
// Make newform 3525.2.a.z in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3525_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_3525_2_a_z();".
// 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_3525_2_a_z();".
// 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 := [5, -6, -15, 16, 10, -9, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0], [5, -2, -17, -1, 9, 0, -1], [-8, 2, 18, 1, -9, 0, 1], [-7, -3, 18, 2, -9, 0, 1], [10, 0, -24, -1, 10, 0, -1], [4, 8, -15, -9, 9, 1, -1]];
    Rf_basisdens := [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_3525_a();"
function MakeCharacter_3525_a()
    N := 3525;
    order := 1;
    char_gens := [2351, 1552, 2026];
    v := [1, 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_3525_a_Hecke(Kf)
    return MakeCharacter_3525_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], [1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0], [1, 0, 1, 1, 0, -1, 0], [0, 1, 0, 1, 0, 0, 0], [1, 0, -1, 0, 1, 1, 0], [0, 1, 0, 0, 1, 0, 1], [-2, 0, -1, 0, 0, -1, -1], [1, 1, 0, 0, 0, 0, -1], [2, -2, -1, 0, 2, 0, -1], [0, 1, 0, -1, 1, 0, 2], [0, 1, 1, 0, 1, 1, -1], [2, 2, 2, 1, -1, -1, 1], [0, 1, -1, -2, -2, -2, 2], [1, 0, 0, 0, 0, 0, 0], [2, 4, 1, 0, 1, -2, 0], [1, -2, 0, -4, -1, 0, 0], [0, -1, 1, 1, -1, 0, -3], [2, -3, 0, 0, -1, 2, -2], [3, 0, 1, 2, 2, 1, -1], [0, 2, -1, -1, -2, 0, -1], [0, 2, -1, -1, 3, 3, 0], [9, -1, 3, 3, 0, -1, -3], [5, -5, 1, -2, -1, 0, 0], [-3, -2, -2, 0, -3, 0, 3], [-2, -1, 0, -3, -3, -3, 1], [9, -1, 0, 3, 0, 2, -4], [3, 4, 1, -3, -1, -1, 3], [-2, 1, -3, 1, 2, 1, 1], [-3, 2, -2, -4, -3, 0, 1], [0, 1, -1, -6, 2, 3, 2], [1, 1, 0, 0, 3, 1, 1], [6, -1, -1, 3, 0, -1, -3], [2, -3, 2, 0, -2, 2, 1], [6, 0, 3, 3, -1, 2, -1], [-1, -3, 2, 0, -4, 0, -3], [6, -4, 3, 0, -3, -3, 1], [2, 0, -4, -4, 2, 3, 3], [8, -4, -4, -1, 0, 0, 2], [7, -1, 1, 1, -2, 4, 0], [-2, -5, -1, 2, -3, -4, 3], [-1, 3, 2, 2, 2, -3, 2], [1, 1, 0, -2, -3, 0, 2], [3, 0, -1, 0, -1, -4, -4], [-3, -2, -5, -3, -2, -1, 2], [5, -6, -1, 4, 1, 2, -6], [-3, 0, -1, -2, -4, 0, 1], [-3, 1, -7, -5, 3, 5, 0], [12, 2, -2, 0, 1, 4, 1], [-6, 2, 4, 0, 0, 0, 3], [0, -3, 2, 4, -1, 2, -4], [-2, 2, 0, -4, 1, -1, 4], [-6, 0, 0, 2, -5, -2, 0], [-4, 7, -5, 2, 1, 3, 1], [1, 2, 0, 