
// Make newform 6025.2.a.h in Magma, downloaded from the LMFDB on 19 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_h();".
// 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_h();".
// 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, 18, -45, -328, 105, 444, -123, -219, 65, 44, -14, -3, 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], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-8, -213, 94, 768, -94, -611, 54, 184, -13, -23, 1, 1], [7, 34, 168, -422, -663, 538, 488, -273, -119, 57, 9, -4], [85, -745, -2230, 2774, 2625, -2491, -1046, 869, 172, -130, -10, 7], [-169, 951, 4634, -470, -6481, 597, 3330, -569, -690, 160, 48, -13], [141, -1335, -5698, 1798, 7333, -1629, -3426, 829, 662, -180, -44, 13], [93, -741, -2750, 1654, 4025, -1631, -2086, 773, 432, -158, -30, 11], [-109, 737, 3354, -1178, -4717, 1223, 2398, -669, -492, 150, 34, -11], [201, -1643, -6602, 2718, 9809, -2777, -5178, 1513, 1086, -340, -76, 25], [381, -2497, -10758, 5014, 14249, -4819, -6822, 2237, 1340, -450, -90, 31]];
    Rf_basisdens := [1, 1, 1, 8, 8, 16, 16, 16, 8, 8, 16, 16];
    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 := [[0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 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, 0], [2, -1, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 1, 0, 0, -1, 0, -1, 0, 0, -1, 1], [1, 0, 0, 1, -1, 1, 0, 0, -1, 0, 0, 1], [-1, 1, 0, 0, -1, 0, -1, -1, 0, -1, -1, 0], [-5, 1, -1, 3, -1, -2, -1, 0, 0, -1, -3, 1], [-1, -1, 0, 1, 0, 0, -1, 0, 1, 0, -2, -1], [0, 1, -1, 0, -3, 0, 1, 0, 0, -1, -1, 0], [1, 0, -2, 0, -1, 1, -1, 1, -1, -1, -1, 0], [-2, 1, 1, 0, -2, -1, -1, -1, 0, 2, 0, 1], [0, -1, 0, -1, 0, 0, 1, 2, 1, 1, 0, 0], [-2, 0, -1, 0, -1, 1, 0, 0, -1, -2, 1, -2], [-1, 1, 0, 0, -1, -1, 0, 1, 0, -1, 0, 0], [4, 2, 0, -3, 0, 3, 0, 0, -1, 0, 3, -2], [-1, 0, 0, 1, 0, 0, 1, -1, -2, -2, 1, -1], [-3, 0, 1, 2, -1, -3, -2, -1, 1, 2, 0, 2], [9, -2, 1, 1, 2, -1, 1, -1, -2, 1, 0, 2], [1, 0, -2, 1, 0, 0, 1, 0, 0, -1, 0, -3], [-2, 0, -1, 0, 0, -2, 0, 2, 0, 0, -4, 0], [1, -2, 3, -2, 2, -1, 1, -2, 2, 3, -1, 3], [-1, 0, -2, 2, -1, -1, 1, 3, 0, -3, -3, 0], [4, 1, -2, 1, -1, 1, 0, 1, -3, -3, 1, -1], [0, 4, -1, -1, -3, -1, -1, -3, 0, 1, 0, 1], [0, -3, 3, -1, 4, -2, 2, 0, 2, 0, 0, -2], [-2, 0, 2, -1, -1, -1, 0, -3, 1, 2, -1, 3], [-2, 2, 0, 2, -2, 1, -2, -1, 1, 2, -1, 1], [-1, 2, 1, -5, 2, 2, 3, -1, 2, 2, 3, -3], [0, -3, -2, 1, 2, 0, 1, 0, -1, -1, -1, 0], [1, 0, -2, 1, 0, -4, 1, -2, 2, 1, -1, 3], [-2, -4, 0, 3, -2, 0, -2, 1, 1, 0, -2, -1], [1, 0, 2, -2, -2, 3, -2, 0, 4, 0, -1, 1], [2, 2, 0, 0, -1, -2, 2, -1, 1, 0, -4, 3], [-2, -4, 0, -1, 4, -2, 1, 3, 2, 3, 3, -3], [3, 