
// Make newform 1925.2.a.ba in Magma, downloaded from the LMFDB on 26 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1925_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_1925_2_a_ba();".
// 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_1925_2_a_ba();".
// 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 := [20, -35, -25, 47, 10, -13, -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], [-36, 21, 60, -3, -15, 0, 1], [-52, 21, 64, -3, -15, 0, 1], [-30, 7, 46, -1, -13, 0, 1], [-28, 19, 46, -3, -13, 0, 1], [80, -17, -138, -11, 39, 2, -3]];
    Rf_basisdens := [1, 1, 4, 4, 2, 2, 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_1925_a();"
function MakeCharacter_1925_a()
    N := 1925;
    order := 1;
    char_gens := [1002, 276, 1751];
    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_1925_a_Hecke( : Kf := Kf)
    return MakeCharacter_1925_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]];
    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, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0], [1, -1, 0, -1, 1, 0, -1], [0, -1, -1, 1, 0, 0, 1], [3, 0, 1, -1, 2, 0, 0], [1, 0, 0, -2, 1, 0, 0], [0, -1, 0, 0, 0, -1, 0], [3, 1, 2, 0, 0, 0, 1], [-3, 1, 1, -1, 0, -2, 0], [-2, 1, 0, -1, 0, 0, 1], [1, -2, 2, 0, 1, -1, 0], [0, -1, 0, 0, -1, 0, -1], [1, 0, -3, 1, -2, 1, 0], [-1, -1, -1, 0, -1, 3, 2], [4, 0, 0, 1, 0, -1, 1], [1, 0, 1, -1, 2, 0, -2], [-1, 1, 1, 1, 0, -3, 0], [2, 1, 0, 1, 1, -1, 0], [2, -1, -4, 0, 0, 2, 2], [3, -3, 0, 2, -1, 1, 0], [0, -4, 0, 0, 0, 2, 0], [-2, 3, 0, 0, 0, 3, 0], [-2, 3, 2, -3, -1, -1, 0], [3, -3, -2, 0, -2, -2, -3], [0, 1, -1, 1, 0, 3, 4], [-1, 2, 2, -2, 1, -2, 2], [-4, 2, 2, -2, 1, 4, 2], [-3, 1, 3, 1, 1, 1, 2], [2, -3, 0, -2, 2, 1, 0], [7, 2, -3, 3, 0, -2, 0], [10, 0, 1, -1, 1, 0, -2], [-4, 1, 2, 2, 0, -3, 2], [2, 0, -2, 0, -4, 0, -2], [5, 2, 0, 0, -1, -2, 2], [-3, 7, 0, 0, -1, 3, 2], [2, 2, 1, 3, 3, 0, 0], [-3, 3, 1, -4, 0, 0, -1], [5, 2, -3, 3, -4, 2, 2], [2, -1, -4, 2, 0, 1, -2], [6, 1, 4, -2, 0, -3, -2], [-3, -2, -2, 2, -1, 2, -2], [-5, 0, 0, -4, 1, 0, 0], [6, -2, -1, 2, -1, -5, -3], [9, -1, 3, 1, 1, -5, -2], [0, 3, 1, 0, 1, -2, 3], [8, 1, 3, -3, 1, -3, -2], [0, -1, -2, 2, -6, 3, 2], [1, 3, 2, -4, 5, -1, -2], [2, 