5, -3, -2, -3], [3, -4, 1, -1, 1, -1, -3], [5, 4, 5, 6, 1, -5, -2], [-12, 5, 1, -3, 0, 0, 5], [0, 0, 0, -1, 1, -2, 4], [12, 0, 2, 0, 0, 1, -3], [12, -1, 4, -5, 4, 2, 2], [3, -2, 0, 3, 2, 4, 1], [14, -1, 3, 1, 5, 0, -2], [-2, 3, -4, 5, 6, 3, -7], [-3, 1, -1, -3, 4, 4, -1], [-1, -4, -3, -4, 5, 5, -2], [0, 5, 6, 1, -1, 0, 4], [-3, 5, 6, 2, -1, -7, 3], [-2, -3, -1, -2, -7, -2, 6], [-7, 7, 2, -1, -2, -1, -3], [-7, -3, -7, -8, -1, 1, 1], [6, 0, -6, -7, -2, 2, -3], [5, -4, -3, 0, 2, 2, -4], [0, -7, -3, 4, -3, 1, 0], [-13, 5, -3, -8, 1, 2, 7], [0, -3, 1, 3, 0, -2, 0], [-2, 2, -3, 1, -6, -1, 2], [0, 1, 10, 7, -1, -1, -2], [16, 0, -3, -5, 2, 5, 1], [-14, 3, -9, -1, 2, 3, -2], [5, 0, 6, 7, 0, 1, -9], [-1, -3, -3, 1, 3, -1, 4], [5, 2, 1, -6, -4, 4, 0], [-6, 7, 3, -2, -1, -6, 2], [0, 6, 6, 1, 0, 2, -5], [3, 11, 0, 6, 0, 1, -2], [-1, 3, -3, -3, 6, 0, 4], [1, -2, 2, 10, -5, -1, -3], [7, 7, 0, 2, -4, 0, 1], [-2, 1, 2, -1, -3, 1, -3], [4, -1, 3, 4, -7, -1, -1], [0, -5, -9, -13, 3, 6, 2], [-4, -3, -2, 1, 0, -4, 1], [-4, 5, 1, 2, 1, -3, -5], [-4, 14, -2, 5, 5, -3, 2], [-5, 4, 6, 5, 2, -3, -2], [-8, 8, 4, 0, 3, 0, 3], [2, -11, -6, -1, -1, 5, 1], [7, 9, 2, 9, 5, -4, -2], [-1, 1, 4, -1, -3, -8, -4], [11, -10, 2, 1, -3, 2, -4], [8, 3, 5, -3, 6, 2, 2], [5, -6, -4, -8, 3, 15, 0], [3, -6, -3, -2, -4, -2, 7], [-1, -10, 4, 1, 3, 5, -5], [7, -1, 0, 3, 8, 5, -3], [-9, 5, -4, 3, 4, -5, -3], [-6, 7, 5, 9, 2, -3, -4], [-4, -6, 1, -2, -6, 5, -1], [7, -11, -8, -8, -2, 3, 0], [5, 11, -7, -1, 3, 7, 4], [10, -4, 6, -2, 0, 7, -4], [-3, 4, 7, 3, 6, -4, -1], [0, 5, 10, 6, -2, -6, 5], [-5, -1, -4, 5, -6, -3, -4], [5, 9, -3, -7, 4, -2, 3], [-3, 8, -9, -1, -6, 1, -4], [8, -7, -1, 8, 9, 8, -6], [12, -10, 1, -4, -1, 3, 1], [6, -2, -4, -1, 6, 4, 1], [8, -3, 0, -7, 0, 4, 9], [10, -5, 4, 6, -4, 4, -8], [-13, 5, 3, -6, -3, 2, 3], [8, 0, 7, -1, 3, -1, 2], [-18, 3, -7, -2, -4, 0, 3], [8, -9, -11, -4, 1, 4, -2], [-5, -7, -1, -2, 7, -5, -1], [-7, 0, 8, 3, 0, -1, -7], [0, -8, -7, -3, -9, -4, 10], [5, 4, -2, -3, -3, -2, 7], [3, 3, 14, 7, 4, -4, -3], [5, -4, -1, 2, -7, -2, 6], [-5, -9, -1, 2, 3, -4, 0], [-10, -4, -3, -11, 3, -2, 6], [-1, 13, -10, -1, 7, 1, 6], [-12, -8, 10, 2, -7, -5, 1], [-5, -19, -3, -9, 4, 0, -1], [-2, -6, 3, -4, 1, 1, 8], [-6, -6, -1, -1, -10, 