2, 4, -3, 2, 0, 2, -6, 1, 4, 5, 1], [3, 0, 0, 0, -2, 4, 0, -1, -2, 0, 4, -2], [-2, 1, 0, 0, 1, -2, 1, 0, 0, -1, 2, 2], [-3, 0, 0, 2, 0, -4, -1, 1, 1, 3, -2, 0], [2, 2, 5, -4, 3, -2, 3, -3, 6, 1, 0, -1], [2, -4, 2, -3, 1, 4, 1, 0, -3, -1, 3, 0], [4, 4, -3, 2, -4, 1, -1, 1, -3, -1, -2, 3], [1, 0, -2, 3, -2, -1, 0, -2, 1, -5, -5, 2], [2, 2, -2, 2, -5, 1, -4, 2, 0, -4, -5, 0], [-2, -3, 1, 2, 5, -7, 1, -1, 2, 5, -2, 5], [2, -4, -4, 0, 0, 1, 3, 3, 0, -1, -2, -1], [2, 4, 0, -2, 1, 0, 4, 3, -1, 0, 0, -1], [1, 2, 3, -5, 0, -1, -1, -1, 6, -1, 0, -2], [-3, -3, -1, 1, 7, -9, 2, -2, 1, 3, 2, 1], [-2, -2, 2, -2, 0, 1, -3, -1, 6, 3, -2, 1], [10, -2, 0, 2, -3, 1, -2, 0, 4, 0, 1, 2], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-5, -2, -1, 7, -2, -7, -3, -1, 0, -3, -3, 4], [-2, 6, -1, -4, -2, 0, 1, -2, 2, 1, -1, -2], [4, -4, -2, -2, -1, 4, 3, -3, -2, 3, 6, -1], [-4, 6, -3, 2, -5, 2, -3, -1, -1, -3, -3, -3], [7, -4, 4, -8, 3, 3, 3, -3, 1, 5, 7, 0], [0, -4, 3, -3, 1, -1, 1, 1, 1, 1, 2, -3], [2, -4, -1, -3, 4, 1, 2, 3, -2, -3, 0, -2], [-4, 3, -6, 2, -4, 3, -2, 4, -1, -4, -4, -2], [-1, 2, -1, 7, -2, -3, -7, 1, -3, -3, -3, 2], [4, 1, 0, -1, 3, 3, -3, 2, 0, -3, 0, -2], [7, 0, -5, 4, -3, -3, -1, 5, -3, -3, -6, 0], [1, 1, -1, 1, -3, 4, 0, -1, 2, -5, 1, 0], [6, -6, 3, 1, 4, -4, 5, -3, 1, 3, 5, -1], [-4, 0, 1, 5, -1, 3, 0, -3, -1, 0, 1, -1], [1, -2, -1, 4, 0, 1, -1, 3, -2, 0, -2, 3], [-1, -2, 0, 5, -2, -1, -4, -1, -4, -2, -2, -2], [-8, -3, -2, 5, -1, -8, -5, 0, 4, 4, -5, 6], [-8, 2, -4, 4, -1, -6, -1, -1, 2, -5, -5, -1], [12, 2, 2, -3, 2, 2, 3, -1, 1, -3, 3, -3], [-7, 0, 0, 5, 0, -4, 0, 0, -5, -2, -4, 5], [8, -4, 7, -5, 1, 2, -1, 0, 4, 1, 3, 0], [-1, 2, -1, 5, -2, 1, -2, 1, 5, 0, -3, -2], [2, 7, 1, -3, -1, -1, 3, -8, -3, 5, 6, 6], [-5, 4, -1, 7, -1, -4, -2, 1, -1, -2, -3, -2], [-5, 4, -6, -2, -5, 4, 0, 0, -4, 0, 5, 1], [-6, 0, 0, 2, -3, -5, -1, -3, 5, 2, -3, 4], [-6, 4, -3, 2, -3, 3, -3, 2, 0, 1, -4, -2], [7, 1, 1, -5, 5, 6, 0, 0, 4, 2, 6, -1], [7, -3, -1, 6, -1, -2, -2, 2, 0, -1, -5, 3], [21, -3, 2, -2, 2, 4, 1, 0, -4, 5, 7, 1], [1, 2, 1, 2, 0, -12, 2, -5, 2, 0, -3, 2], [-10, 7, -1, 1, -5, 1, -9, 2, 3, -1, -7, 2], [4, 5, -6, 3, 1, 0, -4, 1, 0, -4, 0, -1], [11, -8, 1, -6, 0, 6, 4, 5, 0, 2, 9, -4], [-3, 7, -1, 6, -5, 1, -7, -1, -2, -4, -5, 4], [11, -4, 2, -7, 2, 9, 4, 5, 4, -2, 6, -8], [-11, 3, 3, -3, 1, -1, -4, -3, 8, 8, 3, 2], [5, 0, -4, 4, -7, 8, -4, 0, -1, -4, -1, -1], [3, 4, 4, -1, 3, -3, 0, -6, 6, 0, -4, 3], [-2, -4, -3, -5, 1, -4, 2, 6, 2, 0, -2, -2], [-15, -6, 2, 6, 2, -7, -3, -2, 3, 7, -3, 3], [-5, -1, -3, 11, -8, -6, -7, -3, -3, 5, -7, 10], [7, 7, -2, 1, 3, 2, 2, 2, -1, -4, 1, -1], [4, 4, 5, 0, -8, 3, -4, -4, 2, -1, 