2, -7, 3, -2, 2, 2], [0, 3, 4, -3, 1, -1, 0], [5, -8, 4, -2, 2, 1, -1], [1, 0, 0, 2, -5, -4, 0], [-1, -3, -3, -1, 1, 1, -2], [4, 3, -4, 2, 2, 1, -4], [-4, -3, 3, 3, -1, -1, 0], [-5, -4, 2, -2, -1, 4, 0], [-11, -6, 0, -2, 1, 2, -2], [-5, 0, -4, 4, -7, 4, -2], [-2, -2, 3, 0, 6, 4, 0], [6, 1, -1, 1, -1, 3, 4], [3, -4, -3, 0, -1, 0, 0], [-10, 1, -8, 2, -2, 3, 0], [1, -3, 5, -1, 3, -3, 0], [14, -1, 1, -1, -5, 1, -2], [2, 3, -2, 2, -4, -3, 0], [-3, -2, 4, 4, -1, -1, 2], [1, 4, 7, -1, -3, -1, 3], [-1, -3, 2, -2, -1, 1, 2], [-3, -4, 0, -4, 3, 0, -6], [-1, 4, 3, -6, -2, 3, -1], [-2, -5, -2, 4, 4, 5, 4], [8, -4, 0, 0, -2, -2, 2], [5, 2, 0, 2, -2, -5, -3], [-2, -1, 0, 2, -7, 1, -2], [11, 0, -4, 2, -3, 2, 0], [-2, -5, -2, 2, -1, -1, -4], [1, -4, 3, 7, -1, 1, 1], [0, 5, -11, 6, -8, 3, 2], [6, -3, -3, -3, 3, 3, -6], [-8, 2, -2, 2, -6, 3, 4], [6, -7, 4, 0, 4, -1, -2], [0, -6, -5, -3, 1, 2, 0], [-2, 3, 5, 1, -1, 1, -2], [-3, 5, 1, -7, 4, 2, 2], [8, -3, 4, -4, 4, 3, 0], [-5, -5, 5, 5, -1, 4, -1], [6, -1, 1, -1, 7, -5, -4], [-6, -1, 4, 0, -1, 2, 7], [-18, 0, 5, -7, 1, 0, 0], [-11, -4, 9, -1, 2, -2, 2], [-3, 3, -6, 6, 3, 2, 0], [9, -8, -7, -3, 4, 0, -4], [-6, 3, 7, -1, -1, 1, -2], [9, 5, 2, 2, 3, 1, -2], [10, -3, 2, 6, -6, -1, 2], [7, -11, 3, -1, -2, 3, 2], [1, -3, -4, -6, -3, 9, 0], [3, 6, 3, 5, -3, -4, 4], [-1, 3, 6, -2, 1, -1, -2], [1, 2, 9, 1, 0, -8, 0], [-4, 3, -2, 6, -4, 1, 8], [2, 4, -4, 4, -2, 0, -4], [17, -1, -2, -4, 1, 7, 4], [-6, 3, -3, 6, -1, 0, 1], [15, -2, -1, 0, 2, -5, -9], [-5, 8, -1, 1, 2, -6, -2], [-10, -3, 0, 4, 1, 0, 1], [3, -2, 4, 4, 1, -4, -6], [-25, 3, 0, 0, -3, 1, 2], [2, -6, -3, 1, 1, 9, -2], [8, 7, -1, 2, -8, 5, -2], [2, 5, 6, 0, -4, 3, 2], [0, -11, -7, -1, 1, 4, -2], [-1, -3, 1, 0, 3, 9, 2], [-10, 0, -3, 2, -6, 2, 4], [-16, 9, 4, -4, -5, 1, 4], [22, 5, 2, 4, 2, 3, 0], [2, -1, 6, -8, -2, -1, -2], [-11, 0, 4, 0, -1, 0, 8], [10, 1, 10, -7, 11, -3, 0], [2, -5, -8, 6, 0, 7, 6], [2, -9, 2, -2, -1, 0, 5], [1, -3, -6, -4, 5, 3, 2], [4, 7, 4, 4, 3, 3, 4], [7, 7, -1, 2, -4, -4, 1], [-1, -7, -3, -4, 7, 5, -2], [1, 9, 4, 6, 2, -8, 3], [7, 0, -1, -5, -3, 0, -2], [12, 1, -5, 5, -4, -1, -4], [12, -5, 1, 1, -7, -1, 2], [-6, -15, -3, 11, -2, -1, 2], [-18, 0, 4, -7, 3, -4, -2], [5, 12, 