
// Make newform 1875.2.a.p in Magma, downloaded from the LMFDB on 19 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_p();".
// 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_p();".
// 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 := [[0, 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, 0, 1, 0, -1, 0, 1], [0, 1, 0, 1, 1, 0, -1, 1], [2, 1, 1, -1, 1, 2, 0, 0], [3, -2, 1, -1, -1, -1, -2, -1], [-1, 0, 0, 0, 0, -1, 1, 2], [2, 1, 0, 0, 1, 1, 2, 2], [0, 0, 2, -1, -2, -1, 2, 0], [-4, 2, 0, 0, 0, 1, -1, 0], [5, 0, 1, -3, 0, -1, 0, -1], [2, 1, 2, -3, 1, 1, -1, -1], [0, 0, -1, 2, -2, 0, -2, -2], [2, 3, -2, 0, 4, 2, 0, 3], [5, 0, -1, 3, 1, -1, 2, 1], [2, 2, -2, 0, 2, 1, 2, 2], [-1, 0, 2, -4, 2, 4, -4, -2], [2, 0, -3, 2, 0, -4, 0, 0], [1, 0, -4, 1, -1, -1, 1, -1], [0, 4, -2, 0, 0, 4, 1, 0], [-4, -3, 1, 2, -2, -3, 1, -2], [3, -1, 1, -3, -1, -1, -4, -1], [3, 0, 4, -2, 1, 1, 1, -1], [4, -2, -1, -2, 0, 0, 3, 0], [-1, -2, -2, 1, -7, -1, -1, 1], [2, -1, -1, -1, 1, 0, -2, 2], [5, -3, 1, 1, 1, 1, 2, 0], [2, -1, -3, 0, 6, -1, 1, 0], [4, -2, -2, 0, -5, -3, 2, -2], [3, -2, 3, -5, -4, -1, -2, -3], [8, -6, 0, 1, 4, -2, 0, 2], [-1, -1, 3, 1, 2, 2, 6, 2], [-4, -2, -4, 4, 0, -2, 2, 4], [5, -5, -2, 2, 2, -1, 0, 0], [-1, -4, 2, -2, 3, -2, 4, -2], [1, -1, -3, 0, -3, -5, 1, 1], [-1, -6, -2, 1, 2, -1, -2, -3], [-2, -1, -2, 6, -2, -2, 2, 0], [-1, -2, 1, 1, -3, -3, -2, -4], [6, -4, 0, -6, -4, 1, 0, -2], [-6, -4, 0, 4, -6, 1, -1, 0], [3, -6, -2, 8, 3, -3, 1, 1], [0, 7, -1, -1, 7, 4, 6, 6], [9, -3, -5, 3, -3, -1, -4, 0], [-3, -2, 4, -10, 1, 3, -5, -4], [-1, -4, -4, 2, -3, -7, -1, 2], [6, 3, 7, -3, 13, 0, 2, 0], [4, 3, -4, -4, 4, 2, 2, -1], [-3, -2, -2, 2, 3, 3, -1, -2], [4, -3, 4, 0, 1, -1, -6, 2], [0, -3, -4, -2, -7, 4, -7, -5], [-5, 7, 3, -2, 5, 5, 9, 4], [12, -3, 2, 0, 9, 4, 1, -5], [0, 2, 8, -6, -3, 1, -6, -1], [4, 2, -6, 0, 4, -2, -2, 1], [6, -6, -4, -2, 6, 0, -2, -2], [-8, 5, -1, -2, 4, 5, -3, 0], [2, 1, 4, 5, -1, -4, -1, 2], [2, 6, 2, -6, 0, 3, 0, -6], [4, -1, 5, -3, 13, 4, 4, -2], [10, 2, 0, 4, 8, -8, 2, 4], [-7, 5, 1, -2, -3, 7, 3, 1], [5, -11, -6, 1, -4, -5, -8, -8], [2, 2, 4, 0, 4, 2, 1, 2], [0, 7, 6, -6, 0, 4, -4, 1], [1, 1, 5, 0, 2, -1, 3, -2], [13, -1, -2, 6, 11, -1, 8, 3], [-1, 8, 5, -5, 1, 7, -4, -6], [-5, 1, 9, -4, -10, -8, 0, 0], [-4, -5, -8, 4, -7, -9, 2, -1], [-3, 5, 4, 1, -4, -2, 8, 4], [-4, -7, 5, -1, -15, -4, -5, -2], [3, -3, -4, 0, -11, -1, -10, -3], [4, 