, -2, -3, -4], [0, 0, -3, 3, 0, 1, -4, 4, -3, -3, 0], [-7, 6, -1, -3, 0, -2, -1, -2, 4, -2, -3], [0, 3, -4, -1, 0, 6, 0, 0, 3, 2, 5], [-3, 3, 5, -2, -4, -2, -4, -1, 3, -5, 1], [-3, -2, 3, 5, -4, 4, -5, 4, -4, -2, 3], [-5, -3, 3, 3, -4, -1, -6, -1, -1, -5, -3], [-1, -1, 1, 7, 0, 2, -2, 4, -1, 0, 1], [3, -5, 1, -3, -4, 3, 2, 3, -1, 4, 4], [4, -5, 7, 2, -2, -3, 1, 0, 3, -8, -1], [-5, 1, -6, 1, 0, -1, -1, 0, 4, 5, 4], [-2, 1, 1, -1, 7, -7, 3, -2, -2, -3, -8], [1, 7, -3, -4, 0, -1, 2, -3, -2, -2, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-4, 1, -2, 1, -4, 3, -5, -1, 4, -1, 6], [5, 3, 4, 3, 4, -1, 1, 3, -1, 2, -2], [-11, 10, -13, -4, 0, -4, 4, -1, 2, -3, -4], [1, 7, -8, -9, 1, -1, 2, 3, 1, 3, -2], [6, 1, 1, -4, 3, 1, 7, 3, -1, 1, -3], [4, 1, -3, -2, 1, -6, 3, 4, -5, 0, -5], [5, -6, 2, -1, 2, 2, 1, 1, -7, -3, 0], [5, 0, -1, 0, 4, -1, 10, -1, 4, 4, 1], [-6, 2, -5, -3, 1, -1, 4, 0, 3, 8, 1], [3, -1, 0, 0, -5, 1, 0, 0, 7, -3, 1], [5, -4, -5, -4, 4, 0, 0, -5, -1, 5, -2], [1, -2, -1, -5, -3, 2, -2, 2, -1, -1, 3], [6, -1, -4, 2, 5, 2, 0, 0, -6, 6, -1], [4, -1, 3, 2, 5, -6, 3, -1, -3, -9, -3], [1, 1, -1, -3, 0, 1, -2, -2, 1, -6, -7], [6, 6, -1, 3, -1, 0, 0, 0, 1, -2, -3], [0, -2, 8, 1, -1, -5, 1, 0, 2, -3, -2], [0, -4, 1, 9, 0, 0, -5, -1, -4, 0, 1], [2, -2, -7, -5, 4, 5, 3, 3, -3, 5, 2], [3, -1, 2, 2, -5, 2, -3, 6, -3, -3, -2], [-3, 4, 0, 3, 8, -6, 1, 2, -3, -6, -4], [-4, -5, -4, 3, 0, 8, -5, 0, 1, 3, 3], [4, -3, -2, -3, -5, 3, 2, -1, 2, 3, 1], [4, 3, 0, 5, 4, 2, 3, 5, -1, 4, -3], [6, -7, 8, 3, -6, 1, 2, -4, 4, -1, 0], [10, 2, -2, -4, 4, -14, 3, -3, 5, -4, -8], [0, 2, -4, -2, -1, 6, -6, 5, 3, -1, 0], [1, 3, 2, -2, -4, 1, -1, 3, -8, -1, 1], [-9, 4, -5, 8, 4, 1, 0, 3, 2, 5, -2], [-4, 14, -2, -1, -3, 2, -2, 2, 2, 2, -4], [-6, 4, -9, -4, 4, -1, -3, -3, -2, 1, -6], [8, 0, 6, 5, 1, -5, 4, -4, -3, 3, 1], [8, 0, 6, 2, -4, 8, -5, 5, -2, 6, 8], [-2, -2, 4, 5, 0, 5, 1, -4, 4, -2, -1], [-3, 2, -3, -4, -1, 1, 4, 5, 3, 10, 1], [-11, 4, -5, -2, 0, 2, -9, 0, -5, -6, 3], [0, 6, -3, 9, 2, -4, -7, 1, -1, -3, 0], [5, -1, -8, 5, 2, -5, 2, 2, -1, 1, 1], [6, 5, -6, 2, -2, 13, -2, 11, -8, 7, 1], [-2, -1, -11, -3, 5, -3, 6, 1, 4, 2, -3], [7, 8, -7, -8, -4, -7, 7, 2, 8, 