7, -4, 0, 1, 3], [4, -1, 4, 2, -2, -1, -2], [-4, 7, 9, -3, 1, 1, 6], [-2, 8, 2, -6, 0, -2, -2], [-20, 9, -4, -4, 2, 7, 4], [4, 13, -4, 6, -4, -5, 8], [-17, 6, -8, 4, -3, 0, 0], [-4, 3, -2, -12, 2, -3, 2], [-22, 10, -6, 2, 4, 3, 0], [-7, -4, 6, -4, 1, 4, 2], [-2, 12, 1, -2, -3, -5, -1], [18, 0, 10, -3, -3, -2, -2], [22, -10, -2, -3, 8, 3, -5], [0, -7, 3, -8, -4, 5, -2], [20, 4, 1, 7, -3, 2, 2], [-26, 11, -4, 2, -6, 3, 4], [3, 5, 8, 6, -7, 3, 2], [2, -3, 1, 3, 3, 1, -2], [9, 0, -3, -7, 0, 0, 0], [-2, 5, 6, -8, -2, -1, 0], [12, -5, 5, 3, -5, -5, 0], [-18, 4, 1, -3, 2, 2, 8], [1, 2, -8, -4, -1, 8, 4], [2, -4, 8, -5, 3, -10, -2], [-6, -9, -10, -2, 7, -2, -5], [-4, -7, -5, -5, -1, 1, -8], [-11, 1, -4, 2, -11, -3, 0], [-8, -9, -2, 0, 8, 1, 0], [13, 11, -3, -12, 4, 0, -3], [-12, 6, 7, 1, -1, 4, -2], [-24, 13, -5, -10, -2, 1, 6], [4, 3, 4, -10, 0, 1, -4], [-17, 4, 0, 7, -14, -4, 6], [4, -2, 4, -6, 2, -6, -4], [24, 5, -4, 0, -2, 5, -2], [-4, 5, 15, -1, -2, 5, 6], [5, 6, 8, -6, 7, -6, 2], [13, -9, 6, -2, 7, 0, 0], [-13, -10, -12, 2, -7, 8, 4], [-20, 2, 4, -4, -3, 1, -1], [11, 2, -8, 2, -3, -6, -6], [4, 4, 8, -6, -4, -4, -2], [16, -12, 7, -5, 5, -1, -6], [4, -3, -3, 3, 1, -5, 4], [-9, 2, -1, -5, -7, -5, -5], [-8, -10, -7, -1, 1, 0, -4], [8, 7, -7, -1, -5, 3, 0], [-4, 19, -4, 8, -4, -3, 6], [14, -7, -5, -4, -7, 2, -1], [6, -10, 8, -10, 12, 2, -2], [-1, -3, -6, 0, 5, -9, -6], [-25, -3, -6, -2, -7, 1, -2], [2, 6, -3, -14, 6, 0, 4], [-8, 4, -10, 0, -2, 0, -4], [-3, -9, 5, -1, 4, 3, 2], [14, -8, -8, 0, -8, -6, -2], [18, -1, 12, -8, 8, 5, 6], [13, 12, 2, -6, 1, 4, -4], [-16, 9, 2, -2, 0, 1, -2], [1, 8, 6, -6, -3, -3, -6], [-15, 4, -4, 4, 3, 2, 2], [2, 1, 2, -4, 0, -9, -6], [-17, -3, 11, 3, 4, 6, -2], [7, 14, -6, 8, -10, 5, 5], [24, 11, -1, 5, -9, 7, 2], [-4, -1, 0, 12, 0, 3, 6], [-1, -1, 7, 1, 4, 5, -2], [4, -7, -12, 10, 4, 9, -2], [-10, 8, -5, -4, 3, 3, 1], [14, 6, 2, -10, 3, 3, -1], [-30, 11, -4, 0, -2, 1, 8], [7, 14, 3, -2, 6, -15, -1], [-35, 2, -8, 6, -5, 2, 0], [-10, 3, -4, 0, 2, -1, 2], [34, 7, 1, -4, 2, -1, -6], [-6, -12, -2, 6, -4, -2, 6], [-18, -2, 10, -6, 2, 6, 8], [10, -4, -11, 1, -3, 10, -4], [5, -7, 11, -10, 14, -2, -3], [8, -11, -1, 15, -3, -5, 0], [2, -3, -2, 4, -6, 15, 4], [-3, 6, -6, 