
// Make newform 7497.2.a.cd in Magma, downloaded from the LMFDB on 22 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7497_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_7497_2_a_cd();".
// 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_7497_2_a_cd();".
// 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 := [4, 0, -32, 0, 40, 0, -12, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [0, 22, 0, -10, 0, 1, 0, 0], [4, 0, -22, 0, 10, 0, -1, 0], [-16, 0, 26, 0, -10, 0, 1, 0], [-20, 0, 38, 0, -12, 0, 1, 0], [0, -18, 0, 32, 0, -11, 0, 1], [0, -20, 0, 38, 0, -12, 0, 1]];
    Rf_basisdens := [1, 1, 4, 4, 4, 4, 4, 4];
    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_7497_a();"
function MakeCharacter_7497_a()
    N := 7497;
    order := 1;
    char_gens := [1667, 5050, 6616];
    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_7497_a_Hecke(Kf)
    return MakeCharacter_7497_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], [-2, 0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -1, -1, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 2], [-1, 0, 0, 0, 0, 0, 0, 0], [0, -3, 0, 0, 0, 0, 1, 0], [0, 1, 1, 0, 0, 0, -2, -1], [0, 1, 1, 0, 0, 0, 1, 1], [0, -2, 3, 0, 0, 0, -1, -2], [2, 0, 0, -2, 1, -2, 0, 0], [-2, 0, 0, -3, -1, 1, 0, 0], [-4, 0, 0, -1, -2, -1, 0, 0], [-5, 0, 0, 1, 0, -4, 0, 0], [0, 2, -3, 0, 0, 0, -1, -1], [-3, 0, 0, 3, -1, 1, 0, 0], [0, 1, 0, 0, 0, 0, 0, 1], [7, 0, 0, -1, -3, 4, 0, 0], [0, -1, 1, 0, 0, 0, 5, 3], [0, 1, -3, 0, 0, 0, -1, -3], [10, 0, 0, -2, 1, 2, 0, 0], [-4, 0, 0, -2, -2, 2, 0, 0], [0, 0, 0, -4, 1, -1, 0, 0], [0, 0, -4, 0, 0, 0, 2, -2], [3, 0, 0, -1, 3, 1, 0, 0], [0, 0, -3, 0, 0, 0, 2, -3], [0, 2, 0, 0, 0, 0, -5, -3], [6, 0, 0, 0, -2, 3, 0, 0], [0, 3, -3, 0, 0, 0, 0, 8], [2, 0, 0, 3, 3, -8, 0, 0], [-8, 0, 0, 1, 2, 0, 0, 0], [0, 1, -2, 0, 0, 0, -4, 4], [0, -1, -3, 0, 0, 0, -1, 1], [0, 5, 3, 0, 0, 0, -3, 1], [-1, 0, 0, 3, -3, 2, 0, 0], [0, 1, 0, 0, 0, 0, -1, -4], [-6, 0, 0, 0, 1, -7, 0, 0], [-11, 0, 0, 0, -4, 3, 0, 0], [0, 0, 0, -3, -6, 2, 