5, 0], [-10, -4, 4, -2, -4, 1, 0, 8, -3, 1, -3], [6, 1, 4, -5, 1, -5, 7, -2, 5, -3, -1], [1, -2, -5, 7, 2, -3, 5, 4, -6, 2, -7], [0, -5, 2, 7, -6, -1, -5, -2, 2, 1, 3], [-3, -5, -10, 3, 1, 5, -7, 2, -3, -1, 0], [-5, 1, -5, 4, 3, -9, 2, -1, 3, -7, -5], [0, 7, 6, 0, 1, -3, -1, 1, -6, -6, -3], [-9, 11, -7, -2, 5, -4, 6, 2, 7, 5, -2], [-3, -2, -2, 12, -5, 5, -6, 3, 2, 5, 3], [4, 3, 6, -1, -5, -7, -2, -2, 5, -12, 3], [-15, 4, 1, 10, -7, 11, -5, -2, -3, -1, 8], [-5, -1, -16, -3, -3, 7, -3, -4, 2, 5, 6], [9, -8, 7, -1, -3, -2, -1, -4, -7, -5, 2], [11, -8, 8, 2, -9, -2, -2, -1, 0, -2, -6], [0, 3, 4, 10, 10, -11, -5, 1, -5, -13, -2], [0, 0, -1, 3, -2, -5, -4, 5, -3, 2, 4], [13, 0, 1, -2, 7, -10, 12, 2, -2, 10, -6], [14, -3, 5, 6, -1, 1, -3, -2, 1, -4, 1], [8, 1, -2, 2, -1, 4, 0, 10, -4, 11, 2], [-6, -2, -3, -8, 1, 5, 6, 6, 1, 5, 2], [-5, -3, 6, 2, 5, -5, 5, -4, 10, -4, -5], [9, 0, 6, 8, -4, 7, -10, 2, -5, -10, -3], [-2, 6, -7, -7, -1, 6, -6, -3, 1, -2, -2], [8, -11, 0, 0, -7, 17, -4, 5, -3, 10, 6], [6, -8, 8, 2, -2, 1, 4, -6, 1, -2, 0], [3, -5, -1, -2, -1, 1, 2, -7, 4, 6, 0], [8, -3, 5, 2, -2, -5, 4, 7, -3, 4, -1], [1, -6, 11, -7, -6, -10, 1, -1, -1, -5, -1], [-4, -2, 2, 5, 15, -12, 10, -3, -4, 3, -8], [-5, 5, -7, -5, 11, 0, 4, 0, -2, -5, -10], [-17, 4, -15, -1, -12, 13, -7, 2, 5, 2, 3], [-8, 4, -6, -5, -3, 2, 0, -7, 7, -4, -5], [-3, 3, -3, 4, -3, 10, -9, 10, -10, 1, 13], [7, -1, 6, -5, -13, 4, -1, 2, 5, 2, 5], [7, 10, -5, -8, -4, -8, 4, -6, 5, -6, -8], [7, -5, 6, -2, -9, 10, -2, 6, 3, 1, 5], [-6, 6, 0, 2, -12, 3, -7, 2, 7, -1, 4], [-11, 4, 4, -3, -4, 2, -1, -1, 0, -15, 2], [0, -5, -2, -3, -12, 8, -1, 3, 0, 3, 8], [-14, 9, -6, 5, 8, 6, 0, 0, -6, 2, 1], [-1, 4, -15, 2, -5, 13, -9, 5, -2, 7, 11], [-4, 3, -6, 5, 6, 8, -13, 4, -9, 4, 6], [-2, 5, -4, 3, 6, -2, -2, -4, 9, 0, 3], [-2, 7, -4, -2, -3, 0, 11, 1, 8, 4, -1], [-6, 4, 7, 6, 1, -7, 3, 0, 0, -5, -3], [4, -5, 3, 13, 2, -4, 3, 4, 4, 3, -2], [-6, -1, 9, -8, -4, 2, -1, 3, -9, -1, -3], [-5, 13, -11, 6, 4, -3, 4, 6, 0, 7, 4], [-11, 5, -2, 5, -12, 2, -7, -1, 4, 8, 6], [3, -9, -1, -7, -2, -10, -1, -4, 2, -1, -3], [-7, -4, 3, 0, 3, 13, -7, 1, -7, 0, 0], [-3, 4, -1, 0, 0, -3, -4, -10, 11, -2, 8], [3, 8, -11, 