1, -1], [-7, -2, -4, 4, -1, -7, -10], [-3, 3, -2, -7, 3, 7, 0], [17, -4, -2, 8, 2, 1, -8], [4, 7, -1, -2, -1, 2, 8], [-8, -1, -7, -10, 5, 2, -2], [-13, -8, 0, 0, -7, 0, 3], [-9, -7, 3, -1, -11, 0, 3], [0, 7, 5, 5, -4, -4, -3], [-9, -3, -5, 6, 3, 0, -5], [11, 2, 8, -2, 0, -1, 6], [-10, -7, 2, 4, 5, -5, 2], [3, 7, 6, 11, -2, -3, -11], [9, 14, -3, 0, -5, -4, -1], [10, 5, 5, 15, 4, -1, -2], [-8, -3, 8, 1, 6, -2, -1], [18, 1, -5, -2, -7, -2, -6], [-8, -9, -3, -12, -11, 1, 5], [-7, -15, -4, -10, 1, -3, 5], [3, -11, -2, -4, 6, 10, -3], [-2, -4, 1, 3, -5, 0, 6], [12, -3, 11, 7, 1, -2, -5], [14, -6, -2, 6, 2, 2, -2], [-13, -4, 5, 8, -4, 0, 0], [15, 6, -7, -5, -5, 3, -7], [-1, -8, 10, 7, -5, -8, -6], [1, 8, 2, -3, 0, 0, 0], [12, -8, -6, 0, 0, 2, -5], [-12, -12, 4, 3, 0, 2, -6], [-9, -5, -3, -7, -3, -5, 4], [-8, 1, -4, 2, 11, 7, -10], [4, -5, 7, -1, 1, 5, -6], [-7, 3, 1, -5, -7, 7, 3], [10, 1, -8, -7, -8, 3, -9], [-2, 3, 17, 10, 1, -7, 5], [-6, 11, -4, -3, 4, 4, 1], [-13, 15, 3, 13, -2, -5, -4], [-2, -16, -2, 0, -8, 4, -3], [-8, -4, -7, 2, -4, 4, 10], [4, 0, 9, 0, -2, -3, 2], [-14, -3, 0, -2, 6, -3, 4], [-5, -9, 2, -1, 8, -1, -5], [15, 5, 12, -2, 6, 1, -1], [-13, 7, -5, -6, -2, -1, 1], [-11, 9, 12, -1, 1, -9, -2], [3, -10, -2, -5, -4, -2, 8], [-5, -1, -2, 10, -11, -1, -4], [-24, 0, -13, -8, -7, 5, 4], [-18, 3, -5, -16, -9, -6, 11], [8, -2, -4, 5, 7, -5, 0], [-12, -5, -4, -12, -5, 10, -4], [2, -6, 0, 3, 0, -7, 6], [-6, -5, 5, 4, 7, -8, 3], [13, -11, 1, -8, -2, 2, -1], [-27, -8, -10, -21, -1, -2, 17], [11, -1, -6, 4, 8, 5, -4], [11, -5, 3, 6, 4, 8, -6], [8, 9, 1, 6, -2, -7, -2], [-5, -15, -10, 1, -5, 1, 6], [-11, 6, 5, -2, -1, -5, -2], [4, -17, -5, -13, -1, -1, 5], [-17, 4, -6, -5, -10, -2, 7], [-4, 4, 9, 5, -5, 7, 6], [1, 0, 2, -1, -2, -4, -9], [9, 12, 9, -2, 2, -6, -2], [-4, -6, -8, -14, -1, 13, 1], [3, 0, 0, 4, -2, -9, -10], [-2, -2, -13, -17, 1, 9, 6], [-1, -12, -4, -7, -1, 0, -2], [13, -13, -11, -3, 9, 10, -2], [12, 12, -6, 1, -2, 5, -8], [-21, -4, 0, 3, -1, -7, 8], [4, 18, 7, 11, 3, -5, -8], [12, 5, 4, 4, -4, -13, -2], [5, -9, 4, 5, 1, -2, -13], [16, -4, 2, -10, 3, -4, -4], [1, 7, 1, 7, 10, -10, 3], [8, -3, -3, -8, 7, 19, -2], [11, 15, 6, 14, 1, -1, 4], [28, -9, 9, 0, 4, 2, 2], [-16, 5, -7, -8, -2, -1, 0], [-10, -7, 