
// Make newform 1875.2.a.m in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1875_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_1875_2_a_m();".
// 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_1875_2_a_m();".
// 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, 12, 7, -30, -4, 20, -2, -4, 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], [-2, -1, 1, 0, 0, 0, 0, 0], [1, -3, -1, 1, 0, 0, 0, 0], [1, 4, -3, -2, 1, 0, 0, 0], [-1, -3, 7, -1, -3, 1, 0, 0], [-2, -9, 3, 11, -3, -3, 1, 0], [2, 5, -16, -5, 16, -1, -4, 1]];
    Rf_basisdens := [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_1875_a();"
function MakeCharacter_1875_a()
    N := 1875;
    order := 1;
    char_gens := [626, 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_1875_a_Hecke(Kf)
    return MakeCharacter_1875_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, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, -1, 1, 0, -1, 1, 1], [1, -1, 1, -1, 0, 0, -2, -1], [-2, 1, 0, -1, 1, 2, 0, 0], [-1, -2, 0, -1, -1, -1, 1, -1], [-3, 0, 0, 0, -1, 1, -1, -2], [-1, 1, 0, 0, 2, 1, 0, 2], [-1, 0, 1, 1, -1, 1, 2, 0], [-2, -2, 0, 0, -1, -1, -1, 0], [-3, 0, 2, -3, -2, -1, -1, -1], [1, -1, -1, 3, 1, -1, 0, 1], [-4, 0, -1, 2, 0, 0, 0, -2], [-2, 3, 2, 0, 1, 2, 3, 3], [-7, 0, -2, 3, -1, -1, -1, 1], [2, -2, -2, 0, -1, -1, 0, -2], [-3, 0, -2, 4, 0, -4, -2, 2], [-4, 0, 1, 2, -4, -4, 0, 0], [3, 0, -3, -1, 1, 1, 2, 1], [-4, 4, 2, 0, 4, 4, -1, 0], [-3, 3, 3, -2, 3, 3, 3, 2], [0, -1, 2, -3, -1, -1, 3, -1], [2, 0, 2, 2, 1, -1, 2, 1], [0, -2, 3, -2, 0, 0, -3, 0], [-3, 2, -1, -1, -7, 1, -2, -1], [2, -1, 2, -1, 1, 0, 4, 2], [-3, -3, -2, 1, 0, 1, -2, 0], [1, 1, -3, 0, 7, 1, 1, 0], [-4, -2, 2, 0, 0, -3, -4, -2], [1, -2, 2, -5, 0, -1, -1, -3], [1, 6, 1, -1, 4, 2, -2, -2], [3, -1, -4, 1, 2, 2, -4, 2], [-6, 2, 0, -4, -2, 2, -2, -4], [2, 5, 0, -2, 3, 1, 0, 0], [-5, 4, 0, 2, 7, 2, 6, 2], [1, -1, 3, 0, -1, -5, 0, 1], [3, -6, 1, 1, -6, -1, -1, -3], [-3, -1, -4, 6, 0, -2, -2, 0], [-2, -2, -2, 1, -4, -3, -2, -4], [-2, 4, -6, 6, -3, -1, 2, 2], [-6, 4, 4, -4, -7, -1, -1, 0], [4, 6, 6, -8, 5, 3, 0, -1], [0, 7, 2, -1, 3, 4, 0, 6], [-9, -3, 2, 3, 2, -1, 4, 0], [-11, 2, -6, 10, 