8, -14, 24, -6, 7, 5, 4, 15], [-6, 3, -5, -3, 0, 9, -5, 0, -13, -7, 2], [-11, 6, 1, -4, -8, 9, -1, 12, -8, 5, 2], [8, -4, 6, -4, -6, -8, 7, 10, 1, -3, -6], [-3, 0, 13, -1, -8, 4, -3, -12, 8, -7, 2], [-17, 5, -5, -8, 1, 3, -8, 8, -5, 2, -3], [-4, 1, 4, 5, 5, -6, 4, -9, 8, -2, 2], [-8, -3, 4, 1, -15, 19, -7, 0, 0, -6, 12], [-14, -3, -7, 9, -11, 6, -4, 1, 10, -1, 12], [-9, -1, 12, 15, 1, -10, -6, -4, -1, -11, -7], [16, 0, -2, -5, -6, 10, -7, 1, -1, 9, 13], [1, -11, 21, 8, -8, -11, 6, -3, 0, -8, -2], [-12, 14, -4, 8, 4, 0, -6, 3, 1, -11, -4], [18, -8, 9, -9, -5, 0, 3, 0, 2, -2, -2], [4, 1, -12, -8, 1, 7, 4, 4, 3, 1, -2], [7, 0, -10, -6, 10, 9, 1, 0, -8, 14, 4], [-24, 1, -6, 3, 6, -10, 3, 2, -5, -3, -5], [2, 4, -4, 1, 12, -5, 12, -4, 0, 9, -5], [-11, -2, -3, -2, 4, 5, -8, 7, -7, 4, 1], [1, 4, -6, -9, 4, -10, 15, -1, 4, 1, -8], [14, -2, 2, 10, 3, -8, 12, 2, 10, 6, -6], [-3, -8, 12, 0, -7, 6, 3, 1, 2, 9, 8], [18, -13, 14, 6, -3, -2, 5, -1, 13, 4, -1], [14, -2, 10, -8, -18, 1, 0, -6, 15, -9, 7], [-5, 3, -6, -1, -1, -15, 9, -6, 9, -5, -2], [2, 5, -10, -11, 7, 2, -4, -7, -6, 0, 0], [11, 15, -14, -12, 11, 0, 3, -5, 3, 5, 1], [-11, 15, -6, -8, 3, -11, 7, -3, 7, -2, -10], [13, -16, 25, -10, -6, -8, -1, 0, -6, -7, 0], [15, 1, 3, 0, 9, -12, 18, 2, 0, 8, -17], [0, -21, -1, 8, 2, 2, -3, -2, -4, -4, 9], [15, -1, -5, 4, -2, -1, 0, 0, 5, -1, 6], [-7, -7, -7, -2, 9, -18, 11, -5, 4, 5, -14], [-15, 7, -8, 3, -10, 3, 1, -6, 3, 0, 3], [5, 6, 9, 2, 8, 4, 2, 11, -15, 7, 6], [15, 5, 8, -8, -2, 0, 3, -8, 1, -5, -4], [-3, -2, 6, -6, 2, -8, -5, 3, -10, -5, 0], [-17, 2, 0, 12, -3, 12, -14, -1, -4, 0, 10], [19, 3, -8, 4, 5, 5, 4, 7, -3, 5, 4], [1, -3, 8, -2, 1, -2, 5, -6, 2, -10, -3], [-20, -1, -16, 9, 0, 10, -5, 2, -1, 3, 3], [18, 6, 21, 2, -7, -6, -1, 0, 4, -3, 5], [10, -5, 5, 14, -8, -3, 3, 3, -2, 7, -1], [-3, 3, -8, 5, -4, 7, -1, 8, -9, 3, -6], [8, 8, 7, -6, -1, -7, 4, -2, -5, -9, -10], [-3, 3, 1, 1, -3, -4, -6, -1, -4, -15, -3], [-9, -5, 5, -3, -9, 5, 0, 3, 3, 5, 13], [19, -8, 12, -15, -5, -17, 12, 1, 4, 6, -8], [-21, 0, -12, -1, 5, 2, -1, -1, 8, 3, -6], [-1, -10, 15, 3, -5, -15, -3, -1, 2, -8, 0], [-2, 7, 1, -3, 8, -1, -7, -8, -1, -16, -1], [12, -8, 20, 9, 3, -5, -7, -11, 