10, 10, 3, -3, -1], [2, -5, 4, 20, 2, 4, -10], [7, 23, 1, 10, 6, -5, -5], [38, -1, 10, -4, 6, 3, 0], [-19, 3, -2, -17, -1, -3, 8], [6, 20, 6, 13, -5, 1, -2], [-2, 25, -6, -5, -1, 11, 5], [4, -7, -7, -1, -11, 7, -1], [-1, -3, -5, 3, -7, 1, 12], [-3, 19, 5, 7, 9, -3, 5], [-10, 15, 0, -4, -1, 1, 14], [19, 7, 6, 17, 7, -10, 0], [34, -15, -6, -1, -10, -2, -6], [-15, 3, -4, 8, -2, -4, -3], [23, -15, -11, -2, 1, 4, -6], [7, -7, 6, 14, -9, -5, -9], [-7, 10, 1, -15, -1, 2, 15], [1, 23, 6, 9, -7, -6, 4], [-44, 6, -6, -6, -2, -1, 6], [3, -11, -5, -10, -4, 0, 5], [-24, -4, -2, 7, 3, -4, 4], [34, -1, -16, -7, 11, 8, -1], [-11, -3, -2, -4, -2, 3, -14], [4, -21, -5, -2, -10, 7, -11], [1, -14, -9, -23, -4, 2, 0], [-21, -7, -14, -23, -10, 8, 1], [27, 2, 16, 14, 2, -9, -4], [7, 11, -2, -2, 0, -3, -4], [7, -15, 17, 2, -7, -11, 2], [12, 6, -4, 2, -4, -8, 4], [9, -3, -4, -15, 9, 11, 9], [-6, 7, 0, 0, 7, 5, 3], [-5, 10, -17, -11, -2, 8, -3], [-10, -8, 4, 3, 4, 6, 6], [10, -6, -10, -4, -10, 2, 4], [15, -10, -2, 0, 7, -1, -9], [11, 6, 4, -14, 4, 5, 19], [28, 2, -2, 0, 2, 15, 4], [10, 8, 5, -1, -6, -4, 16], [7, -6, -12, -1, -2, 1, -11], [10, -10, -1, 7, -5, -11, 7], [4, 11, -8, 2, 2, 12, 4], [10, -8, -1, 11, 3, 12, -3], [-6, -5, -7, 5, 5, 5, -5], [9, -17, 0, 6, 14, 14, -10], [-5, 16, 11, 14, -2, -1, -8], [2, 6, 3, 11, -2, -8, -18], [-1, -1, 1, -8, -9, 3, 1], [16, -6, -16, -10, 2, 5, -8], [18, -5, 5, 2, 5, -5, 3], [-4, -3, -9, -9, 2, 4, -2], [-17, 2, 6, 5, -9, -7, 1], [28, -17, -5, 2, -4, 0, -1], [-10, -19, 4, -1, 4, 7, -4], [32, 13, -10, -11, -1, 5, -3], [2, 20, -2, 1, -2, -6, -5], [-25, -14, -15, -16, -4, 2, 12], [8, 5, -7, -7, 3, -10, 3], [-7, 6, -1, 5, -1, 0, -13], [-10, 7, -9, -12, -9, 0, -1], [26, -6, -1, 1, -3, -10, -9], [-5, 3, 4, -7, -2, -13, 6], [-6, -9, -6, 2, 0, -2, 9], [5, 0, -6, -23, -8, -3, 7], [10, 5, -3, -6, -10, -9, 2], [3, -7, -1, -15, -10, 2, 9], [-27, -4, 6, 15, 3, 3, 2], [-21, 10, 1, -6, -9, -11, 11], [-12, 5, 3, 11, 3, 0, 4], [27, 1, -8, 2, -4, -3, 7], [4, 2, -3, 2, -1, 2, -5], [13, 9, -8, 0, -2, 0, -1], [21, 9, -7, -7, -4, -10, 3], [-25, -10, -1, -16, -2, -9, 4], [-18, -14, -11, -13, -14, 8, 5], [-27, 6, -12, -3, -15, -7, 1], [-15, -1, 2, 2, 7, 14, 3], [-14, -9, -6, -9, -3, 2, -4], [5, 6, -10, 3, 13, 8, -4], [4, 