0, 0], [0, -5, 3, 0, 0, 0, -3, -7], [0, -3, 3, 0, 0, 0, 11, 3], [0, -4, 2, 0, 0, 0, -4, -8], [-8, 0, 0, 6, 2, 1, 0, 0], [0, 2, 2, 0, 0, 0, -1, 0], [0, -3, -2, 0, 0, 0, 2, 7], [4, 0, 0, 0, 6, -5, 0, 0], [0, 10, -3, 0, 0, 0, -6, -10], [-8, 0, 0, -3, 1, 1, 0, 0], [0, 3, -1, 0, 0, 0, 1, -9], [0, -3, 1, 0, 0, 0, 8, 1], [0, -5, 0, 0, 0, 0, -2, 4], [0, -1, 1, 0, 0, 0, 7, 5], [-7, 0, 0, -1, 2, 4, 0, 0], [-15, 0, 0, 7, 0, 2, 0, 0], [0, 0, 2, 0, 0, 0, -6, 0], [10, 0, 0, -3, 7, 5, 0, 0], [0, 1, 0, 0, 0, 0, -3, 2], [-8, 0, 0, 0, -6, -2, 0, 0], [0, -1, -1, 0, 0, 0, 5, 9], [0, -1, 0, 0, 0, 0, -4, -3], [14, 0, 0, -4, 5, 5, 0, 0], [0, 2, -2, 0, 0, 0, -8, -1], [-10, 0, 0, 4, 2, 2, 0, 0], [0, 8, 3, 0, 0, 0, 1, -4], [0, -1, 1, 0, 0, 0, 1, -8], [2, 0, 0, -3, 7, -2, 0, 0], [-4, 0, 0, 6, -6, 5, 0, 0], [0, -3, -7, 0, 0, 0, 9, -1], [0, 5, 2, 0, 0, 0, -1, -3], [-9, 0, 0, 3, -5, 5, 0, 0], [0, -7, 4, 0, 0, 0, 16, 10], [0, -7, 9, 0, 0, 0, 7, 3], [-5, 0, 0, 3, 3, -8, 0, 0], [2, 0, 0, -12, -6, 6, 0, 0], [3, 0, 0, 7, 4, 2, 0, 0], [0, 6, 0, 0, 0, 0, -12, -2], [0, 12, -3, 0, 0, 0, -5, 0], [0, -9, 5, 0, 0, 0, 0, 4], [0, -8, -7, 0, 0, 0, 0, 5], [13, 0, 0, 5, 2, 3, 0, 0], [-6, 0, 0, -7, -5, 2, 0, 0], [0, 11, 3, 0, 0, 0, -7, 0], [0, 1, 4, 0, 0, 0, -11, 3], [0, 3, -8, 0, 0, 0, 0, -3], [0, -8, 3, 0, 0, 0, 3, 1], [0, 1, -5, 0, 0, 0, -1, 1], [-9, 0, 0, 6, 9, -1, 0, 0], [-9, 0, 0, 7, 7, 3, 0, 0], [-11, 0, 0, 7, -1, 0, 0, 0], [-24, 0, 0, 2, -1, -7, 0, 0], [6, 0, 0, -9, 7, -5, 0, 0], [14, 0, 0, 6, 4, 10, 0, 0], [0, -3, -1, 0, 0, 0, 3, -11], [4, 0, 0, 8, 3, -8, 0, 0], [-18, 0, 0, 5, 10, -4, 0, 0], [14, 0, 0, 0, -6, 6, 0, 0], [-5, 0, 0, 4, -12, 9, 0, 0], [0, -2, -2, 0, 0, 0, 10, 5], [16, 0, 0, -2, 1, 3, 0, 0], [0, 0, 0, -2, -12, 9, 0, 0], [0, -1, 10, 0, 0, 0, 0, 4], [-10, 0, 0, 4, 7, -1, 0, 0], [0, 2, 2, 0, 0, 0, 0, -12], [-6, 0, 0, -12, -5, -3, 0, 0], [0, 12, 1, 0, 0, 0, -6, -14], [-9, 0, 0, 1, 2, -2, 0, 0], [-15, 0, 0, -5, 4, 0, 0, 0], [0, -7, -7, 0, 0, 0, -1, 11], [0, 1, -6, 0, 0, 0, 6, 9], [0, 6, 3, 0, 0, 0, -11, 4], [-1, 0, 0, 0, 8, -12, 0, 