6, -12, -2], [-12, -1, -3, -13, -1, 7, -5, 1, -4, 8, 0], [7, 10, -4, -4, 21, -14, 8, -10, -1, 4, -11], [3, 1, -18, -8, -8, 21, 7, 5, 11, 17, 11], [1, 9, -3, 11, 13, 2, 5, 12, -5, 7, -3], [4, 4, 15, 13, 3, -11, -12, 0, -5, -11, 4], [-6, -16, 13, -1, 14, -17, 15, -7, 4, -7, -8], [3, -4, 4, 2, 2, -5, 5, -8, -4, 2, -4], [-2, -13, 9, 4, 8, 9, 1, 5, -13, 5, -7], [18, 6, -7, -2, 9, -1, 3, 9, -3, 8, 3], [11, -17, -1, -7, 2, 7, 0, 1, -9, 4, 6], [2, 7, -15, -9, 0, -3, -7, -4, 0, -15, -10], [-10, -2, 1, 0, 2, 2, 1, -4, 7, -5, -3], [-18, 7, -9, -1, -11, 5, -7, 3, -7, -5, 5], [7, -6, -6, 3, 2, 4, -7, -1, -6, -3, -4], [8, 5, -22, -12, 0, 4, 8, 0, 4, 27, 10], [18, -1, 3, 2, -4, 18, -16, 5, -9, 10, 7], [-9, 10, -10, -12, -4, 3, -2, 10, 4, -5, -10], [-14, 8, -12, 9, -8, 0, -11, 4, -8, 3, 4], [-8, -12, 12, -1, 1, 1, -6, 0, -6, -9, -7], [6, 7, -8, 8, 7, 0, -7, 9, -7, 12, -1], [17, -17, 14, -4, 0, -13, 9, -3, 4, 7, -15], [17, -7, 6, 7, 7, 11, 8, 9, -1, 12, 5], [2, -14, 5, 2, 2, -19, -8, -4, 0, -9, -10], [-1, 3, -11, -9, -4, 14, 5, 5, 0, 27, 6], [-4, -19, 14, 6, -3, -5, -10, -3, -6, -12, -6], [-6, 0, -8, -10, -5, 11, -6, 15, -7, 1, 10], [9, -6, 17, 4, -2, 2, -7, -6, 8, -17, 4], [0, 0, -2, 3, 0, 20, -10, 2, -11, 10, 8], [-4, -5, -1, 11, -7, 8, -19, 3, -3, -5, 3], [-5, 0, -2, 10, 8, -4, 6, 5, -2, 3, -1], [-2, -17, 16, -10, 4, -19, 10, -9, 4, 8, 0], [-6, 5, 0, -18, 2, -10, 15, -2, 13, 4, -14], [8, 7, -6, 0, 9, 0, 4, -2, 6, 5, 6], [9, -11, 7, -10, -9, 1, 5, -10, 7, 7, -1], [-16, 17, -19, -1, 12, 6, -13, 9, -17, 1, -1], [-3, 0, -16, -18, 10, -7, 16, 7, 0, 11, -7], [21, -3, -14, -4, 8, 8, -5, 4, -14, 8, -1], [4, 9, 9, -6, 16, -20, 12, -8, 0, -11, -11], [-7, 8, 13, -5, -2, -10, 9, 1, 5, -9, -11], [7, 2, 7, 19, -2, 1, -11, 2, -6, 0, 7], [-8, -12, -4, -15, 6, -9, 5, -3, 4, 5, 7], [10, -7, 16, -5, -10, -7, 3, -2, -1, -14, 2], [4, 1, 2, -20, 5, -7, 1, -9, -4, 0, -7], [-16, -9, 8, 1, -18, 5, 2, 0, -1, -7, 8], [10, -17, 17, 2, -10, 1, -9, 1, -7, -9, -5], [-7, 7, 10, -16, -7, 2, 8, -6, 9, -7, 1], [9, 3, -2, 0, 9, -1, 6, 12, -1, -1, -9], [-23, 8, -7, 14, 17, -3, -8, 3, -17, -7, -12], [0, 4, 0, 18, 5, 7, -10, 17, -21, 4, -1], [-10, 15, -9, -7, 0, -1, 4, 1, 7, 2, -3], [10, 