6, 5, 6, 2], [-17, 3, -3, 5, -7, 6, 1], [10, -11, -9, -1, 1, 3, -8], [5, 7, 6, -4, 4, -13, 4], [5, 8, 2, -4, 5, -2, -8], [16, -5, -2, 0, -6, 5, 2], [-4, -8, -4, -2, 8, -2, -6], [0, 4, 3, -11, 1, 0, -2], [-18, -6, 10, 0, 4, -14, -2], [-13, -2, -6, 8, -13, 4, 2], [-13, 10, 0, -4, 1, 2, 8], [-4, 11, 0, 0, 8, 7, 4], [-5, 13, -9, -1, 0, 3, 6], [13, -13, -6, 12, -1, 6, 6], [7, 12, -6, 6, -5, -6, 2], [6, -1, 6, 10, -2, -9, 0], [25, -2, 7, -3, 5, 0, 0], [-12, -3, 0, 8, 2, -5, -4], [19, -12, 12, -2, 4, -11, -9], [-22, 10, 10, 10, -2, -8, 4], [4, -11, -16, -6, 2, 5, -2], [-3, -6, -14, 6, -3, 4, 0], [-4, -3, 9, -5, -3, -7, 2], [6, -7, 5, -8, 8, 3, -2], [0, -2, -14, 4, 0, -4, -4], [12, 1, -6, 10, -2, -9, -2], [-4, -16, -1, 0, 2, -6, 0], [1, -8, 9, 3, 6, 11, 2], [-15, 15, -2, 5, -10, 3, 4], [-7, -1, -4, 6, 9, -5, 0], [-8, -9, 2, -6, -1, -6, -7], [-14, 13, 2, -6, 3, -10, -7], [23, -22, 7, 3, 7, 2, -6], [-15, 10, -13, 5, -9, -4, 0], [0, -3, 10, 2, 6, -1, 6], [-32, 6, 1, 4, -8, 0, 8], [-33, 9, -3, -1, -6, 1, 0], [1, 3, -8, 8, -1, 1, 4], [13, -2, -10, 5, -3, 9, -11], [-14, -9, -6, -2, 6, 5, -2], [-4, 0, 5, -3, -5, 1, 2], [6, -8, -11, 7, 3, 10, 0], [-7, 4, 2, -8, -1, 6, 0], [12, 3, 9, -9, 10, -8, 3], [14, 2, 2, 2, -4, 1, 0], [-1, -4, -10, 4, -17, -4, 0], [-9, 4, -3, 2, -9, -8, 4], [14, -20, 2, -5, -3, 0, -6], [20, 2, 12, 0, 6, -4, 2], [43, -1, 6, -4, 4, -2, 3], [-10, 8, 4, 0, -10, -1, 6], [-17, 19, -2, 8, -9, -3, 10], [2, -2, -8, 6, -14, 2, -6], [-24, 4, 8, 6, -4, 0, 2], [-9, 3, -10, 8, 5, 5, 10], [-3, -19, 7, 1, 0, 7, 2], [-27, 4, 6, -4, -5, 6, 6], [-11, -3, 2, 0, 3, -1, -6], [-6, 3, 18, -3, -2, -6, 1], [-5, -10, 7, 0, -2, -5, -3], [8, 12, -15, 1, -13, 2, 0], [-10, 1, -4, -6, 16, -3, -6], [8, -10, -14, 6, 10, 4, -2], [-6, 7, 9, -9, -9, -1, -6], [-11, 15, 1, -1, 2, -1, 4], [-17, 8, -5, -5, -4, 0, 6], [-21, 12, -7, -9, 0, 8, 6], [-16, -5, 1, 4, 2, -9, -2], [10, 1, 6, 10, -16, 5, 2], [12, -8, 5, -5, 19, -6, -4], [17, 3, 0, 4, 9, 0, 2], [-2, 14, 14, -12, -2, 0, -2], [-30, 11, 5, -8, 3, 2, -1], [-37, 2, 0, 2, -6, -5, -7], [1, 3, -11, 3, -8, 0, 8], [1, 8, -1, -6, 4, 3, -1], [-28, 10, -8, 6, -6, 2, 0], [-26, 2, 7, -13, 13, 0, 0], [9, 10, 0, -2, 5, -4, -4], [8, -2, 3, 0, 17, 5, 1], [12, 12, -3, 4, 