15, -1, 2, 7, 1, 0], [22, -3, -1, -10, -9, 1, 4], [-6, 19, -2, 3, -2, 6, 4], [-7, 3, 3, 5, 10, -2, -5], [12, -7, -17, -7, 8, 11, -7], [20, -15, 3, -7, -6, -6, 2], [-18, -14, -21, -12, 2, 11, -7], [6, -19, -18, -19, 3, 7, 7], [-5, -3, 13, 13, -12, -22, 0], [-11, 30, 18, 13, 2, -4, -2], [12, 16, 2, 5, -3, 7, 0], [23, -12, -4, 6, 4, -4, 8], [25, -12, -17, -10, 13, 3, -2], [-45, -7, -8, -20, -6, -2, 11], [-11, -5, 3, 0, 8, 8, -16], [16, 11, -7, 10, -1, 7, 8], [-8, -3, 14, 13, -6, -6, -6], [-13, -9, 5, -16, 3, -1, 4], [22, -12, -15, -7, 6, 9, 0], [-5, -11, 9, 10, -9, -17, -6], [-6, -21, 8, 2, 0, -4, 7], [-9, 4, 13, 8, -13, -10, 1], [-26, 17, 8, 10, -2, -13, -2], [33, 6, 5, -13, 0, 4, 10], [-12, -6, 3, -4, 0, 14, 3], [9, -10, 7, 20, 0, 8, -15], [16, -5, 1, -13, 7, 9, 10], [-21, -8, 4, 4, -10, -3, 12], [22, -18, 1, 16, 17, 6, -17], [11, -19, 12, 2, 0, -8, -6], [-20, 14, 2, 16, 5, 2, -12], [4, -5, 5, 23, -4, -4, -15], [-9, -6, 15, 7, -3, -5, -9], [1, -9, 15, -5, -12, -6, 12], [0, 4, -3, 5, -4, -8, 11], [12, 9, -14, -14, -7, -1, -6], [31, 17, -4, 6, -7, -2, 3], [-9, 15, -1, -15, 11, -2, 22], [-23, -10, -7, -5, -1, -11, -1], [6, 1, 21, 10, 11, 0, 10], [23, 4, 3, -3, -8, -9, 13], [-9, 1, -2, -8, -11, -4, -7], [-22, 18, -6, -2, -17, -9, 3], [-11, -21, 10, 7, -3, -6, 7], [10, 23, 8, 11, 4, 3, -11], [-10, 5, 3, -8, 17, 14, -7], [-4, 31, -3, -10, -7, -5, 12], [-8, -5, -3, 3, -3, 7, -6], [-15, -6, 14, 8, 6, 7, 1], [4, -2, 16, -1, 6, -6, -8], [14, -7, -2, -5, -5, 7, 8], [9, 26, -1, -5, -1, 1, 5], [23, 6, 15, 17, -1, 10, -10], [19, -4, -17, -5, -3, 14, -3], [2, 13, 2, 0, -3, -1, 18], [-18, 3, 4, -2, -4, -5, 13], [-15, -2, -10, 13, 9, 3, -9], [-13, 19, -7, 4, -5, 7, 11], [1, 18, 5, 10, 15, -7, -8], [-19, 0, 16, 11, 4, 3, -3], [1, 2, 11, 16, 2, -5, 1], [0, 5, 0, -11, -3, -3, 17], [20, -28, 9, 1, -1, -10, -11], [-27, 0, 9, 7, -3, -15, 2], [-20, 7, 12, -7, -2, -10, -7], [17, 1, -4, -15, 7, 18, 0], [-2, -10, 8, 7, -1, -9, -4], [-29, -30, 7, 4, 1, 8, -13], [0, -2, -11, -2, -3, -2, 0], [-37, 13, 4, 1, 14, 12, -1], [4, -8, -16, -4, 4, -1, 5], [-38, -4, -10, -2, 7, -5, 2], [18, -12, 18, 11, 13, 10, -5], [39, 15, 1, 13, -7, 2, 2], [-7, 10, 9, 5, 11, -16, 3], [-22, 10, 13, -3, 6, -4, 13], [-33, -22, -5, -5, -12, 4, 16], [-23, -13, 