11, 3, -3, 0, -7, 5, -3, 0, -5, 9], [15, 3, 1, -11, -6, -3, -4, 3, 6, -6, -4], [0, -2, 18, 12, -2, -3, -10, -4, -1, -1, -3], [30, 3, 3, -19, 8, 3, 10, -2, -7, 5, -5], [2, 1, 11, 11, -10, 8, -15, 6, -9, -4, -1], [0, -15, 9, -6, -15, -4, -5, -14, 9, -9, 10], [-3, 0, 12, 6, -5, -2, 2, -4, -2, -17, -8], [0, -13, 8, 1, -6, 16, -1, 8, -5, -1, -9], [-15, 12, 11, -10, -3, -4, 9, -2, 9, -5, -6], [16, 3, 22, -3, -3, -7, 9, -6, 11, -2, 2], [13, 19, -23, -18, 6, 9, 0, 4, 2, 3, 2], [-4, 18, -10, -21, 0, 10, 8, 13, -12, 3, -6], [-13, 18, -23, -2, 7, 16, -3, 3, 5, 5, -3], [10, -14, -10, -2, 1, 4, 5, 3, 0, 9, 9], [10, -18, 16, 1, -1, -11, 3, -1, -10, 0, -6], [17, -19, 13, 21, -1, -13, -3, -8, 6, 3, 11], [4, 1, 16, 6, -8, 2, -5, -2, 10, 6, 12], [-42, 7, -4, -5, 6, -8, 8, -2, 18, -4, -17], [19, -20, 30, 10, -1, -15, 7, -8, 5, 6, 2], [-8, -7, -2, 6, 11, 16, -9, 0, -8, 3, 3], [-15, -10, -16, -9, 2, 2, 0, -2, -4, -3, -4], [2, -6, -2, 7, 2, 8, -3, -7, -8, 9, 11], [14, -1, 9, 9, 20, -18, 8, 0, -13, -3, -7], [25, 3, 19, 15, -5, -1, -5, 7, -7, -4, -11], [16, -25, 12, 0, 3, -9, 13, -3, 0, 1, -8], [-6, -18, 13, -2, -5, 1, 1, 4, -2, 7, 1], [10, -15, 22, 8, 4, 8, 3, 9, -5, 3, -2], [11, -13, -11, -14, 4, -4, -5, -2, -2, 0, -6], [-10, 4, -1, -3, 8, 1, -4, -7, 5, -10, 8], [-14, -4, -19, 3, -5, 7, -3, 1, 9, 1, 6], [17, -13, 9, 2, 3, 10, -3, 1, -5, 15, 16], [7, 3, 15, 3, -9, -3, -5, 2, 4, -1, 3], [21, -21, 15, 7, -1, -4, -3, -3, 7, 6, -1], [26, 1, 6, 2, -1, 2, 4, 5, -4, 12, 0], [-17, 16, -4, 14, 7, 2, 3, 4, 12, 0, 12], [-28, 12, -9, 7, 0, 3, -3, -7, 18, 6, 5], [4, 13, -14, 1, 4, 15, 3, 11, -14, 4, -11], [1, -5, -4, 11, 2, -7, 9, 2, -8, -2, -9], [-14, 22, -14, 7, 9, 10, -14, 9, -6, -13, 6], [6, -6, -9, 2, -9, 22, -5, 7, 4, 18, 15], [16, -3, 8, 2, -7, 0, -3, 5, 7, 8, -3], [-18, 11, -15, -8, 6, 9, 3, 12, -6, 2, -15], [-6, -9, 0, -3, -1, 14, 7, -4, 12, 2, 0], [1, -1, -10, 2, -12, 11, -17, -6, 15, -9, 4], [19, -4, -5, -5, 15, 7, 23, -3, 1, 18, -12], [4, -13, 18, 29, -5, -1, -15, -12, 7, -9, 11], [4, -7, 6, 13, -5, 4, -17, -5, 5, -7, 14], [25, 17, -12, -14, -6, -3, -11, -12, 3, -14, 7], [-25, 7, -28, 0, -11, 7, -10, 3, 3, 3, 8], [4, 3, 4, 2, 10, 5, 3, 1, 12, 2, 5], [-8, -11, -15, 