0], [0, 13, 5, 0, 0, 0, -5, 13], [0, 0, 5, 0, 0, 0, -11, -8], [5, 0, 0, -6, 0, 0, 0, 0], [0, -2, -6, 0, 0, 0, -9, 9], [0, -7, 5, 0, 0, 0, 15, 5], [-30, 0, 0, -2, -5, -1, 0, 0], [0, 12, -6, 0, 0, 0, -7, -15], [0, 12, 3, 0, 0, 0, 3, -5], [0, -3, -6, 0, 0, 0, -7, -5], [-4, 0, 0, 10, -7, -5, 0, 0], [-18, 0, 0, -7, -9, -3, 0, 0], [0, -4, -8, 0, 0, 0, 5, 14], [0, 10, 5, 0, 0, 0, -5, 12], [0, -16, -4, 0, 0, 0, 16, 4], [-22, 0, 0, -2, 3, -5, 0, 0], [-4, 0, 0, -5, 6, -6, 0, 0], [0, 16, -6, 0, 0, 0, -8, -13], [0, 0, 4, 0, 0, 0, 6, -3], [-2, 0, 0, -17, -10, -5, 0, 0], [0, 21, -7, 0, 0, 0, -9, -13], [-14, 0, 0, -2, 7, 0, 0, 0], [6, 0, 0, -9, -1, 10, 0, 0], [-14, 0, 0, -8, 4, 4, 0, 0], [0, -6, -13, 0, 0, 0, 16, 2], [-23, 0, 0, -5, -7, 3, 0, 0], [0, -7, -8, 0, 0, 0, -2, -15], [-14, 0, 0, 8, -1, -7, 0, 0], [0, -17, -1, 0, 0, 0, 8, -6], [0, 13, 7, 0, 0, 0, 1, -9], [0, 9, 1, 0, 0, 0, -18, -19], [24, 0, 0, 6, -3, 6, 0, 0], [0, -5, 5, 0, 0, 0, 0, -14], [0, 24, -4, 0, 0, 0, -6, -7], [-16, 0, 0, -7, -9, -3, 0, 0], [0, -4, 13, 0, 0, 0, -11, -8], [10, 0, 0, -6, -4, 0, 0, 0], [0, 2, 2, 0, 0, 0, 14, 19], [0, -5, 8, 0, 0, 0, 6, 16], [24, 0, 0, -2, 0, -3, 0, 0], [-6, 0, 0, 14, -4, 0, 0, 0], [-19, 0, 0, 8, 1, -11, 0, 0], [-27, 0, 0, -4, 2, -15, 0, 0], [-2, 0, 0, 10, 8, -4, 0, 0], [0, -15, -1, 0, 0, 0, 2, -8], [-5, 0, 0, 8, 2, -8, 0, 0], [4, 0, 0, 1, 7, -21, 0, 0], [0, 11, 11, 0, 0, 0, -13, 5], [-2, 0, 0, 18, 12, -10, 0, 0], [0, -3, 7, 0, 0, 0, 7, 23], [0, -5, 0, 0, 0, 0, -8, -24], [-2, 0, 0, 11, -1, 12, 0, 0], [-30, 0, 0, 10, -6, -4, 0, 0], [0, 0, 0, 0, 0, 0, 20, 18], [3, 0, 0, 2, 4, 1, 0, 0], [0, 0, 0, -20, -11, 1, 0, 0], [0, 18, 6, 0, 0, 0, -22, -4], [4, 0, 0, 4, 9, -5, 0, 0], [2, 0, 0, 8, 0, 16, 0, 0], [0, 2, 5, 0, 0, 0, 15, 15], [0, 13, -14, 0, 0, 0, -11, -8], [0, 8, 6, 0, 0, 0, -17, -9], [0, 0, 0, -7, 6, 7, 0, 0], [0, -10, -9, 0, 0, 0, 14, -7], [-1, 0, 0, -10, 3, 6, 0, 0], [-8, 0, 0, 16, 11, -2, 0, 0], [0, 17, 1, 0, 0, 0, -2, 7], [0, 15, -16, 0, 0, 0, -7, 0], [0, 10, 13, 0, 0, 0, -18, 5], [-11, 0, 0, 16, 4, -2, 0, 0], [12, 0, 