0, 5], [4, 0, -9, 0, -2, 6, 0, 6, 1, -2, -2, 0], [-1, 2, -1, 4, -6, -1, -8, 0, 4, 4, -2, 3], [0, -2, 7, 0, 1, 3, -1, 0, 1, -1, -4, -2], [6, 5, 4, -6, -1, 3, -4, 1, 2, -4, 1, -5], [0, -1, 5, -6, 3, 1, 0, 0, 2, 5, -1, 4], [1, 2, -2, 6, -2, -7, 2, -2, 3, -4, -8, 3], [0, 4, 3, -3, 2, -3, 0, 1, 2, 1, -6, 4], [-1, 5, 0, -1, 2, 1, 4, 0, 7, -4, 0, -1], [8, -4, 2, -3, 3, 3, 1, -3, -6, 5, 6, -1], [-9, 13, -4, -4, -5, 1, 1, -2, -3, -7, -1, -9], [4, 6, -3, -5, -2, 0, 4, 3, -2, -6, 6, -1], [7, 7, 4, -1, -5, -2, 5, -6, -1, -3, 0, 1], [-10, -2, 8, 2, 2, -2, -3, 0, 8, 1, -5, 2], [-2, 6, 6, 0, -3, 2, -5, -7, 3, 3, -1, 5], [-8, 8, 5, 0, 4, -4, 1, 0, 0, 1, -5, 0], [-3, -1, 12, -1, 6, -7, 3, -7, 8, 10, 5, 8], [-7, 11, 0, 1, -4, -5, 1, -4, -4, -5, -3, -3], [1, 11, -7, 1, 0, 2, 0, 3, -1, -6, 6, -8], [1, 4, -9, 3, -4, -1, -3, 3, 6, -1, -5, -2], [2, -8, 2, 7, -4, -3, -4, -4, -2, 0, -5, 2], [-7, 7, -5, -5, 4, 4, 7, 5, 6, -5, 3, -6], [12, 4, 6, -4, 2, 4, -2, -6, 3, 4, 12, -4], [-8, -2, 5, 2, -2, 7, -7, -1, 0, -3, -2, -3], [7, 0, 2, -10, 6, 9, -4, 4, 1, -2, 8, -7], [1, -6, -4, 5, 1, 0, -1, 4, -5, -2, 1, 0], [-3, -10, -3, 2, 0, 2, -6, 5, -1, -4, -1, 0], [1, -6, 1, 2, 0, -3, -4, 0, -5, 0, 0, 5], [17, 2, 1, -11, 2, 9, 5, 3, -12, 5, 15, -4], [6, -4, 2, -5, 3, 4, 1, 4, 4, 3, 7, -8], [7, -6, 1, -8, -4, 10, 0, 3, -5, 0, 7, -6], [0, 0, 2, 0, 10, -12, 7, -4, 2, 9, 5, 2], [-2, -2, -4, 2, -3, -3, 3, -2, -2, 7, 8, 0], [1, 6, 8, -4, -3, 7, -2, -3, 1, -4, 0, -6], [-12, -2, -7, 3, -4, -2, -2, 1, 5, -4, -4, -9], [2, 4, -4, 2, 6, 5, 1, 7, 0, -3, -2, -5], [11, 1, 5, -12, 8, 12, 3, 2, 1, 1, 11, -3], [2, 2, -6, 3, -6, -1, -1, 2, 3, -7, -10, 4], [-3, 4, -4, 5, -6, -1, -7, 3, -1, -11, -3, -4], [-11, 2, 3, 3, 1, -5, -1, -8, 0, 5, 5, 7], [16, 0, 2, -13, 3, 12, -1, 1, 0, 2, 8, -6], [-1, -4, -1, 8, 8, -9, -1, 2, -2, -3, 1, -1], [-7, 2, 10, -2, 9, -6, -1, -4, -2, 7, 4, 3], [4, -4, 8, -9, 7, 1, 2, 3, 4, 4, 11, -5], [-8, -5, 8, -3, 3, -4, 1, -3, 8, 1, 0, -5], [10, -4, -3, 6, -12, 5, -6, 0, -4, -1, 0, 5], [7, -6, 5, -7, 4, 1, -3, 7, 6, 5, 5, 0], [-1, 3, 3, -2, -2, 4, 2, 6, 1, -6, -4, -5], [-8, 8, -4, 8, -9, 1, -2, -5, -3, -9, -4, -2], [12, 8, -3, 8, -7, 10, -7, 1, -5, -5, -3, 3], [12, -10, 2, -3, 14, 1, 2, 2, 3, 2, 5, -6], [-3, -8, 5, 4, -5, -4, -3, 3, 1, 0, -7, 3], [-12, -8, -6, 15, -7, 1, -9, 3, -4, -5, -7, 3], [-3, 0, -3, -3, -1, 1, 0, 4, 4, -10, -7, -5], [-16, -2, -5, 6, -7, 3, 1, 3, 5, 0, -3, -4], [-9, 14, -3, -5, -8, 6, 1, -6, 6, -1, -8, 3], [7, 6, 0, 2, -10, -6, 0, -7, 5, 2, -5, 6], [9, -4, -1, 8, -3, 1, -5, -1, -6, 3, 1, 6], [8, 0, -2, 2, -9, 5, -1, 0, -3, -11, -3, -4], [21, 0, -2, -2, -2, 2, 2, 5, 6, -4, -3, 0], [-7, -10, 3, -3, 1, 2, 9, -5, -1, 7, 6, 