7, -3, -7], [14, -18, -4, 14, -2, -10, -8], [6, 5, -3, -7, 1, 7, 6], [6, 22, 4, -6, 6, -2, 0], [-12, 0, 5, 10, -4, -8, -4], [6, -1, 10, 8, 6, -9, -6], [20, 8, 0, -2, 0, 12, 6], [19, 3, 17, -3, 2, -3, -2], [-8, 10, -3, 5, -3, 4, 12], [-24, -5, 11, 3, 3, 2, 0], [-8, 3, -14, 6, 4, 5, 4], [2, -8, 8, 0, -6, -4, -8], [-4, 0, -11, -5, -7, 16, 4], [22, 5, 2, -6, 10, -11, -6], [21, 20, 2, -4, -5, -2, 2], [-20, 14, -5, 5, -1, 4, 14], [2, -3, 12, -12, 6, 3, -2], [-1, -2, 1, 17, -12, -6, 8], [-1, 8, 16, -16, -3, 1, -6], [-20, 10, -17, -3, -8, 7, 7], [-5, -9, -9, 6, 0, -8, 1], [4, -9, 1, 5, -2, -1, -2], [36, 15, -12, 4, -10, 11, -2], [4, -6, 2, 8, -2, 2, -6], [20, 6, 3, 11, -3, 10, 2], [-6, -1, -1, 1, 6, 5, -2], [36, -3, -2, -3, 7, -1, -12], [10, 0, -2, -6, 8, 16, 10], [7, 6, -10, 0, 7, -4, -2], [1, 1, 4, 2, -1, -9, -6], [15, -9, -17, -5, 6, 9, -6], [14, 10, -18, 2, -5, -1, -3], [-40, 17, -12, 2, -2, 2, 8], [-22, 18, 0, -12, 6, 14, 4], [-1, -11, 24, -10, 11, -11, -2], [24, 13, 3, 3, -9, -5, 6], [-7, -20, -6, 18, -9, 0, 2], [-14, 12, -1, 1, -11, 4, -4], [1, -3, 10, -14, 3, 10, 0], [33, 0, 4, 6, 10, -7, -1], [12, 3, -1, 9, 3, -2, -2], [-7, 12, 15, 3, -11, -5, 1], [9, 6, 0, -2, 11, 8, -2], [4, -9, -9, 9, -1, -1, 10], [-10, -5, 17, -9, -3, 7, 8], [1, 10, 6, -4, -7, -11, -2], [-15, 17, 0, 7, -6, -5, -4], [30, -15, 1, -1, 11, -11, -2], [-2, -2, 25, -9, 3, -4, -2], [8, 8, 5, -3, 2, 2, -4], [-13, -13, -10, 2, -11, 9, 10], [-8, -7, -15, -1, -11, 7, 2], [-6, 31, -7, -1, -5, 7, 4], [26, -12, 8, 10, -6, -8, 0], [34, -7, 0, -12, 4, 1, -2], [-12, -15, 0, 6, -6, 5, 8], [3, 3, -18, 0, 4, 7, 4], [-1, 18, -3, 3, -2, -6, -2], [-5, 4, 8, 2, 1, -14, -14], [1, -1, 10, -9, 11, -10, -7], [-1, 10, -4, 10, -2, -11, -1], [-6, -1, 9, -15, 5, -3, -12], [0, 7, 4, 12, 6, -7, -4], [54, 9, 2, -2, 2, 1, 4], [-20, 10, 2, 4, -8, 12, -2], [-8, -28, -9, 2, 4, 4, -8], [16, -8, 14, -2, -8, 12, 8], [-2, 3, -14, 12, -14, 5, 4], [-8, 11, -3, 2, 6, 7, 6], [10, -3, 1, 5, -1, -2, -8], [17, -9, -6, 11, 3, 10, 5], [27, -1, 4, -2, -1, 9, 2], [26, -8, -18, 10, 2, -2, -10], [-6, -2, 7, 13, -11, 2, -4], [11, 6, -12, -2, 6, -7, -15], [-10, 9, 8, -4, -8, 11, 2], [6, 2, 8, 10, 2, -12, 6], [24, -4, -13, 1, 7, 16, 0], [2, -6, 9, -13, 5, 9, 4], [17, -15, 1, 