0, -16, -12, 0, 0, 0], [-8, 0, 0, 2, -9, 2, 0, 0], [-2, 0, 0, 2, 5, 1, 0, 0], [0, -4, 8, 0, 0, 0, 8, 12], [-8, 0, 0, -16, -13, -9, 0, 0], [-10, 0, 0, 2, -5, 13, 0, 0], [0, -6, 1, 0, 0, 0, 18, 4], [-16, 0, 0, -10, -11, 12, 0, 0], [11, 0, 0, -1, -2, 3, 0, 0], [0, 17, -12, 0, 0, 0, 2, -1], [0, 9, 9, 0, 0, 0, -5, -5], [-6, 0, 0, -3, 12, -1, 0, 0], [15, 0, 0, -15, 7, -1, 0, 0], [0, 0, -12, 0, 0, 0, -5, -3], [-1, 0, 0, -3, 8, -11, 0, 0], [-8, 0, 0, 14, 8, 6, 0, 0], [-14, 0, 0, 12, 6, 4, 0, 0], [-10, 0, 0, 4, -6, -6, 0, 0], [0, 0, 0, 2, -12, 20, 0, 0], [0, -3, -7, 0, 0, 0, -6, -1], [0, 6, 6, 0, 0, 0, 2, -6], [0, 11, 10, 0, 0, 0, -5, 3], [0, -11, 0, 0, 0, 0, 19, 22], [-9, 0, 0, -11, -1, -1, 0, 0], [11, 0, 0, -13, 1, 3, 0, 0], [0, 0, 14, 0, 0, 0, 4, -13], [0, 1, -9, 0, 0, 0, -13, -7], [0, -4, -10, 0, 0, 0, 5, 13], [0, -9, 5, 0, 0, 0, -1, -3], [18, 0, 0, -8, 7, -6, 0, 0], [15, 0, 0, 5, 7, 9, 0, 0], [-10, 0, 0, 6, -2, 5, 0, 0], [-24, 0, 0, 10, -6, 14, 0, 0], [-14, 0, 0, 7, -6, -16, 0, 0], [0, -9, -5, 0, 0, 0, -3, 9], [-10, 0, 0, -11, -9, 2, 0, 0], [6, 0, 0, 14, -1, 9, 0, 0], [0, 3, 3, 0, 0, 0, 14, 15], [0, 11, -12, 0, 0, 0, -24, -14], [-23, 0, 0, 8, 24, -6, 0, 0], [0, 1, -5, 0, 0, 0, -27, -23], [0, -20, 2, 0, 0, 0, 30, 16], [-28, 0, 0, 14, -8, 15, 0, 0], [-10, 0, 0, -16, -1, -11, 0, 0], [-39, 0, 0, 11, -7, -4, 0, 0], [-8, 0, 0, -5, 5, -21, 0, 0], [0, -14, -18, 0, 0, 0, 7, 7], [0, -4, -1, 0, 0, 0, 14, -19], [0, -7, 17, 0, 0, 0, 3, -15], [-7, 0, 0, -16, -11, 15, 0, 0], [0, 12, 9, 0, 0, 0, -17, 2], [-5, 0, 0, 5, 5, -20, 0, 0], [0, -12, -13, 0, 0, 0, -1, 7], [0, -10, 2, 0, 0, 0, -2, -23], [13, 0, 0, 3, -10, 10, 0, 0], [0, -35, -4, 0, 0, 0, 14, 19], [0, -3, -13, 0, 0, 0, 18, -15], [0, -16, -8, 0, 0, 0, 5, -22], [-32, 0, 0, 6, 6, 2, 0, 0], [0, 0, 0, 0, 0, 0, -7, 17], [0, -32, 8, 0, 0, 0, 16, 17], [0, -14, -7, 0, 0, 0, 8, 18], [38, 0, 0, -1, 4, 11, 0, 0], [20, 0, 0, -1, 10, -12, 0, 0], [1, 0, 0, -24, -5, 0, 0, 0], [0, 17, -9, 0, 0, 0, -9, -9], [3, 0, 0, 6, 3, 8, 0, 0], [6, 0, 0, 5, 4, 17, 0, 0], [0, -9, 5, 0, 0, 0, -11, -37], [-23, 