15, -8, -4, -18, -1], [-6, 8, 26, 19, -5, -18, -2], [0, 10, -3, -3, 12, 10, 11], [-9, 15, -2, -10, -6, 3, 14], [5, 9, -10, 4, 20, 9, -13], [-47, 16, -1, -16, -8, 5, 16], [6, -4, 5, 20, -3, 3, -6], [-43, -4, -4, -1, 6, 3, -11], [8, -12, 21, 8, -2, 5, -19], [2, -5, -12, -3, 15, 6, 2], [19, -21, 7, -3, 6, 15, 1], [-23, 16, -8, 12, 9, 12, 6], [-45, 24, -10, 1, 5, 6, 9], [14, -22, 23, 2, 0, -9, -11], [-10, -3, -26, -16, 9, 19, -2], [-20, -12, -8, -13, -3, -1, -4], [-2, -4, 4, -15, -15, -14, 7], [10, -8, -21, -17, 3, 16, 0], [28, -6, 4, 2, 2, 8, 8], [24, -13, -15, -17, 17, 11, 1], [-27, -6, -14, -29, -2, 0, 20], [-10, -8, 6, -14, 1, 0, -9], [-14, 12, -3, 3, -6, 3, -7], [-3, 0, -12, -9, -6, -13, 10], [3, 11, -13, -8, 20, 14, 4], [-12, -11, -15, -3, -10, 12, 8], [-21, 13, 8, 13, -6, 0, -7], [-22, -3, -13, -18, 0, 6, 11], [-13, -9, 2, -25, -12, 0, -6], [1, -17, 10, 5, 14, 18, -8], [-20, -7, 9, 4, -14, -20, 6], [3, -10, 19, 5, 7, -2, 11], [46, -5, -3, -1, -12, 5, -5], [-28, 16, 11, 11, -10, -17, -10], [3, -25, -9, 8, -2, -5, -5], [4, 7, 15, 7, 2, -3, 8], [-8, 9, 2, 4, 7, 9, 3], [34, -3, 2, -9, 0, 14, 2], [20, -14, -1, 9, 4, -14, -10], [-34, 3, -6, -22, -20, -2, 20], [6, -14, 16, 20, 0, 5, -25], [12, -11, -23, -21, 10, 13, 4], [-17, -2, -3, -7, 6, -12, -1], [48, -2, 6, 13, 0, 7, -10], [15, 0, 11, -9, 4, -1, -8], [21, -17, -12, 7, 16, 1, -17], [-17, 8, 6, -4, -14, 11, 5], [-21, 11, 11, 11, -13, -4, 1], [-33, -14, 5, 7, -3, 0, -16], [-31, 6, 3, 6, 7, 4, 0], [-7, 21, 8, 0, 0, 1, 7], [5, -9, 0, 11, -11, -4, 11], [4, 9, 24, 19, -16, -7, 2], [1, 21, -1, 7, 15, 8, -4], [9, -4, 18, 15, 11, 0, 9]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3525_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3525_2_a_z();".
// 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_3525_2_a_z(:prec:=7)
    chi := MakeCharacter_3525_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_3525_2_a_z();".
// 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_3525_2_a_z( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3525_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-5, -6, 15, 16, -10, -9, 1, 1]>,<7,R![-489, 978, -411, -181, 133, -7, -7, 1]>,<11,R![5, -41, -52, 133, 4, -23, 0, 1]>],Snew);
    return Vf;
end function;