2, -3, -1, 4], [-5, 4, -2, -2, 2, 7, -3, -2], [-6, 3, -4, -3, -13, 0, -2, 0], [-4, 3, 8, -4, -3, 2, -3, -1], [-1, 2, 0, -2, 2, -3, 1, 2], [1, -3, -4, 0, 0, -1, 8, 2], [0, 3, -6, 2, -6, -4, -2, 5], [-4, -7, 1, 2, -4, -5, 5, -4], [14, 3, 2, 0, 10, -4, 6, 5], [3, 2, -2, -6, 3, 1, 5, -1], [-5, 2, 6, 0, -5, -2, 3, 1], [0, 6, -6, 2, 8, 0, 0, 2], [-5, -5, -3, 2, -1, -5, -3, 0], [-6, 1, -9, 5, -1, -4, 3, 2], [8, -6, -4, 6, 3, -3, 6, 6], [-2, -1, -2, -3, -11, 4, -6, -2], [-12, 2, -4, 4, -12, -8, 2, 4], [5, 5, 1, -2, 11, 7, -2, 1], [3, 11, -5, -1, 9, 5, 0, 8], [-2, 2, -4, 0, 0, 2, 1, 2], [0, 7, 0, -6, 5, 4, 5, 1], [4, -1, 5, 0, 5, 1, 5, 2], [-15, -1, -4, 6, -9, -1, -5, 3], [-8, 8, 0, -5, 0, 7, -2, -6], [-8, -1, 5, 4, -2, 8, 0, 0], [4, -5, 4, 4, -3, -9, -3, -1], [-1, -5, 5, -1, -6, 2, 4, -4], [10, -7, -4, -1, 9, -4, 3, -2], [-3, -3, 4, 0, 7, -1, 7, -3], [13, -9, -5, -4, -3, -1, -1, -4], [3, -1, 2, 7, -8, -5, 4, 4], [3, -4, 1, -1, 3, 3, 2, -8], [4, -1, -7, -3, 5, 10, -5, -2], [-1, -5, 3, 1, -2, -4, -6, -7], [-16, 3, -9, 6, -4, 0, 6, 0], [11, -2, 3, 1, -5, -3, -4, 1], [4, 1, 7, -4, 4, 1, -7, -4], [4, -4, -2, -2, -9, -7, -2, -9], [5, -7, -8, 4, 9, -1, -2, -1], [-6, -2, -6, 2, -3, 0, -2, 2], [3, -2, 2, -2, -5, -2, 5, -5], [5, -10, 3, -3, -7, -3, 2, 1], [2, -5, 6, 1, 5, -2, 2, 2], [2, -2, -6, -6, -4, 8, 0, -5], [2, -6, -1, -4, -10, 0, 3, -2], [15, -4, 0, -3, 5, 3, -9, -1], [12, -12, -6, 2, 12, -6, 0, 3], [14, -3, 0, -9, 3, 4, 9, -2], [5, -12, -1, 3, -2, 0, -4, 1], [-4, 16, 0, -2, -2, 6, 4, 4], [1, -1, 0, 5, -10, 3, -2, -2], [8, 2, -4, 4, 7, 7, 2, 0], [4, -7, -2, 0, 1, -1, -8, 5], [3, -9, 6, 2, -11, -5, 1, 5], [15, -8, -2, 0, 1, 2, 8, 0], [-2, -6, -4, 12, 4, -12, -5, -2], [-10, 9, -12, 1, 1, 0, 1, 1], [-1, 5, 10, -9, 6, 5, -4, -8], [3, -5, -5, 3, 1, -5, -10, -2], [-2, -2, -12, 10, -1, 3, 1, 2], [0, 5, 9, -5, 5, 6, 3, 4], [0, -1, 10, -9, -1, 9, -7, -9], [3, -4, 0, -4, 9, 7, -5, -5], [14, -1, 0, 0, 18, 8, 4, 2], [10, -2, 10, -4, 9, 2, 0, 6], [6, -10, 1, 0, 6, 0, -3, 0], [-8, 11, -10, -5, 9, 4, -3, 2], [-13, 10, 4, 1, 3, 0, 3, 1], [1, -8, 12, -8, 5, 8, 4, -2], [0, 12, 0, 4, 8, 9, 5, 12], [5, 10, 7, -7, -13, 7, -2, 0], [3, -4, -1, -1, -4, 3, -6, 3], [20, -1, -6, -3, 