0, 0, -8, -6, -6, 0, 0], [50, 0, 0, 4, 10, 22, 0, 0], [0, 17, -2, 0, 0, 0, -6, -16], [0, -17, -11, 0, 0, 0, 31, -1], [2, 0, 0, 8, 11, -27, 0, 0], [0, -8, -15, 0, 0, 0, 3, 15], [8, 0, 0, -4, -17, 33, 0, 0], [0, 18, -13, 0, 0, 0, 5, 10], [13, 0, 0, 4, -17, 12, 0, 0], [0, 1, 9, 0, 0, 0, -9, 7], [12, 0, 0, 8, 18, 2, 0, 0], [0, -19, -5, 0, 0, 0, -11, -15], [0, -19, -6, 0, 0, 0, 18, 5], [0, 10, -14, 0, 0, 0, 10, 24], [-31, 0, 0, -2, -14, -11, 0, 0], [0, 2, 1, 0, 0, 0, 14, 23], [0, -4, 25, 0, 0, 0, -7, -15], [9, 0, 0, 7, 9, -23, 0, 0], [-2, 0, 0, 16, 14, -8, 0, 0], [0, -4, -9, 0, 0, 0, 7, 29], [0, -15, -22, 0, 0, 0, 4, 13], [38, 0, 0, 1, -9, 12, 0, 0], [0, -17, 2, 0, 0, 0, 3, 8], [-27, 0, 0, -3, 3, -4, 0, 0], [0, -22, -2, 0, 0, 0, 0, -7], [0, -16, -11, 0, 0, 0, -12, -11], [0, -10, -12, 0, 0, 0, 4, 30], [-6, 0, 0, 31, 14, 3, 0, 0], [27, 0, 0, -16, -18, 12, 0, 0], [-50, 0, 0, 6, -6, 4, 0, 0], [5, 0, 0, 5, 3, 17, 0, 0], [-16, 0, 0, 6, 7, -1, 0, 0], [14, 0, 0, -24, -10, 12, 0, 0], [0, 18, 22, 0, 0, 0, -18, 3], [0, -13, 24, 0, 0, 0, -6, 15], [0, 4, -2, 0, 0, 0, 4, 24], [-6, 0, 0, -6, -2, -9, 0, 0], [0, 15, -5, 0, 0, 0, -9, 0], [0, 3, 10, 0, 0, 0, -1, -30], [-16, 0, 0, 2, -10, 6, 0, 0], [0, -6, -6, 0, 0, 0, 3, 11], [12, 0, 0, 1, 5, -13, 0, 0], [0, -2, 12, 0, 0, 0, 8, -14], [15, 0, 0, 10, 6, 13, 0, 0], [-18, 0, 0, -10, 7, 10, 0, 0], [-21, 0, 0, -16, 0, -5, 0, 0], [0, 4, 12, 0, 0, 0, -4, -27], [-6, 0, 0, 4, -18, 8, 0, 0], [-1, 0, 0, 6, -12, 17, 0, 0], [0, -9, 5, 0, 0, 0, 13, 5], [0, -26, -3, 0, 0, 0, 18, 10], [0, 23, -3, 0, 0, 0, -5, 5], [12, 0, 0, -2, -11, -7, 0, 0], [0, -4, 28, 0, 0, 0, -7, -7], [28, 0, 0, 2, -7, -9, 0, 0], [4, 0, 0, 4, 14, -14, 0, 0], [0, -7, 4, 0, 0, 0, 24, -2], [0, -16, 19, 0, 0, 0, 9, -2], [0, -12, 8, 0, 0, 0, 8, 20], [33, 0, 0, -15, -7, -2, 0, 0], [-6, 0, 0, 20, 0, 14, 0, 0], [0, 3, 3, 0, 0, 0, 11, -4], [0, 15, 17, 0, 0, 0, -10, -18], [-22, 0, 0, 1, -7, -20, 0, 0], [0, 40, 12, 0, 0, 0, -22, -12], [0, 13, -16, 0, 0, 0, 3, 16], [-20, 0, 0, 10, -19, 13, 0, 0], [0, -19, 25, 0, 0, 0, 