9, -9, 24, -10, 3, -4, 8, 25], [-2, -12, -5, -15, -1, -7, 5, -12, 4, -1, -9], [-8, -14, 0, 10, 3, 0, -6, 1, 3, -5, 0], [-17, 4, -32, -18, -9, 16, 0, 6, -13, 4, -3], [2, -15, -4, -7, -11, -1, -5, -10, 9, -11, 11], [13, 8, -14, -13, -6, 9, 8, -8, 1, 0, 11], [-3, -23, 12, 9, 5, -16, -5, -18, -9, -6, -9], [-29, 13, -30, 1, -3, 22, -18, 4, -11, -11, 7], [11, 6, -9, -20, -13, 14, -7, 2, 2, 1, 2], [-2, 0, 5, 10, -8, 7, -14, -4, 4, -5, 18], [-18, 10, -10, -12, 9, -5, 8, 5, -12, -7, -14], [-2, -7, 5, -19, 6, -15, 0, -9, -4, -3, -2], [0, 1, -16, -7, -9, -18, -1, 1, 10, -5, -2], [-1, 24, -34, -17, 4, 15, -6, 9, -5, 15, 3], [-7, 1, 8, 7, 18, -30, 10, -8, -8, -22, -22], [-5, 7, -17, -8, 7, 10, -13, 5, -11, 4, 11], [-25, 10, 9, 24, 12, -19, -7, -2, -5, -34, -13], [15, -1, 8, 3, 3, 17, 3, 2, 1, 16, -1], [-13, 9, -10, 7, -2, 8, -18, -1, -14, -1, 14], [2, -4, 6, 17, -18, 11, -16, 4, -7, 0, 2], [3, -10, 20, 4, -17, 10, -9, 1, -1, -2, 3], [20, -21, 25, 4, 2, -15, 15, 0, 11, 6, -5], [9, 1, 5, -10, 5, -1, -6, -3, -19, -1, 1], [-31, -5, -11, -6, 11, -8, 10, 0, -4, 11, -4], [-1, -5, 28, 0, 0, -20, 8, -12, 13, -11, -1], [11, 13, -2, 7, -7, 13, 12, 7, 15, 28, 11], [-21, 23, -20, 11, -4, 22, -22, 11, -12, -4, 11], [-17, 7, 1, 10, 0, -11, 1, -2, 5, -9, -9], [-13, -8, 8, -7, 2, 6, 6, -6, 13, -6, -11], [1, -3, -19, 13, -9, 21, -15, 6, 5, 2, 16], [29, -7, 2, -9, 3, 11, 4, 12, -18, 20, 2], [0, 6, 2, -5, 0, -13, -4, -7, 0, -15, -16], [14, 6, 0, 5, -10, 13, 1, -2, 1, 21, 6], [-3, 10, 3, 21, 14, -10, -1, -1, -11, -13, -3], [-7, -2, -13, -4, -8, 18, -15, -1, -11, 2, 0], [-18, 9, 16, -3, 3, -22, 10, -13, 23, -9, -3], [1, -16, 36, 10, -11, -1, -2, -3, 0, -9, -9], [17, 5, -5, -3, 9, -29, 15, 5, 9, -11, -16], [23, -11, 13, 3, -5, -23, 20, -9, 14, -6, -12], [6, -15, 0, 3, -14, -1, 9, -2, 11, -1, -7], [-38, 30, -24, -17, 1, 6, -5, 5, 1, -4, -12], [14, -9, 9, -10, 6, 5, 13, -5, 9, 5, -7], [24, -17, 7, 17, 9, 7, 6, -3, -5, -3, 8], [-20, 12, -4, -7, 10, -2, 8, 7, -3, -15, -18], [10, -9, 5, 15, 11, 10, -3, 4, -14, -2, 13], [-12, 9, 5, 1, -2, -16, 8, -2, 6, -6, -8], [0, 4, 12, 9, -7, 0, -6, -1, 16, 10, 15], [9, 6, -27, -20, -8, 17, 8, 8, -11, 0, 0], [-8, -3, 4, -3, 5, -5, 9, -10, 14, 