1, -2, -3, 1], [-18, 3, -2, 4, 5, 1, -1, 7], [7, 1, 9, 1, 2, -4, 4, 4], [12, -12, -8, -2, -8, -4, -4, -6], [21, -10, -1, 5, 6, 1, -1, 5], [-9, -2, -2, -9, -5, 1, -15, -11], [-9, 14, 6, -7, 7, 12, -11, -5], [4, 5, 5, 8, 13, -1, -5, 4], [-4, 4, 6, -8, 0, 12, -10, -2], [1, -14, -6, 8, -13, -10, -12, 0], [3, -3, 5, 7, -3, -5, 6, -3], [-5, -16, -1, 1, -8, -7, 12, 1], [11, 9, -1, 0, 13, -3, -2, 3], [9, -7, -9, 2, 1, -12, -2, 0], [9, 3, 12, -13, -6, 3, -12, -4], [0, -1, -7, -12, -7, -3, -1, -6], [-20, -8, -7, 8, -20, -8, 2, -6], [-2, 15, 8, -2, 8, 8, 10, 13], [9, 9, 7, -6, 0, -3, -7, 0], [-4, -4, 2, -5, 6, 12, -6, -10], [8, 0, -5, 10, 10, -4, 10, 8], [-10, 6, 8, -11, 4, 8, -4, -10], [18, -8, -2, -8, 12, -8, -8, -2], [4, 4, 16, -8, 15, -3, -5, -8], [2, -1, 10, -4, -6, 10, 10, -2], [-4, 0, -3, -4, 6, 6, 4, -14], [5, -12, 6, -3, 15, 2, -7, 5], [8, -6, 12, 2, -3, -13, -3, 2], [-4, -5, 0, 2, -11, -7, -14, -2], [-17, -13, -2, 4, -5, -1, 1, -3], [-15, -7, 0, 7, -12, 6, 8, 0], [-2, 9, 1, -4, 4, 1, 1, 0], [12, -3, -2, 8, 3, -1, 15, 1], [-8, -7, 8, -3, -3, -2, 7, -4], [0, -6, 4, 6, -4, -4, -6, -8], [17, -14, -2, -15, 4, 1, -4, -7], [-3, 13, -2, 4, -6, -1, -2, -2], [4, -1, -7, -1, -11, 0, -15, -6], [6, -19, 8, 8, -4, -8, -6, 4], [-12, 0, -2, 6, -7, -10, 2, 8], [-8, -3, 16, -4, 1, 11, 2, 0], [-21, -9, -11, 12, 3, 1, 6, -7], [-13, -8, -1, 1, -9, -9, 16, 4], [6, -22, -8, 4, 10, -7, 2, -2], [10, 10, 2, -6, 28, 10, 0, 0], [1, 7, 4, -10, -1, 13, 0, 1], [9, 2, 9, -15, 12, 0, 12, 2], [-10, 7, 4, -17, -11, 14, -11, -1], [21, -10, 8, 11, 13, -4, -7, 3], [-2, -4, -8, 8, -13, -6, 2, 12], [-18, 6, 2, 4, -14, -8, -1, 0], [5, -2, -16, 4, -7, 0, -10, -2], [-20, 4, 2, 5, -4, -5, 6, 2], [-2, 15, 8, -6, 7, -1, 10, 2], [15, -6, 0, 16, 5, -14, 2, 6], [0, -9, 4, -4, 5, -5, 14, 2], [-16, -6, -5, 20, -12, -12, -1, 0], [-10, -9, -15, 10, 7, -1, 9, -4], [8, 10, -6, 10, 17, -3, 10, 17], [-1, -23, -15, 16, -2, -1, 1, 6], [-16, 0, -4, -8, -28, 0, -10, -20], [-3, 9, -10, 12, 21, 9, -5, 7], [-23, -4, -8, -8, -21, 1, 9, -8], [-1, -10, -6, 9, -4, -11, -12, 5], [14, 1, -4, 6, 0, -10, -4, 4], [-4, 7, 13, -17, 17, 8, 2, -6], [-12, -4, -16, 6, -30, -6, 2, 0], [-2, 6, -6, 0, -4, -4, 8, -2], [22, -22, 