6, -1], [0, -7, -10, 0, 0, 0, -2, 33], [0, -7, 22, 0, 0, 0, -4, 8], [0, 15, -3, 0, 0, 0, -17, -9], [-22, 0, 0, 2, 26, -28, 0, 0], [9, 0, 0, -7, 18, -30, 0, 0], [-19, 0, 0, 4, 0, -10, 0, 0], [0, 11, -3, 0, 0, 0, 3, 17], [0, 13, -12, 0, 0, 0, -9, -33], [4, 0, 0, -6, 0, 22, 0, 0], [0, 6, -29, 0, 0, 0, -7, 22], [0, -23, 25, 0, 0, 0, -5, 7], [0, 11, -5, 0, 0, 0, -12, -18], [-12, 0, 0, 14, -11, -9, 0, 0], [0, -1, -9, 0, 0, 0, -2, 29], [0, -1, -8, 0, 0, 0, -8, -5], [-44, 0, 0, -4, -15, -13, 0, 0], [48, 0, 0, 2, 2, 15, 0, 0], [-3, 0, 0, -4, 18, 1, 0, 0], [-11, 0, 0, -8, -1, -15, 0, 0], [-37, 0, 0, 14, 12, -15, 0, 0], [0, -5, -14, 0, 0, 0, -26, -7], [0, 27, -14, 0, 0, 0, 11, 21], [-38, 0, 0, 5, 22, -7, 0, 0], [0, 6, 5, 0, 0, 0, -1, 13], [-11, 0, 0, -2, -10, 13, 0, 0], [-11, 0, 0, 10, 13, -9, 0, 0], [0, 17, 1, 0, 0, 0, -13, -39], [0, 6, 6, 0, 0, 0, 11, -29], [0, -14, -16, 0, 0, 0, 0, 34], [-14, 0, 0, -31, -17, 4, 0, 0], [35, 0, 0, 4, -6, 23, 0, 0], [-30, 0, 0, -6, 2, -16, 0, 0], [0, -19, 4, 0, 0, 0, 7, 11], [16, 0, 0, -8, -16, 11, 0, 0], [0, -3, -12, 0, 0, 0, 18, 8], [0, -24, 8, 0, 0, 0, 26, 34], [37, 0, 0, -14, -26, 14, 0, 0], [-7, 0, 0, 23, 20, -18, 0, 0], [-1, 0, 0, -13, -11, 35, 0, 0], [-23, 0, 0, 25, 8, -11, 0, 0], [0, -25, 15, 0, 0, 0, 0, -4], [0, -13, -29, 0, 0, 0, 7, 9], [16, 0, 0, 18, -1, -9, 0, 0], [-5, 0, 0, -6, -24, 3, 0, 0], [0, -30, 5, 0, 0, 0, 5, 1], [0, -9, -27, 0, 0, 0, -5, 16], [0, 18, -11, 0, 0, 0, -8, -8], [-34, 0, 0, 23, -6, 9, 0, 0], [31, 0, 0, -12, -17, -4, 0, 0], [-22, 0, 0, 4, 26, -18, 0, 0], [2, 0, 0, 8, 14, -26, 0, 0], [0, -28, -3, 0, 0, 0, 25, 9], [0, -10, 16, 0, 0, 0, 6, -12], [0, -4, -14, 0, 0, 0, -1, -30], [0, -2, -27, 0, 0, 0, 5, -7], [0, -31, 15, 0, 0, 0, 17, 17], [-16, 0, 0, 6, -2, -20, 0, 0], [23, 0, 0, -22, -17, 6, 0, 0], [0, -11, -1, 0, 0, 0, 13, -22], [0, -17, 20, 0, 0, 0, -17, -17], [13, 0, 0, -19, -18, 0, 0, 0], [0, 15, -2, 0, 0, 0, -11, 4], [-3, 0, 0, 2, -8, 5, 0, 0], [0, -29, -6, 0, 0, 0, 16, 8], [24, 0, 0, 2, 2, 15, 0, 0], [0, 20, -16, 0, 0, 0, 19, 13], [0, -31, 2, 0, 0, 0, 22, 