0], [-7, 2, 8, 9, -2, -7, -11, -5, -1, 5, -9, 4], [-10, -8, -5, 14, 6, -12, 1, 9, 0, 2, -6, 8], [19, 0, -3, -4, -1, 9, 5, 1, -4, -11, 5, -4], [-3, 3, 10, -11, 13, -9, 10, -7, 6, 14, 8, 2], [15, 4, -1, 6, -8, -2, 0, -3, 5, 2, -9, 4], [6, -2, 1, 8, 0, 1, -1, -5, 3, -5, -2, -1], [-1, -9, 4, -3, 4, -4, 3, -7, 5, 5, 1, -4], [0, -2, -5, -1, -1, -1, 5, 1, -8, -1, 0, 3], [8, -4, -1, 1, -8, 2, -1, 5, 3, -5, -4, -5], [9, -2, 5, -6, 0, -5, 2, -14, 10, 6, 4, 7], [1, 4, -1, -6, -5, -2, 1, -4, 2, 11, 0, 9], [-7, -7, 1, 2, -2, -11, 4, 0, -5, 5, 0, 7], [19, -8, -5, -2, 0, 5, 6, -4, -2, 4, 8, 1], [5, 3, -4, -1, 9, 8, -1, 8, 8, -1, 6, -9], [-10, 6, 0, 0, -1, -12, 0, -9, 7, 8, -4, 3], [9, 0, 5, -7, 4, -5, 1, -9, 11, 11, 5, -4], [5, -6, -1, -1, -5, 4, -1, -7, -1, 3, 8, -4], [6, 4, 3, -1, 6, 0, 0, -9, 6, 4, 6, 3], [6, -6, 3, 3, 0, -6, 5, -9, -9, 9, 5, 7], [1, 6, -6, 1, 2, -6, 12, 4, -3, 2, -5, -1], [0, 4, -7, 7, -14, 15, -3, 1, -9, -10, -5, -2], [8, -3, -6, -1, -5, 9, 1, 4, 1, -1, -1, 4], [19, -6, -5, 1, -9, -1, 0, -2, -8, -2, -2, 3], [8, -2, 12, -7, 14, -9, 8, -14, 12, 12, 5, 6], [-1, 0, -8, 5, 2, -4, 0, 8, -2, -10, -4, -7], [14, -6, -7, 3, 4, 0, 1, 5, -12, 5, 5, -1], [-17, 6, 9, 0, 0, -2, 1, -5, 9, -3, -8, 0], [-6, 7, -1, -10, 5, 6, 4, -2, -6, 0, 6, -6], [-18, 16, 8, -2, -3, 3, -4, -4, 0, 0, -3, 2], [0, -6, 11, -13, 12, 4, 11, -5, 10, 3, 9, -5], [-15, 8, 1, -1, -1, 5, -5, 0, 7, 5, 3, -9], [1, -16, -3, 7, 8, -3, 0, 5, -4, -6, 6, -6], [20, 4, -18, -3, -5, 13, 1, 3, -9, -5, 6, -13], [6, 4, -1, -14, 5, 8, -1, 9, 6, -7, 5, -7], [-7, -4, -6, 0, 12, -6, -5, 7, 5, 5, 1, -8], [3, 1, 0, 1, 0, -4, -2, 1, -8, -4, 1, 6], [-5, -7, 2, -1, -5, -1, -2, 0, 5, 7, -6, 3], [17, 8, 10, -15, 9, 9, 8, -6, -3, 2, 12, 1], [10, -5, 1, 11, 2, 1, 0, 2, -13, -3, 3, -2], [3, 0, 1, 7, -11, -1, 0, 1, -1, -11, -7, 7], [-3, -8, 11, -2, 9, 2, -1, -9, 9, 12, 7, 1], [2, -9, 3, 0, 3, 5, 5, -2, 6, 0, 2, 2], [2, 0, -6, 2, 8, 3, -5, -1, -2, -1, 0, -3], [1, 6, -12, -3, -3, 12, 5, 7, -3, -15, -4, -8], [-12, 14, -4, 9, -14, 3, -18, 6, 2, -4, -12, 2], [-5, 3, 3, -2, 0, 1, 3, -2, 10, 0, 1, -5], [10, 0, -9, 6, 4, 3, 1, 7, -4, -3, -4, -5], [0, -4, 7, -8, -2, 4, 2, -2, 4, 0, 8, 2], [5, -5, 4, 11, -7, 1, -3, -4, -5, -2, -7, 7], [13, 1, 2, 0, -1, 0, -5, 1, -11, 3, 5, 2], [1, -3, 6, 2, 3, 1, -1, -6, 10, -1, -1, -1], [-9, -2, 0, 8, -6, 3, -1, 4, -12, -9, -9, 1], [13, -6, 1, 0, 8, 16, -7, 7, -5, -11, -5, -6], [-11, -2, 5, -6, 3, 4, -2, -6, -1, 14, 3, -3], [-5, -7, -1, -3, -5, 9, 0, -2, -4, 9, 12, -3], [-1, 2, 2, 5, -14, -4, 2, -6, 2, -10, -15, 9], [1, -5, -2, -2, 6, -9, 6, 1, 6, -2, 5, 0], [-1, -3, 3, -1, -7, 10, -4, 4, 7, 6, 2, -7], [-15, -4, -5, 7, 5, -12, -4, 9, 5, -8, -7, 