9, -9, 4, 2, 1, 3, 4], [-5, -1, -9, 7, 7, -5, 0, 4], [6, 4, 0, 1, 8, -3, 10, 8], [-2, -1, 10, -3, 3, 10, 3, -2], [2, 5, 4, -1, 9, 4, 1, 7], [-7, -3, -3, -6, -4, 0, 6, 0], [9, 2, 4, -1, -3, 3, -5, -1], [5, -1, 3, 4, 7, -1, -3, 4], [7, 4, -4, 2, 7, 7, 7, 9], [-3, -7, 4, 4, -11, -1, 1, -1], [-8, 2, -4, -2, -5, 0, -4, -2], [-4, -2, 0, -2, -2, -2, -10, -5], [-9, 10, 0, 3, -5, 3, 1, -1], [4, -5, -7, 1, -5, -2, 0, 2], [-1, 2, -12, 6, -7, -8, 5, 5], [6, -6, 5, -4, 8, 0, -5, -2], [-9, -4, 3, -3, -3, 3, 8, -1], [-1, 12, -4, -2, 15, 6, -3, -3], [-4, -3, 9, -9, -1, 4, -11, -2], [-5, 12, 2, -3, -3, 0, -5, -1], [6, -16, -2, 2, -12, -6, 0, -4], [-7, -1, -5, 5, 11, 3, 0, -2], [14, -2, 0, -4, 0, -7, 2, 0], [-8, 7, -2, 0, -3, 1, -13, -5], [9, -9, -8, 2, 11, -5, 4, 5], [7, 8, -2, 0, -1, -2, 8, 0], [-6, -6, -8, 12, -18, -12, 3, -2], [1, 9, 11, 1, 0, 0, 0, 1], [-3, 5, -1, -9, -9, 5, -4, -8], [3, 5, -2, -3, 8, 5, -8, 2], [4, 2, -2, -10, -6, -3, -1, -2], [4, 5, -4, -5, 5, 6, 1, 4], [1, -1, -1, -9, 1, 9, -2, -9], [0, -4, 4, -4, -7, 7, 0, -5], [11, 1, 0, 0, 8, -8, 2, -2], [-2, 2, 6, 4, 1, -2, -6, -6], [4, -10, -1, 0, -6, 0, 3, 0], [4, 11, 15, -5, -3, 4, 5, 2], [3, 10, -5, 1, -2, 0, -2, 1], [-13, 8, 4, 8, -1, -8, 6, 2], [4, -12, 4, -4, -13, -9, -7, -12], [8, -10, 0, 7, -20, -7, -2, 0], [5, -4, 2, -1, 10, 3, 9, 3], [-15, -1, 9, -3, -2, -2, 4, 1], [18, 3, -2, 4, 3, 1, 8, 7], [5, -1, 10, -1, 2, 4, 0, -4], [4, 12, -10, 2, 2, 4, 2, 6], [-11, -10, -4, 5, 0, 1, 6, 5], [9, -2, 11, -9, -5, 1, 4, -11], [-3, 14, 1, -7, 0, 12, 6, -5], [13, -5, 13, -8, 10, 1, -9, -4], [-6, -4, -2, 8, -10, -12, -8, 2], [-5, 14, 2, -8, -3, 10, -12, 0], [10, 3, 12, -7, 5, 5, 9, 3], [21, -16, 0, 1, 2, -7, -11, 1], [-17, 9, 1, 0, -13, -3, 5, 3], [4, 7, -7, -2, 13, 12, -2, 0], [-3, 3, 1, -13, 5, 3, 8, -4], [-7, 1, -19, 12, 2, 3, 5, 6], [14, -8, -1, 8, 6, -8, -8, -6], [-2, -15, 6, 2, -13, -8, -3, -13], [12, -9, 1, 6, 3, 3, -7, 0], [3, -4, 3, -5, -4, 12, -4, -10], [-10, 0, -5, 10, -6, -4, -2, 8], [5, 6, 3, -11, -6, 8, -6, -10], [4, 8, -10, 8, 22, 8, -6, 2], [8, -4, 8, 8, 26, 3, 3, 8], [-1, 1, 6, 4, -14, -10, 12, 2], [-6, 0, 7, -4, -14, 6, -18, -14], [15, -12, -3, -3, -8, 2, 12, 5], [2, 6, 14, -2, 8, 13, -5, -2], [-5, 5, 2, -2, -2, 7, -12, 2], [23, -13, -2, 4, 1, -1, -4, -3], [15, -7, -7, 7, 18, 6, -8, 0], [3, -9, -3, 4, 3, -1, 1, 0], [-16, -3, -6, 8, -3, -1, -14, 1], [14, -7, -5, -3, -3, -2, -11, -4], [8, 6, 10, -6, 