4, -2, 8, -2, -18, -3], [-10, -12, -9, 10, 12, -12, -8, 8], [27, 4, -3, -1, 30, -7, -2, 5], [4, 2, 2, -4, 2, 4, 20, 6], [-19, 13, 5, -9, -3, 15, 14, 11], [2, 15, 10, -9, 15, 10, -9, -10], [-10, 1, -4, -9, -19, 3, 5, 1], [-24, 14, -2, 10, -22, -1, 3, 10], [-25, 18, 1, 1, -10, -12, 10, 10], [-11, -8, 6, 3, -7, -5, -9, 3], [2, 10, -8, 8, 22, 1, 2, 2], [23, 8, 14, -4, 14, -4, 2, 2], [-13, 19, 15, -8, -9, -3, 2, 1], [-2, 2, 2, 5, -8, -2, 10, 0], [19, -22, 0, -4, -9, -6, -1, -5], [-4, -19, 0, 14, 10, -22, -4, 3], [-9, -4, 8, -16, -1, 2, -8, -2], [-2, 4, -4, -4, -14, 8, -1, -6], [-19, -4, 2, 4, -9, 2, 8, -4], [-5, 5, 5, -15, 6, 24, 0, -7], [27, -3, -4, -8, 0, 1, 0, 6], [-10, 14, 4, -8, 23, 25, -11, -2], [-3, 4, 12, -7, -13, -5, -5, -3], [-2, 2, -4, 14, 22, 6, -4, 7], [3, 3, 5, -14, 3, 17, -7, -6], [-11, 8, -4, -3, -6, -5, 2, 15], [10, -2, -4, -2, -24, -6, -10, 6], [-14, 1, 7, 7, -23, -12, 6, -4], [30, -4, 0, 5, 26, -3, 14, 0], [12, 10, -2, -10, 20, 10, 10, 6], [0, -15, 5, -10, 3, 6, -14, -10], [-11, 20, -5, 9, 14, 1, -7, -1], [1, -15, 11, -11, -10, -2, 22, 0], [14, -6, -8, -2, 8, -1, -14, -6], [-5, 4, -16, 1, 23, 12, -9, 1], [-2, -7, 1, 10, -13, -12, 2, 0], [5, -14, 4, 6, -12, -2, 2, 0], [-17, 22, 1, -7, 20, 18, -2, -5], [-10, 7, 11, -5, -23, 6, 6, -8], [11, 15, -10, 10, 8, -12, -4, 6], [4, 1, 3, -12, 2, 3, -3, 8], [7, 6, -10, 10, 35, -1, 5, 3], [38, 1, 0, 2, 10, -4, 10, 6], [6, 1, 21, -13, 15, 6, -16, -4], [8, -2, -4, -2, 12, 2, -12, -10], [0, -15, -8, 1, 7, -8, -12, -8], [7, 0, 8, 12, 26, 4, 6, 16], [-3, -4, 10, -12, 4, 4, 4, -14], [-4, -6, -16, 16, -17, -3, -10, 5], [0, 20, 10, -2, -4, 14, 9, 4], [4, -2, -12, 7, -12, -3, 0, 10], [1, -3, 3, 6, -16, -8, 2, -2], [-10, -3, 0, 0, -25, 8, -3, -9], [-5, 1, -11, 17, 3, 5, 0, 2], [-7, 10, -3, 11, -13, -13, 8, 2], [4, 23, 6, -4, 7, 6, -1, 7], [3, -18, -8, 0, -11, -5, -7, -14], [-9, 4, 4, 9, -26, 7, -7, -7], [-3, 16, -2, -10, 16, 10, 6, -6], [9, -3, 12, 3, -6, -7, 6, 0], [-10, 13, 4, 0, -2, 12, -14, 5], [9, 2, -4, 13, 8, -1, -11, -5], [12, -4, -8, 22, -4, -6, -4, -3], [-9, 6, -4, -4, 1, 1, -23, -10], [6, -21, 16, -9, -29, -10, -13, -19], [-2, -5, 15, -3, -17, 2, 3, -4], [-18, -12, 20, -6, -11, -9, -4, 2], [-18, 