2], [8, -18, 7, 3, 11, 1, -3, 7, -6, 5, 0, 1], [0, -8, 7, -1, 2, -7, -3, -8, 0, 9, -6, 14], [-9, 2, -2, 4, -7, -4, 2, -2, 1, 12, 1, 15], [5, 4, 3, -1, -10, -2, -3, -6, -1, 1, -6, 7], [9, 6, -1, 5, -7, 9, -5, -4, -11, -15, 3, -3], [14, -2, 6, -10, 1, 12, 0, -7, -8, 8, 16, 5], [0, 12, -11, -7, 2, 4, 6, 1, 4, -2, 2, -11], [9, 7, 4, -9, 4, 10, 4, 1, 1, -6, 15, -10], [-8, -4, -7, -8, -2, 3, 13, 9, 4, -3, 9, -13], [10, -6, 7, 5, 6, -8, -5, 5, 0, -5, -3, 5], [12, -12, 3, -4, 10, -3, -3, -1, 3, 9, 2, 1], [26, -1, -16, 2, -3, 2, 2, 10, -7, -14, -3, -8], [13, 6, -10, 3, 6, 5, -1, 3, -5, -1, 3, -8], [21, 6, 6, -5, 1, -6, -2, -3, 6, -2, -11, 6], [-8, 14, -6, 7, 5, -4, -2, -2, 3, -4, -6, -4], [0, 3, -2, -16, 3, 19, 13, 1, -6, 3, 16, -5], [-31, 6, -1, 11, -1, -12, -11, 1, 9, -1, -10, 2], [10, -8, -14, -2, -3, 14, 5, 15, 0, -11, 3, -9], [6, 0, 8, -13, -7, 8, -5, -2, 4, 9, 12, 4], [0, 0, 0, -6, -4, 4, 6, 4, 6, 8, 0, -4], [-6, -10, -4, 4, -3, -6, -3, 11, 1, 1, -3, 1], [1, -6, -5, 4, 3, -1, 7, 11, -2, 7, 7, -2], [0, 14, 6, -8, -7, 1, -1, -6, 6, 7, 8, -2], [-21, -1, -2, 12, -8, -8, -5, 0, 2, -5, -7, 3], [24, 4, -8, -2, -3, 11, 3, 0, -7, -9, 0, 0], [0, 1, -9, 1, -7, -3, -3, 0, 8, 3, -12, 2], [10, -4, -6, 12, -6, -5, -2, -5, -3, -6, -5, 1], [4, -18, 13, 2, 5, 3, 2, -4, 2, 12, 5, 2], [-16, 3, 9, 11, 2, -24, -2, -15, 7, 17, -10, 19], [15, -4, -1, 10, -4, -1, -9, 4, -9, -3, -3, 7], [-2, 14, -9, -6, 4, -7, 7, -5, 8, 1, 2, -5], [-7, 6, 3, 3, -15, -5, -6, -6, 1, 16, -6, 17], [10, -18, 4, -2, -4, 4, -12, 2, 8, 12, 0, 6], [-3, 4, -6, 1, -2, -5, 5, -7, 0, -5, 8, -6], [7, 4, -3, -7, 1, 4, 7, 2, 5, 6, 7, 4], [2, 4, 6, -13, 6, -8, 6, -9, 8, 14, 6, 9], [4, -2, 1, 1, -10, 6, 3, -5, -4, -1, 7, 5], [-3, 8, 6, -3, 5, 0, 2, -5, -1, -4, -5, 2], [-1, -2, -2, 13, -4, 1, -10, -11, -2, 0, -14, 12], [-8, 3, -10, 12, 1, -19, -5, 1, 3, -9, -13, -5], [-5, 6, 1, -7, 6, 3, 8, 7, -1, -6, 4, -6], [11, -12, 8, -8, 4, 3, 2, 2, 11, 10, 6, 5], [-12, 13, 0, -2, -9, 3, -13, -6, 1, 0, -2, -4], [4, -9, 4, 12, 7, -11, 1, -6, -8, 0, 2, 14], [-15, -4, -9, 16, -16, 4, -14, 7, -4, -10, -13, -6], [15, -8, 2, 4, -8, 16, -10, 5, -4, -8, -9, 2], [-5, 4, -3, 3, -5, -12, -1, -4, 9, -8, -7, 8], [-8, 2, -8, 3, -13, -6, -9, 8, 1, -13, -19, 2], [-24, -8, -1, -2, -2, 2, -1, 8, 10, -3, -3, -10], [10, -4, -2, 1, 10, -13, 5, 2, 9, 1, 2, 4], [2, 6, 8, -10, 14, 5, 12, -5, 4, 8, 17, -5], [-25, -6, 6, -4, 15, -9, 5, 1, 15, 19, -3, 2], [4, 0, -2, -1, -1, -16, 0, -2, 6, 4, 2, 6], [14, 8, -13, 8, -16, -2, -5, 2, -7, -3, -5, 2], [15, 8, -1, 2, 7, -5, -4, 2, 3, 7, 1, 6], [7, -2, 6, 2, 0, -2, 1, 1, -6, -3, 2, 2], [-3, 4, -1, 5, -5, 2, 4, 7, -4, -4, -11, 2], [-3, -12, 6, 22, -5, -3, -7, 1, -2, 