8, 4, 2, 8], [5, -14, 17, -15, -10, 1, -3, -7], [-16, 13, -2, 4, 3, -1, 0, -2], [-8, -1, 8, -1, 5, 0, 9, -6], [-9, 19, 16, -8, 0, 8, -10, -4], [-14, 0, 4, -6, -5, 10, -6, -8], [-15, 3, 12, 4, -10, -11, 2, 0], [11, -9, -1, 12, -9, 1, -13, -7], [-24, 8, 0, -1, -4, 9, 12, -4], [10, -22, 4, 4, -19, -7, -4, -2], [-14, 10, 4, -6, -18, 10, 0, 0], [3, 7, 6, -10, 15, 13, 1, 1], [-6, -2, -6, 15, 10, 0, 10, -2], [19, 7, 13, -17, 24, 14, 10, -1], [-19, -10, -19, 11, -14, -4, 10, 3], [-10, 4, 0, -8, -19, 6, -10, -12], [8, 6, -6, 4, 6, -8, 1, 0], [9, 2, -12, -4, -5, 0, -8, 2], [13, 4, -7, 5, 1, -5, -4, 2], [5, -15, 2, 6, 6, 1, 8, -2], [19, 6, 16, -16, 13, 14, -4, -6], [-15, 9, 0, 4, 8, 5, 12, -2], [2, -6, -15, 20, 0, -12, 1, 0], [-5, 9, -5, -10, 12, 1, 13, 4], [11, -10, 4, -10, 3, 3, -7, -17], [-14, 23, 1, -16, -7, 1, -5, -6], [-4, 0, -12, 8, -8, 0, 10, 20], [-11, 9, -2, 12, -5, 9, 12, 7], [-27, 4, -16, 8, -14, -1, 17, 8], [7, -10, -3, 9, -2, -11, 17, 5], [17, -1, 2, -6, 6, 10, -8, -4], [8, 7, 4, -17, -15, 8, -8, -6], [10, -4, 10, 6, 24, -6, -2, 0], [-6, 6, 6, 0, -2, -4, -10, -2], [1, 22, 2, 2, 13, 2, -15, 3], [20, -12, -1, 10, -16, -12, 16, 8], [-25, 4, 4, -1, -32, -7, 7, 5], [-4, -2, -2, 4, -8, -4, 14, -6], [-26, -13, -4, 9, -29, -15, 3, -11], [-18, 15, -1, -9, -15, 10, -1, -10], [19, 1, 13, -9, 23, 3, -4, 1], [-10, -14, 8, -10, -31, 1, -7, -10], [-16, -18, 2, -1, -8, 12, 0, -10], [19, -8, -9, 3, 5, -5, 12, 3], [-18, 10, 0, 8, -19, 1, 0, 2], [25, -8, 10, 4, 16, 4, 0, -2], [3, 19, -7, -8, 7, -3, -1, 1], [-5, 2, -7, 5, 6, -2, -10, 0], [2, -22, 4, -4, -2, -6, -4, -5], [-12, 19, 14, -14, 29, 22, -7, -3], [-27, 4, -8, 16, -1, -2, -6, 2], [-4, 4, 8, -4, 16, 8, -5, -6], [-15, 4, 6, -4, -7, -2, 12, 4], [-8, -5, -10, 15, -11, -24, 7, 7], [-10, -3, 12, -8, 7, 1, 6, 6], [-2, -14, -4, 8, 0, -25, -9, 2], [3, 4, -5, -7, 5, -5, 2, -3], [7, -2, 10, -14, 9, -6, -11, -7], [-2, -3, -9, 14, -8, -17, -1, 6], [21, 8, 7, -3, 16, -5, 13, 15], [0, 2, -6, 2, -24, 6, -16, -6], [2, 1, -14, 7, 7, -12, -10, -4], [-31, -4, -5, 5, -29, -3, -14, 0], [6, -10, -12, 10, 4, -10, 4, -6], [-15, 15, -5, 10, 7, -6, -4, 10], [-19, 20, -4, 9, -14, 1, 6, -1], [-25, 15, 0, 11, -8, 2, 22, 0], [-12, -6, 10, -2, -15, -1, 8, -6], [1, 4, 15, 1, -10, 12, 10, 1], [1, 7, 11, -10, -1, 12, 2, 0], [-3, 14, 10, -6, -10, 2, 2, 0], [3, -22, -6, 7, 7, -18, 3, 5], [0, 7, -6, -5, 21, 6, -14, -8], [-30, 15, 0, 10, -14, -12, 10, 