16, 8, 0, -16, 4, -4, -10], [-23, 5, -9, 8, 1, 0, 10, 10], [19, -9, 14, -9, -4, 4, -22, -14], [1, -9, 3, -3, -15, 1, 2, -5], [9, -9, 15, -4, -5, -11, -7, -14], [9, 9, -10, -8, 1, 15, 19, -1], [25, -1, 4, 6, 6, -1, -6, -2], [11, 12, -2, 3, 4, 1, 14, 13], [4, -5, 6, -19, -11, 10, 6, -10], [4, -9, -12, 17, 11, -2, 17, 7], [18, -3, -5, -20, -19, 2, 0, -10], [-41, 20, 1, 11, -4, 5, 12, 7], [6, 2, -2, 12, 0, -15, 4, 20], [3, 5, 21, -18, 1, 4, 8, 4], [-28, -2, 10, -14, -16, -6, -12, -16], [-22, 0, 20, -4, -14, -4, -10, 3], [-12, 14, 5, 12, -20, 2, 15, 8], [9, 4, -2, -21, 1, 4, -1, -5], [9, -3, -1, 6, 6, -17, 11, 16], [22, -4, -18, 0, -16, -8, -8, -6], [29, 20, 5, -11, 22, 9, 7, 3], [-1, -3, 25, 0, 3, 2, -4, -10], [1, 6, -1, -11, -24, -7, -4, -5], [25, -4, -22, 1, 12, -7, 13, 7], [18, -2, 4, 12, 20, 5, -12, -2], [-12, -10, 8, -12, -30, -17, 19, -4], [-25, 10, -6, 16, -12, 1, -1, 6], [15, 4, 13, -13, 11, 3, -6, -11], [21, 10, 0, 14, -1, -4, -5, 7], [6, 14, -2, -2, 12, 11, 17, 8], [-3, -4, 14, -10, -21, -1, -15, -5], [4, -3, -7, -4, 27, 7, 1, 4], [35, -4, 6, 1, 12, -13, 0, -3], [8, 13, 4, 10, -6, -6, 4, 5], [-15, 25, -5, 10, 19, -3, -4, 3], [-10, -5, 10, -13, -5, -6, -19, -15], [-13, 10, 2, 4, -3, 2, -6, 8], [-12, 15, 16, 4, 22, 18, -6, 9], [-36, 7, -2, 8, -1, 5, -2, 2], [20, -2, -8, 5, -6, -4, 18, 4], [-7, 15, 6, 8, 22, -2, 12, 6], [39, 7, -1, -1, 32, -2, 12, 15], [-9, -11, 22, -8, 9, 11, 5, -7], [-27, 7, 5, -1, -21, -5, -6, 1], [-18, 16, 16, -8, 0, 1, -3, -18], [-28, -3, -4, -1, -35, -7, 17, -9], [23, -2, 5, 13, 15, -17, -4, 12], [-10, 13, -5, 21, 1, 2, 12, 10], [11, -9, 4, 16, 7, -23, 3, 7], [-6, 9, 8, -26, 13, 30, -9, -7], [-21, 8, -6, 2, 5, -10, -10, 0], [-13, 14, -18, 21, -13, -15, -5, 5], [-6, 8, 14, -16, -9, -2, 12, 6], [-22, -9, -5, 7, -27, 4, -21, -6], [15, 0, 0, -8, -7, 11, 9, -11], [9, 12, 10, 2, -6, 8, -4, 6], [-16, 24, -4, 0, 18, 6, -14, -14], [-31, 5, -2, -4, -17, -11, 13, 11], [13, -13, -8, 9, 16, -7, -12, -10], [-12, 13, -18, 6, -14, -6, 6, -1], [-39, 1, -5, -9, 2, 14, -22, -11], [8, -15, 6, 1, -13, -8, -6, -24], [16, 0, 2, -8, 22, -7, 20, 12], [15, 20, 0, -4, 33, 7, 3, 10], [12, 15, 2, -6, -4, 6, 4, -11], [6, 23, 4, -6, -1, 