4, -10], [-11, -1, 11, -1, 8, -4, 14, 9, 1, -2, -12], [-4, 19, -29, -13, 19, 9, -4, 14, -18, 11, 3], [9, 3, -23, -18, 6, 5, 10, 14, -3, 8, 1], [2, -8, -19, -29, -1, 11, 12, 4, -8, 9, -6], [0, -4, 2, 11, -3, 17, -6, 0, -8, 2, 6], [27, -5, 27, 6, 0, -3, -5, -8, -20, -15, -1], [-21, 7, -18, -11, 1, 9, 10, 16, -2, 22, 2], [13, -16, -5, -8, 4, -16, 15, -10, 23, 6, -11], [19, -9, 11, 9, 4, -2, -5, -8, -12, -9, -9], [9, 0, -6, -3, 15, -10, -2, -7, 7, -15, -7], [20, 7, -11, -4, 6, 10, 10, 3, 1, 2, -5], [-32, 25, -18, -19, 12, 4, 7, -7, 2, 0, -3], [11, 7, 11, 1, -4, 13, -1, -11, 4, 6, 19], [12, 3, -14, -14, -11, 11, 0, 3, 6, 13, 15], [16, -9, 2, -10, -23, 4, -14, 11, -6, 0, 2], [16, 6, -4, 8, -2, 13, 7, 5, -6, 26, 0], [9, 3, 14, -8, -10, 6, -2, 11, 1, 0, 3], [11, 2, -7, -3, -12, 9, -7, -3, 4, -12, -2], [6, -17, 4, -1, -4, 7, -15, 6, -9, -9, -1], [-4, 2, -9, -5, -1, 16, -8, 0, 0, 14, 5], [-8, -6, 10, -13, 8, -15, 6, -15, 6, -30, -9], [-26, -2, 4, 27, 6, -18, 11, -3, 26, -14, -2], [-14, -6, 5, -4, 4, -14, 8, 4, -6, -12, -9], [-37, 2, 7, 6, 8, -6, -6, 4, -18, -16, -6], [8, -7, 6, 10, 12, -5, 5, 9, -1, -4, 3], [-6, 8, -3, 4, -14, 37, -20, 10, -2, 18, 27], [-16, -8, -21, 6, 2, -15, 14, 2, 0, 7, -6], [6, 19, -13, -2, -12, 1, 4, 7, 10, -1, 2], [-15, 10, -41, 0, -4, 18, -13, 4, 1, 9, 16], [14, 11, -12, -19, -3, -13, 6, 5, 7, 13, -5], [24, 7, 22, 0, 7, -16, 6, 2, 5, 3, -6], [-13, 10, 8, 3, 13, -3, -1, 6, -8, -2, 2], [-4, 19, 4, 17, -12, 2, 1, 8, 7, -4, -6], [1, -25, 36, 11, -11, -7, -3, -2, 12, 3, 4], [18, 4, -18, 7, 9, 2, -10, 10, -14, -1, 5], [29, -5, 5, 17, 0, -13, 17, 6, 3, 6, 6], [2, 7, 5, 9, 21, -20, 12, -6, -6, -9, -3], [-35, 5, 2, 18, -14, 0, -20, -2, 6, -4, 14], [-11, -11, 1, 0, 15, -38, 18, -13, 8, -11, -23], [34, -4, 18, 11, 10, 1, -9, -2, 0, -5, 4], [22, -18, -13, -16, 1, -4, 13, 9, -2, 10, -1], [-31, -17, -8, 11, 4, 3, -12, 3, -7, -7, 12], [-28, -2, 0, 17, -11, 4, -16, -6, -5, -3, -3], [18, -12, -5, 24, -6, 15, -8, -1, 2, 14, 10], [3, -5, -9, -3, -13, 19, -22, 10, -5, -5, 4], [26, 2, 12, 8, -17, 22, -24, 1, -8, -9, 26], [0, 19, -1, 1, -16, -6, -7, 3, 9, -9, 1], [2, 7, -10, 5, -7, 21, -9, 5, -17, -3, 10], [2, -7, 5, -1, -19, 28, -10, 6, -1, 6, 