6], [-15, -1, -9, 12, -9, -3, -11, -8], [-20, 6, 0, 10, -33, -1, -2, 3], [35, -1, 2, -2, 8, 4, 4, -6], [2, 1, -8, -13, -13, 6, 12, -4], [14, 2, -6, 2, 20, -2, -2, 10], [6, -15, 7, 1, -23, -8, 4, -8], [3, 0, 20, -12, 6, -4, -10, -16], [-5, 4, -2, 12, 14, -4, 18, 14], [1, 6, 0, -16, -19, 3, -15, -5], [-14, 20, -8, -2, 22, 14, -5, 4], [1, -2, 5, 7, 19, -3, 10, 10], [6, 3, 9, -6, -6, 8, 4, 2], [4, -3, 0, 0, 24, 8, -6, -9], [11, -1, 6, -17, -4, -5, -2, -2], [12, -10, 8, -11, -2, 13, 6, -2], [-16, 23, -2, -4, 6, 6, 8, 7], [-1, 18, -8, 0, 8, 5, 7, 14], [-11, 4, -13, 9, 26, 7, 0, -7], [9, -16, -12, 10, 12, -10, 12, 6], [-9, -3, -15, 3, -1, -7, -6, 0], [-2, -13, 4, 0, -19, -12, -19, -5], [-29, 2, -9, 13, -14, -1, 6, -5], [33, 4, 14, -22, 5, 6, -1, 3], [3, -6, -8, 4, 10, -1, -13, 10], [5, -21, -7, -9, 0, -10, -6, -19], [6, -5, -12, -3, 15, 2, -7, -4], [-38, 12, 14, 6, -4, 9, -6, -2], [-8, 16, -8, 0, 10, 4, -6, -10], [-20, -5, -1, -8, -9, 0, 0, -10], [-15, -9, -5, -9, -6, 4, 8, -14], [-6, 9, 0, 3, -11, -1, 7, 5], [10, 9, 11, 4, 20, 11, 7, 14], [-11, 9, 18, -8, 13, 15, -20, -1], [-32, -1, -10, 6, -9, -1, 4, -2], [-13, 12, -1, 3, 10, 1, -1, 13], [10, -5, 13, -19, 11, 10, -16, -10], [-5, -9, -5, 17, -6, -2, -10, 7], [5, 3, -25, 20, -11, -2, 10, 10], [17, 20, -12, 11, 16, 5, -5, 7], [0, 2, -10, 12, 5, -15, 16, 20], [-14, -5, 3, 18, -7, -4, 4, -4], [-28, 2, -4, 14, 6, 6, 4, 16], [-29, 0, 16, 4, -13, 4, -13, -3], [-6, 14, -17, 12, 30, 2, -7, 8], [3, 4, 23, -21, -2, 4, -4, -5], [-4, 3, 5, -6, 7, 17, -5, -16], [-24, -4, 18, 0, 2, -8, 2, -6], [-35, 20, 6, -11, -10, 9, -4, 3], [6, 3, 25, 0, 11, -2, 6, 10], [-1, 6, 12, -11, 12, -7, -1, -5], [-15, -4, 21, 1, -12, -7, -6, 7], [-30, -2, -16, 12, -17, 5, 10, -2], [-30, 10, -4, 12, -9, 17, 23, 4], [-5, -10, 10, -16, -19, -1, -7, -6], [17, -4, 0, 13, 19, -3, 5, 11], [-38, 10, -14, 14, 4, -4, 12, 7], [10, -14, -4, 2, -7, -11, 9, -8], [12, -4, -4, -10, 15, -1, 10, -5], [-7, 3, -11, 4, 16, -7, -3, -4], [-35, -4, -7, 1, -28, -13, -3, -3], [26, -13, 14, -10, -5, 6, -1, -5], [-17, 25, -5, 10, -19, -3, 7, 3], [13, -5, 3, -13, -16, -6, 4, -15], [-7, -10, 6, -4, -13, -2, -14, -8], [-2, -15, 20, -4, -5, -18, -15, -9], [-23, -7, 6, -8, -8, -5, -4, -2], [-19, -2, 3, 5, 6, -4, -14, 4], [-10, 15, -14, 8, -18, -2, -6, 6], [30, -7, -2, 1, 19, 2, -3, -15], [21, -11, -14, -8, -5, 11, -12, -7], [-22, -7, 4, 1, -17, 5, -7, -1], [8, -16, 8, 8, 17, -1, 15, 18], [23, -3, 