13], [64, 0, 0, 3, -4, 20, 0, 0], [-35, 0, 0, 14, 9, -1, 0, 0], [0, -24, 12, 0, 0, 0, 14, 33], [48, 0, 0, -12, 0, 5, 0, 0], [-40, 0, 0, -8, -9, -7, 0, 0], [-42, 0, 0, 12, 16, -11, 0, 0], [1, 0, 0, -14, 19, -2, 0, 0], [0, 12, 11, 0, 0, 0, -5, -29], [0, 27, -2, 0, 0, 0, -19, -24], [0, 29, -16, 0, 0, 0, -24, -18], [-16, 0, 0, 2, -1, -10, 0, 0], [0, 0, 20, 0, 0, 0, -18, -23], [-5, 0, 0, -15, -7, -19, 0, 0], [-16, 0, 0, 6, 0, -5, 0, 0], [0, 32, 11, 0, 0, 0, -13, 9], [0, -24, -11, 0, 0, 0, 5, -18], [0, -13, 4, 0, 0, 0, 0, 6], [6, 0, 0, 7, 10, 17, 0, 0], [28, 0, 0, 0, -18, 28, 0, 0], [-5, 0, 0, -1, 9, -15, 0, 0], [0, 16, -12, 0, 0, 0, -12, -22], [-5, 0, 0, 7, 3, -2, 0, 0], [0, 38, 7, 0, 0, 0, 1, 1], [0, -10, -5, 0, 0, 0, 16, 43], [15, 0, 0, -26, 7, -16, 0, 0], [0, 11, 20, 0, 0, 0, -20, -1], [0, -5, 5, 0, 0, 0, -15, -11], [0, -24, -8, 0, 0, 0, 2, -28], [-34, 0, 0, 28, -3, 17, 0, 0], [4, 0, 0, 19, -4, 1, 0, 0], [-11, 0, 0, 3, 22, -17, 0, 0], [0, -6, -10, 0, 0, 0, -15, 22], [0, -3, 23, 0, 0, 0, -7, -3], [8, 0, 0, 1, 20, -12, 0, 0], [-23, 0, 0, -15, -5, -29, 0, 0], [0, 3, -5, 0, 0, 0, 1, -49], [0, 24, 1, 0, 0, 0, -1, -2], [0, -31, 11, 0, 0, 0, 6, 21], [17, 0, 0, 3, -7, 11, 0, 0], [0, -41, 4, 0, 0, 0, -6, -19], [8, 0, 0, -10, -17, 7, 0, 0], [0, 29, -7, 0, 0, 0, -31, -19], [0, -3, -5, 0, 0, 0, 2, -10], [0, -2, 7, 0, 0, 0, 6, -9], [-4, 0, 0, 1, 7, 9, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7497_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7497_2_a_cd();".
// 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_7497_2_a_cd(:prec:=8)
    chi := MakeCharacter_7497_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_7497_2_a_cd();".
// 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_7497_2_a_cd( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7497_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![4, 0, -32, 0, 40, 0, -12, 0, 1]>,<5,R![-13, -10, 6, 6, 1]>,<11,R![625, 0, -1000, 0, 306, 0, -32, 0, 1]>,<19,R![32041, 0, -14352, 0, 1990, 0, -88, 0, 1]>],Snew);
    return Vf;
end function;