13, -10, -10], [-9, -8, -16, 33, 10, -11, 9, 5], [-15, -23, -23, 3, 2, -2, 0, -12], [-13, -1, -9, -4, -30, -10, -6, -16], [17, 10, -11, 3, 4, -7, 2, -7], [-5, 4, -15, 25, -5, -5, -2, 7], [19, -19, 0, 16, 11, -3, 6, 5], [0, 4, 8, -19, 8, 12, -14, 6], [-3, 11, -5, 4, 0, -7, 15, 10], [16, 7, -21, 4, 12, 2, 10, 4], [13, -5, 4, 4, 21, 5, -6, 7], [-13, -7, 10, -6, 10, 16, -8, 0], [-22, -7, 6, 6, -15, -4, 3, -3], [-27, 30, 6, 12, 3, 12, 16, 24], [-14, 3, 2, -9, -1, -4, -7, -7], [2, -20, 16, -12, 18, 3, 3, 6], [16, -20, -12, 23, 14, -4, 20, 6], [2, -5, 13, -20, -29, 0, 2, -14], [-23, 13, -30, 16, 6, -7, 18, 24], [-13, 22, -30, 4, 12, 3, 17, 10], [-6, -4, -12, -8, 6, 13, -8, -26], [7, 17, -15, 14, 8, -11, 5, 18], [-30, -10, -5, 10, -30, -4, -10, -16], [-8, 2, -14, 6, -8, 2, 3, 12], [-33, 4, -4, 4, -27, 10, -1, -3], [9, 12, -5, 5, 7, 3, -12, 2], [-20, -2, 0, 4, 18, 4, -6, 14], [-2, -10, 6, 6, -12, 9, 20, 8], [-2, -3, 4, 0, -52, -8, 0, 2], [10, -3, 19, 14, 2, -8, 4, 10], [-18, 17, 21, -5, 7, 18, -16, 2], [16, -3, -14, 2, -17, -7, -4, 5], [-14, 10, 6, -25, 18, 1, -4, -10], [19, 0, -2, 6, -4, -24, 10, 16], [-9, -14, 1, 19, -16, -1, -1, 1], [-3, -35, 9, -1, -2, 2, 0, -2], [-21, 8, 11, -15, -16, -19, 5, 3], [11, -6, 2, 4, 15, 6, -2, -2], [23, 1, 8, -20, 18, 16, -6, -10], [49, -5, -3, -11, 4, 4, -4, 2], [16, -3, 8, -16, 10, -8, -4, 1], [-2, -10, 20, -7, 6, -11, 6, 6], [0, 29, -14, 15, 25, -1, 13, 23], [-44, 11, 8, 3, 15, 18, 4, 8], [-20, 8, -2, 18, -19, -3, 14, 10], [-8, -5, 8, 15, 9, -10, -3, 1], [10, 4, 2, -8, -3, 7, 12, 13], [-22, 18, 5, -6, -22, 6, -10, -24], [-7, -4, 8, -27, -20, 5, -7, -13], [-41, 3, 8, 10, -4, 9, 0, -8], [41, -26, 8, -11, 0, -1, -23, -15], [-17, 3, -29, 20, -10, -16, 32, 16], [27, -4, 5, 5, 5, -27, 2, 11], [-11, 33, 9, -18, 6, 15, 1, 6], [-46, -18, -12, 19, -42, -1, -8, -2], [4, 9, 8, -16, -18, 12, 2, -3], [-19, -1, 17, 2, -13, 8, 24, -2], [17, -4, 23, 9, 34, 11, 15, 15], [-11, 4, -1, 11, -3, 13, 8, 12], [41, -19, -19, -5, 13, -11, 22, 9], [-14, 28, -6, 10, 12, -10, -7, 4], [-22, -10, -14, 14, -17, -3, 8, 2], [-11, 1, 11, -24, -13, 19, -13, -5], [47, -19, -13, -3, -13, -17, 0, -10], [-19, -7, -12, -3, 2, 7, 12, -18], [32, 0, -6, -14, 14, 18, 