5, -15, 6], [-12, 6, -16, 8, -12, 1, -11, 2, -6, -8, -11, -1], [3, 8, 3, 5, 5, -1, -5, -6, -1, 7, 7, -1], [14, -16, 2, -5, 14, 3, 1, -2, -6, 11, 14, -4], [-6, -2, 7, -7, 12, -7, 14, -6, -11, 0, 9, 6], [2, 0, -6, 0, -11, 7, -5, -10, -2, -1, 2, 4], [1, -3, -2, 18, -6, 2, -14, -4, -14, 2, -7, 13], [20, -7, 3, -3, 12, 4, 3, -2, 11, -7, 4, -6], [11, -8, 6, -13, 5, 12, 7, -9, -7, 3, 10, 0], [-4, -6, -13, 14, 3, -5, 5, 16, 5, -1, -8, 0], [20, 8, -3, -2, -2, 3, 6, 1, -4, 8, 1, 13], [-1, 6, 5, -2, -3, 13, -8, -7, -3, -8, -4, 4], [8, 1, 7, -2, 7, -7, 4, -3, 10, 4, -2, 9], [-15, -4, -3, -2, 1, 4, 7, -6, -3, 7, 0, 5], [16, 0, -3, -7, -6, 15, -3, -2, -9, 11, 12, 2], [-14, 4, 3, 20, -6, -11, -13, -1, -3, -1, -18, 7], [-20, 0, 2, -11, -3, 1, 0, -11, -2, 8, 1, 15], [28, -7, -3, -4, 10, -3, 3, 3, -7, -2, 4, -7], [-13, 15, 4, -10, -4, 2, 6, -2, 20, -2, -1, -7], [24, -16, 0, -4, 5, 13, -8, 15, 3, -9, -4, -4], [3, -8, 1, 2, -3, 8, 2, -2, -6, 10, 13, -3], [-10, -7, 4, -1, 4, -6, 10, -11, 2, -1, 8, 5], [-16, -14, 3, 10, -10, 4, -6, -4, -16, -4, -2, 8], [11, -6, 8, 6, -1, 3, 0, -1, -5, 14, 9, -2], [32, -1, -7, 0, -7, 27, -8, 10, -5, -19, 5, -16], [-2, -13, 6, -8, 3, 12, -2, 0, -2, 0, 8, -6], [17, 20, 5, -19, 0, 13, 6, -7, -5, 0, 12, -4], [5, 14, -9, 8, -14, -7, -3, -6, 0, -7, -13, 7], [4, 6, 1, -18, -2, 11, 9, 1, 4, -3, 10, -9], [14, -5, -1, 4, -7, 9, 1, 7, 5, -11, -10, 3], [-9, 6, -6, -1, -8, -5, -3, -9, 10, 3, -1, 0], [-1, 8, -11, 9, 4, -2, 3, 8, -18, 1, 8, -5], [-9, -4, -3, 0, 8, -8, -5, 17, 8, -3, -2, -6], [6, -20, -7, 4, -8, 11, -13, 9, 0, -13, -3, -3], [19, -2, 0, -13, -6, 15, 2, -1, 0, -4, 10, -8], [44, 1, -3, -20, 5, 18, 4, 0, -6, 6, 19, -8], [3, 2, 8, -10, 8, 5, -4, -2, -6, 4, 16, 1], [18, -2, 13, -9, 7, 11, 6, 5, -6, -4, 9, -7], [-34, 13, 4, 10, -12, -3, -13, -1, 0, 6, -12, 11], [-4, 4, 4, -13, 1, 11, 1, 9, 4, -3, 4, -5], [10, -3, -13, 10, -2, -10, 12, -1, -22, 6, 5, 3], [-22, 10, 8, -5, -2, -5, -7, -6, 26, 1, -10, -4], [10, -7, 0, -11, -1, 13, 14, 11, 2, -5, 9, -7], [1, 1, 3, -13, 15, -1, 17, -3, -10, -1, 18, -8], [25, -10, 0, -3, 6, 5, 7, 3, -7, 5, 1, 12], [-6, 8, 2, 6, 0, 6, -4, 8, -4, -8, -8, -2], [-22, 4, -14, 6, -9, 15, -6, 9, -7, -13, -8, -6], [22, -2, 6, -7, -2, 7, 2, 4, -5, 8, 14, 0], [8, 13, -5, 4, 1, -14, 8, -1, 0, -7, -4, 3], [-21, 12, -3, -5, -4, -1, -4, -7, 10, 6, 2, -10], [-17, 5, 0, -9, -1, 5, 7, -7, 8, 7, 8, -12], [39, -16, 2, -3, 6, -1, 4, 0, -6, 5, 7, 10], [-5, -3, 7, 10, -4, -4, -5, -5, 14, -4, -4, -4], [20, -8, -7, -3, -5, 6, -10, 8, -1, -10, -14, -4], [31, -15, 6, -11, 15, 13, 14, 1, 5, 12, 15, -8], [5, -6, 8, 1, 1, 0, -12, -3, 1, -6, 0, 0], [2, 4, -9, 5, -8, -10, 8, 15, -6, -16, -12, -7], [-13, 2, 