5, -1, 19, -7, -26, -9], [22, 2, 18, -13, 20, 17, -16, -12], [-14, 13, -16, 21, 11, 2, -2, 10], [-11, -9, -20, 16, -23, -23, 4, 7], [16, 9, 18, -26, 10, 30, 2, -7], [-11, -8, -4, -2, 15, 10, -10, 0], [-17, 14, -3, 21, 3, -15, 10, 5], [-20, -8, -2, 16, -13, 2, 6, -6], [18, -9, -2, 7, 25, 4, 15, -6], [-18, 0, 8, -8, 7, 11, -20, -11], [17, -12, 12, -2, -20, -8, -10, -6], [22, -24, -4, 0, 26, -6, 0, 14], [41, 5, 6, -4, 17, -11, -2, 11], [-19, -13, -1, 9, -33, -7, 2, -10], [8, -13, -12, -6, -7, 6, 7, 1], [-36, -1, -14, 9, -1, -14, -11, 11], [-18, -15, -7, 1, -19, -8, -18, -24], [4, 0, 6, -8, -17, -7, -8, 12], [21, -20, -4, 4, 16, -7, -7, -10], [32, -15, -4, 6, 1, -6, 15, 11], [33, -23, -2, 6, -4, -13, 0, 10], [-11, -8, -17, 33, -16, -11, -4, 5], [-23, 23, -20, -3, 16, 2, 12, 12], [-2, 1, -13, 4, -4, 10, 10, 16], [-37, 10, 8, 3, -18, -7, -9, -7], [17, -4, 10, -25, -7, 5, -9, -7], [-11, -19, -16, 16, -9, -3, -1, 5], [21, 4, 11, -19, 10, 12, 20, 6], [2, -11, -1, -4, -3, 7, 5, -10], [23, -7, -17, -4, 6, -2, 6, -4], [-5, -5, -8, 4, -9, 5, 13, 7], [26, -7, -4, -6, 6, 16, 8, 0], [20, -7, -12, 6, 8, -4, -6, -3], [-9, -30, 18, -12, -33, -12, -8, -24], [13, 3, 7, -9, -10, -4, 0, -7], [-36, 20, 4, 12, 9, -3, -3, -6], [-13, -20, -11, 23, -12, -4, -14, 6], [-9, 5, -7, 20, -15, 0, 16, 14], [18, 13, 14, 16, 11, -7, 6, 24], [3, -22, -26, -4, -1, -3, 7, -10], [-8, -4, 20, -8, -19, 13, -18, -26], [20, -17, -1, -14, 1, 11, -13, -18], [14, -10, -5, 10, 10, -4, -6, -16], [12, 2, 8, 6, 22, 2, 9, 12], [22, 4, 0, 4, 34, 10, -2, -3], [24, -12, 0, -5, 2, -3, -14, -2], [-32, 2, 4, -4, 0, -4, -20, -14], [14, -10, -12, 6, 29, 9, -12, 8], [-7, 3, 4, 0, -46, 8, -2, -2], [11, 3, 33, -14, 0, 8, -6, -10], [8, 17, -16, -5, 13, 18, 18, 2], [10, 3, -12, -2, -15, 7, -9, -5], [19, 10, 19, -25, -27, 1, -6, -10], [9, 0, 4, -6, 4, 24, -6, -16], [5, -14, -20, 19, 16, -1, 2, 1], [-37, 35, 8, 1, -2, -2, 2, 2], [31, 8, 4, -15, 0, -19, -2, 3], [11, 6, 6, -4, 11, -6, 0, 2], [-14, 1, 12, -20, -12, 16, -4, -10], [31, 5, -14, 11, -2, -4, -6, -2], [-4, 3, -8, 16, 17, 8, -5, -1], [25, -10, -13, -7, -11, -11, 0, 6], [-21, 29, -1, 15, -3, -1, 10, 23], [38, 11, -11, 3, 11, 18, 4, 8], [-4, -8, 16, -18, -26, 3, 4, -10], [-1, -5, -23, 15, -18, -10, 4, 1], [-7, -4, -6, 8, -23, -7, -1, -13], [-14, 18, 1, -6, 4, 6, -14, -24], [25, -4, 19, -27, 12, 5, -6, -13], [20, 3, -18, 10, 5, 9, -8, -8], [-19, -26, 3, -11, -16, -1, 