4, -8, -10, -3], [-11, 2, -6, -4, -3, -9, -8], [-3, -7, -4, 11, 9, -6, 3], [39, -7, -6, -9, 0, 3, 4], [10, 8, 3, -3, 4, 3, -5], [-20, -8, -2, -6, -8, -6, -6], [0, -12, -13, 7, 9, 16, -4], [-9, -9, -10, 0, 1, -16, -14], [11, -16, 2, 2, 1, 2, -14], [33, 5, -10, -10, -1, 1, -2], [-20, 0, -13, 4, 4, -4, -4], [10, 2, -3, 1, 11, -8, -12], [-29, 3, 2, 2, 5, -5, 2], [-6, 0, 1, 13, 10, -10, 8], [-19, -5, 7, -14, -3, 9, 2], [47, -10, -2, 2, -5, -4, -10], [-11, 0, 14, -6, -7, -8, 4], [-14, 2, 18, -12, 2, -4, -12], [3, 14, 21, -16, 0, -11, -5], [6, -9, -5, 9, -9, -1, 12], [18, -5, 12, 2, -8, -5, -2], [16, 9, 1, 7, -5, 11, 4], [10, -6, -10, 10, -2, 14, -6], [46, 2, 7, -11, 7, 2, 4], [3, -4, -2, 2, -13, 4, 6], [2, -10, -6, -5, -9, -4, -10], [-27, -8, -4, 0, -8, 4, -4], [-19, 11, -13, 3, 1, 1, -4], [-22, -1, -12, 12, -4, -1, -12], [-2, -2, 5, -14, -10, 6, -4], [30, -19, 14, -10, 0, 1, -6], [25, 0, 11, -5, 10, -10, -22], [8, 9, 12, -8, 0, -7, 14], [5, -1, -5, -16, 14, 6, -1], [24, -3, 1, -6, -4, 15, 2], [45, 4, 5, -3, 5, -3, -1], [-28, 0, -14, 0, 0, 2, 4], [14, -2, -24, 6, 0, 2, -6], [31, 6, 4, 0, 1, -14, 10], [-26, 6, -8, -14, -2, -6, 2], [-1, -5, 0, -2, -1, -13, 0], [3, -30, -7, -7, 15, 2, -10], [-14, 4, 7, 2, -4, 0, -4], [45, 5, 8, -6, -11, -1, -8], [7, -2, -16, 22, -11, 2, 14], [20, -22, -5, 7, 12, 20, -2], [-24, -17, -17, 12, 2, 3, -2], [10, 19, 23, 3, 1, -1, 6], [9, 8, -18, 12, -15, 0, 0], [23, 10, 6, -18, 9, -6, 6], [-2, 5, -2, 0, 2, 15, 12], [10, -10, 15, 7, 5, -4, -12], [-17, 13, 10, 0, -15, 7, -4], [-37, 4, -4, 14, -22, -5, 1], [10, -17, 5, -15, 13, 5, -4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1925_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1925_2_a_ba();".
// 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_1925_2_a_ba(:prec:=7)
    chi := MakeCharacter_1925_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_1925_2_a_ba();".
// 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_1925_2_a_ba( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1925_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-20, -35, 25, 47, -10, -13, 1, 1]>,<3,R![34, -85, -20, 77, 2, -17, 0, 1]>,<13,R![1324, -2404, -460, 668, 75, -49, -3, 1]>],Snew);
    return Vf;
end function;