18], [14, -8, -4, -5, -7, -1, -9, 7, 2, 2, 4], [26, -2, 1, -17, -13, -7, 6, 1, 0, 12, 6], [2, 27, -14, -7, 1, -12, 6, 7, -9, 10, -2], [19, -1, 16, 16, 6, 1, -12, 7, -11, 3, -8], [4, 3, 3, 25, 6, 24, -4, 3, 1, -4, 13], [23, -29, 13, 1, 9, -15, 11, -8, -10, 12, 0], [-34, -2, 10, 13, 2, -9, -10, 6, -4, -7, -13], [11, 0, 14, 4, -13, -1, -7, 8, -9, -10, -4], [-4, -24, 19, 20, -16, -2, -17, 4, 2, -4, 10], [-6, -14, 12, 14, -6, -3, -25, -6, 2, -16, -3], [3, 5, 30, -2, -6, -21, 10, -14, 15, -16, -8], [-5, -14, -5, 34, 17, 2, -16, -3, -21, -7, 14], [-20, -1, 10, 4, 0, -30, 16, -19, 18, -21, -15], [-15, -7, 6, 9, -8, -3, -23, 0, 3, -28, -5], [3, -2, -15, 4, 11, -3, -15, -2, -16, 1, -2], [-5, 3, 2, -14, -9, -13, 20, 0, 7, 15, -13], [3, 9, -22, -12, 11, 11, 19, 18, -24, 36, -5], [-22, 25, -22, -23, 6, -6, 7, 4, -5, 10, -14], [17, -4, 13, 3, -7, -10, 12, 8, 3, 1, 9], [13, 19, -1, -4, 9, -19, 16, -3, -5, -8, -16], [10, -6, 0, -32, -8, 10, -2, 2, -4, 3, -3], [13, -2, -13, -12, -14, 22, 10, 8, 11, 12, 0], [-6, 8, -2, 4, 3, -7, -7, -7, 1, -9, -3], [-34, 39, -30, -23, 3, -2, 9, 7, 7, 4, -17], [-10, -9, 1, -2, -5, 11, 3, 11, -3, 10, -4], [10, 3, -27, -1, 1, 0, 10, 12, -1, 18, 10], [10, -24, 16, 0, -21, 14, -16, 0, 4, -4, 18], [25, -11, 9, -6, -9, 10, -4, 12, -5, 13, -4], [-23, 1, 0, -10, -4, -1, 8, 17, 1, 12, -7], [-3, -10, -5, -3, 13, -9, -3, 0, -24, -17, -16], [-7, -1, 17, 3, -20, 11, -17, -10, -9, -15, 9], [-15, 4, -1, -16, -11, 14, -5, 8, -2, -1, 2], [-16, 17, -17, -2, 20, -11, -3, -2, -23, -12, -9], [14, 9, -15, 3, -2, 13, -3, 14, -19, 1, 8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6025_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6025_2_a_g();".
// 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_6025_2_a_g(:prec:=11)
    chi := MakeCharacter_6025_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_6025_2_a_g();".
// 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_6025_2_a_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6025_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![19, 11, -120, -31, 185, 15, -121, 7, 36, -6, -4, 1]>,<3,R![-4, 65, 72, -298, -69, 371, -58, -148, 56, 11, -8, 1]>],Snew);
    return Vf;
end function;