0, -3, -9, -9, -5, -8, 13, -15, -19, 7], [-33, 2, -6, 3, 3, -16, -5, -5, 5, 5, 2, 2], [11, -19, -2, 9, -13, 1, -4, -3, -1, -4, -6, 10], [-19, 5, 8, -7, -6, -13, 6, -10, 3, 12, -1, 13], [13, -2, 11, -14, 16, 0, -4, -7, 13, 2, 9, -8], [-1, 18, -3, -2, -4, 0, 4, -15, -14, -4, 7, 14], [-14, -2, -4, -3, 15, -11, 13, 1, 16, 11, 4, -1], [-15, 8, -12, 19, -23, -5, -7, 16, -5, -13, -25, 9], [-22, -2, -1, 0, -10, -7, 2, -7, -1, 12, -9, 9], [-9, -10, 10, 13, 0, -9, 1, -3, 3, 11, -3, 18], [-2, -16, 6, 0, 9, -4, -1, -3, -8, 5, -3, 1], [18, 6, -7, -15, 9, -3, 9, 6, 9, 2, 5, -9], [-11, -12, 10, -3, -1, -1, -3, 0, 12, 15, -3, 9], [6, 6, -19, 9, -11, 9, 0, 3, -2, -12, -2, -3], [-1, 14, -3, -7, -13, -7, 2, -2, 16, -14, -6, -1], [0, -11, 17, -6, 11, -12, -4, -6, 13, 10, 1, 8], [5, -14, -1, 4, 6, 6, -4, 11, -5, -4, 5, -6], [6, 0, 15, -15, 6, -2, 10, -21, 9, 16, 21, 5], [12, -4, -9, -7, -3, 6, 9, 8, 6, -13, -5, -18], [11, -6, 12, -1, 0, 14, 1, -8, -1, 11, 18, 9], [-10, -12, -4, 19, -1, -4, -6, -6, -15, -2, -5, 12], [0, 6, -5, 8, 12, -14, -6, 6, 12, -4, -18, 2], [29, 8, 2, -10, 0, 14, 1, 0, -5, 0, 25, -6], [0, -2, 10, 7, -5, 2, -16, -4, 10, -2, -16, 8], [34, -22, -3, -10, 2, 14, 4, -2, 0, 8, 20, -4], [-8, -10, 14, -1, -2, 1, -5, 2, 14, -3, -20, -2], [13, -6, 3, 0, -6, 12, 9, -7, -18, 5, 9, -2], [0, 2, -2, 1, 2, 8, -4, -5, -13, -4, 0, -1], [2, 8, 2, -6, -7, 10, -1, 3, -7, -9, 1, -7], [-19, 7, -10, -8, 14, -5, -1, 9, 2, 1, 10, -10], [13, 11, 1, 0, -1, 6, 7, -4, -9, 4, 2, 3], [-16, -24, 7, 4, 5, -4, 3, 1, 15, 9, -3, 1], [10, -11, -5, 11, 11, 1, 3, -2, -25, -7, 0, 0], [2, -10, 2, 2, 10, 2, 0, 16, 7, 2, -10, 2], [5, -1, -4, -5, -2, 16, 5, 7, -6, -11, 2, -8], [2, 10, 6, -2, -4, 9, -13, 5, -6, -13, -4, -7], [9, 2, -3, -6, -2, -1, 11, 1, -13, 8, 4, -1], [-6, -7, 5, 14, 10, -21, -2, 2, 4, -8, -13, 8], [-2, 12, 5, -3, 5, -7, 10, -5, 13, 12, 7, 9], [25, -8, -4, 1, -2, 10, -10, 10, -12, -16, -5, -13], [20, 9, -8, 1, 4, 7, 0, 5, 0, -18, 9, -19], [-5, -12, 3, -8, 1, 21, 5, 7, -12, 3, 13, -6], [3, -12, 12, -4, 2, -8, 6, 3, 6, 6, 13, -2], [-33, 4, -6, 10, -8, -6, -6, -3, -4, -8, -7, -4], [-4, 0, -19, 5, 0, 9, 1, 5, -9, 0, -5, -12], [-12, -8, 0, 14, 5, -4, -9, 7, -1, -3, -7, 3], [20, -2, 5, -4, -8, 12, 6, 4, -9, -6, 10, -10], [-2, 6, -6, 8, -2, -6, -1, -5, -6, 22, 4, 12], [1, -2, 17, -9, 7, 5, -5, 4, -9, 1, 9, -13], [-26, 28, -6, 8, -21, -6, -16, -1, 8, -16, -14, 1], [35, -2, 8, -2, 19, 13, 1, -2, -7, 2, 12, -12], [-13, -4, -11, 17, -7, 11, -2, 11, -11, -1, -5, -3], [2, -6, -23, 11, 4, -6, 0, 17, 3, -4, -16, -15], [1, -12, -15, -5, 7, 3, 7, 5, -3, 8, 8, 5], [3, -2, -14, 11, -12, 8, -10, 6, -15, -2, -1, -7], [-2, 8, 5, -25, 20, 0, 4, 7, 10, 6, 