8, -15], [-10, -3, -9, -20, -10, 16, 16, -16], [17, 4, 10, -5, 21, 27, -9, -11], [-2, -33, -9, 18, -15, -15, -5, -6], [43, -18, -7, 19, 39, -1, 6, -2], [0, -9, -8, 16, -27, -12, 5, 3], [-16, 1, 19, -2, -19, -8, 26, 2], [-7, -4, -32, 9, -8, 11, 0, 15], [-8, -4, 10, -11, -28, -13, -4, -12], [8, 19, -24, 5, 15, 11, 13, -9], [-20, 28, -4, 10, -18, -10, 11, 4], [-20, 10, 0, -14, -16, 3, 6, -2], [29, 1, 13, -24, 27, 19, 8, -5], [35, 19, -16, 3, 14, 17, 10, 10], [11, -7, 15, -3, -13, 7, -30, -18], [-22, 0, 20, -14, 0, 18, -16, -4], [-11, 7, 7, 5, 32, -2, 18, 11], [8, 22, 2, -30, 11, 19, -7, 12], [5, -1, 5, -1, -19, -11, 8, -2], [4, 9, -11, 10, 12, -6, 2, 22], [9, 18, -13, 13, 14, 17, 0, 7], [3, -27, -5, 26, -17, -19, 9, -5], [-15, -6, 7, 1, -40, -10, 2, -5], [-40, 25, -3, 12, -4, -10, 0, 10], [10, -13, -2, -6, 18, -10, 10, 0], [2, -3, -6, -24, -11, 22, -15, -13], [45, 4, 9, -1, 42, 17, 12, 15], [-11, 4, -14, 25, 5, -16, -11, 13], [-4, -4, -2, -14, 2, 0, -3, 10], [-27, 8, -8, 30, 12, -4, -4, 10], [8, -16, -20, 23, 4, -30, 30, 16], [13, 12, 22, -12, 7, 3, -3, 2], [-23, 11, -6, 22, 18, -5, 18, 16], [1, -4, -2, 4, -10, 6, -18, -4], [0, -14, 18, -12, -37, -15, 6, -21], [1, -18, -36, 7, 8, 25, -18, -3], [-5, 8, 1, 11, 29, -5, 8, 1], [27, 15, -12, 4, 4, 9, 4, 4], [-59, -6, 12, 2, -33, 4, -14, 2], [15, -19, -3, -6, -27, -21, 13, -15], [7, -23, -4, -10, 4, -24, 14, 4], [15, 24, -4, -5, 18, -5, 6, 3], [8, -2, 0, -12, -42, -6, -8, 0], [-6, 0, -39, 6, 0, 8, 12, 8], [47, -23, 3, -1, 6, -4, -10, -1], [9, 11, -19, 9, 20, 8, -2, 7], [-5, -8, -14, -8, 14, 1, -7, -20], [7, 6, -5, 17, 12, -19, -4, 11], [2, -1, 8, -25, 9, 0, -15, -19], [11, -19, -2, -7, -24, -10, 20, -8], [8, 7, -1, 5, -11, 0, -2, -22], [0, -4, -18, 2, -18, 6, -12, -11], [-6, 17, 6, -32, 5, 39, -2, 2], [-16, -21, -26, -3, 7, 1, -15, -19], [16, -4, -3, -4, -20, -8, 12, 4], [20, 6, -22, -8, 21, -7, 8, 16], [-14, 14, -4, -6, -14, 4, 8, 20], [2, -30, 11, 18, -26, -28, -18, 0], [-1, -30, -5, -5, -28, -14, 2, -22], [33, -44, -19, 3, 4, -4, -8, -9], [-19, 5, 16, 14, 6, 21, 0, 6], [1, 24, -22, 10, -11, -7, -1, -4], [39, -2, 8, 4, 31, -2, -5, 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_p();".
// 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_p(: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_p();".
// 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_p( : 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;