12, -4], [10, 7, -12, 5, -23, -2, -7, 11], [-12, -22, -28, 30, -20, -19, -19, -12], [-5, -1, -4, -1, 6, -11, -10, -2], [1, -9, -1, -10, -4, 6, -20, -22], [-33, 18, 0, 13, 10, 17, 7, 7], [-7, -27, -21, 26, -7, -19, -14, -5], [15, -6, -8, 1, 25, -10, -7, -5], [-13, -25, 9, -12, -4, 10, -10, -10], [9, -13, 8, -6, -28, -10, -10, 0], [-12, 3, -30, 24, -20, -22, -2, 13], [-33, 4, -8, -1, -10, 17, 3, 15], [5, -4, 11, -25, 8, 16, -24, -13], [32, -4, 16, -14, 8, 0, 13, 10], [1, -8, 22, -30, 6, 4, -14, -10], [-1, 16, 3, -23, 18, 30, 14, -16], [11, -12, 10, 12, 2, -3, -5, -2], [-6, -11, 16, -22, 7, 5, 2, -16], [5, 4, 2, -4, -12, -6, -14, 4], [5, -14, -6, -12, 1, -15, -27, -21], [7, -18, 29, 7, 14, 25, 15, -3], [-13, 8, -12, 11, -33, -5, -7, 1], [42, -15, -8, -4, -9, -9, 0, -4], [-65, 6, 14, -2, -39, -4, -16, -2], [-5, -19, 9, -6, -9, -21, -28, -15], [-30, 23, -14, 10, 24, 24, 10, -4], [-31, 24, 9, -5, -20, -5, -3, 3], [-6, 2, -12, 12, -36, 6, -8, 0], [8, 0, 33, 6, 16, 8, -4, 8], [-24, -23, -2, -1, -11, -4, 9, -1], [-22, 11, 10, 9, -5, 8, 9, 7], [-1, 8, -22, 8, 33, -1, 13, 20], [-19, 6, -12, 17, -20, -19, 15, 11], [-5, 1, -17, 25, 28, 0, 4, 19], [-7, 19, -9, 7, -6, 10, 28, 8], [-42, 7, -4, 5, -11, 0, -20, -22], [-9, -4, 16, 2, 13, 6, 1, -11], [23, 17, 26, -32, 36, 39, 4, 2], [-21, 21, -29, 3, 25, -1, 4, 19], [-4, -4, 7, -4, 16, -8, -8, 4], [-2, 6, 30, -8, -12, -7, 8, 16], [-26, -14, -10, 6, -38, -4, -12, -20], [10, -30, -29, 18, -2, -28, 18, 0], [14, -30, 10, -5, -8, -14, -24, -22], [-1, -44, 16, 3, -17, -4, -1, -9], [-6, -5, 30, -14, -21, -21, -6, -6], [39, -24, -12, -10, 0, 7, 3, 4], [22, 2, 12, -4, 14, 2, -24, -19]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1875_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1875_2_a_m();".
// 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_1875_2_a_m(:prec:=8)
    chi := MakeCharacter_1875_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_1875_2_a_m();".
// 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_1875_2_a_m( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1875_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, -12, 7, 30, -4, -20, -2, 4, 1]>],Snew);
    return Vf;
end function;