18, -19], [-16, -2, -1, 5, 11, -3, 1, 3, 11, 15, 0, 9], [-22, 6, -10, -7, -3, 0, 3, 0, -8, -1, -9, -2], [-33, 4, -7, 3, -15, 3, -3, -8, -10, 1, -7, 11], [22, -6, -16, 15, -8, 4, -6, 9, -5, -18, -14, -1], [-1, 8, -10, -11, 3, 4, 6, -5, 3, -8, 1, -4], [-2, -10, -6, 7, -6, -5, -4, 2, -16, -6, -1, -6], [27, 10, 2, -4, 0, 9, -4, 14, -5, -12, 4, -13], [11, -13, 4, 6, 3, 8, -11, 3, 1, 15, 6, 4], [36, 0, 4, -18, -5, 11, 4, -6, 5, 0, 11, -6], [-1, 8, 12, -2, -3, -10, -5, -16, 19, 11, -4, 11], [4, 15, -1, -20, 9, 1, 18, 3, 4, 2, 14, -17], [-17, 4, -16, 13, -20, 4, -1, 10, -15, -11, -12, -9], [-11, 12, 2, -18, 21, -1, 14, -1, -4, 4, 12, -4], [-6, 10, -16, 5, 1, -11, -6, 1, 5, -6, -19, 3], [-23, 24, -12, 1, -14, -2, 6, -11, 0, -1, -4, 3], [10, -4, 16, -19, 14, 1, 5, -8, 11, 15, 18, -12], [14, 14, -1, 5, -14, 10, -9, 3, 3, -11, -3, -5], [-9, -26, -5, 11, 8, -7, -1, -3, -4, 5, -5, 8], [9, -24, 8, 2, 9, -6, -8, 2, 12, 0, -9, 7], [-3, -10, 2, 3, 1, 0, -9, 9, 7, 3, -17, -2], [-61, -6, 0, 1, 1, -9, -2, 0, 4, -6, -4, -3], [-30, 0, -12, 17, -21, -8, -17, -1, -1, -4, -16, 14], [-9, -14, -1, 9, 4, -4, 6, -8, -8, 6, 4, 7], [22, 12, -8, -11, 3, -1, 13, 3, -4, 5, 17, 1], [-5, -2, 23, -5, 4, -5, 6, -22, 0, 17, 11, 14], [-3, -2, 18, -22, 4, 20, 6, -5, 5, 10, 19, -4], [-16, -6, -6, 6, -15, 10, -12, 3, 0, 2, 2, 1], [7, -2, 7, -11, 17, 0, 15, 3, 19, 5, 4, -8], [4, 8, 2, -4, 1, 3, -11, -2, 15, -9, 0, -6], [4, -10, -2, -2, 7, -1, -4, 12, 0, -10, 3, -6], [27, -5, -9, 9, -6, -3, -4, 12, -3, 6, -2, 5], [-15, 16, -5, 7, -9, -2, -8, -1, 16, 0, -19, 10], [-28, 11, -8, 13, -15, -16, 1, -10, 8, -4, -27, 12], [22, -5, -12, 8, 1, -2, 12, 0, 3, -8, -9, -10], [18, -12, 14, -18, 20, 10, 12, 10, 4, 10, 22, -10], [-6, 3, 9, 7, 16, -3, -7, -2, 9, -4, -8, 4], [-18, -1, 3, 8, -6, 7, -4, 2, -5, 4, -16, 4], [5, 2, -9, -8, 6, 7, -1, -6, -5, 7, 17, -7], [-19, -2, -14, 12, 4, -11, -1, 12, 12, -15, -19, -9], [26, 4, -10, 5, 3, 5, 5, -5, -16, -5, 12, 1], [20, -21, 13, 5, 4, 1, 0, 9, -13, -6, -4, 19], [26, -19, 2, -1, 13, 1, 10, -9, 5, 3, 13, 3], [-10, 8, 8, -12, -7, -11, 16, -14, 7, 24, 7, 14], [-11, -10, -1, 21, 1, -22, -8, 9, 8, 0, -13, 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_h();".
// 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_h(:prec:=12)
    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_h();".
// 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_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6025_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, -18, -45, 328, 105, -444, -123, 219, 65, -44, -14, 3, 1]>,<3,R![64, -400, -992, 960, 1540, -725, -888, 210, 224, -25, -25, 1, 1]>],Snew);
    return Vf;
end function;