
// Make newform 3432.2.a.x in Magma, downloaded from the LMFDB on 09 November 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_3432_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_3432_2_a_x();".
// 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_3432_2_a_x();".
// 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 := [28, 70, 8, -20, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [8, -60, -17, 4, 1], [-8, -1, 1, 0, 0], [-109, -58, 51, 5, -3]];
    Rf_basisdens := [1, 1, 17, 1, 17];
    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_3432_a();"
function MakeCharacter_3432_a()
    N := 3432;
    order := 1;
    char_gens := [2575, 1717, 1145, 937, 2641];
    v := [1, 1, 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_3432_a_Hecke( : Kf := Kf)
    return MakeCharacter_3432_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, 0, 0, 0, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [-1, 0, 0, 0, -1], [1, 0, 0, 0, 0], [1, 0, 0, 0, 0], [1, 1, 0, 0, -1], [2, 0, 1, 0, 0], [-1, 0, 0, 0, 1], [-1, 0, -1, -1, -1], [-1, -1, -1, 0, -1], [2, 0, 0, 0, 2], [3, 0, 0, 0, 1], [0, -1, -1, 0, -2], [1, -1, 0, -1, 1], [1, 1, 0, 1, 1], [4, 1, -1, 1, 0], [-3, -2, -1, 0, -1], [2, -1, 0, 0, 0], [-1, -1, -2, 1, -1], [4, -1, 0, -1, 0], [2, -2, 2, -1, 0], [0, -2, 0, 0, 0], [3, 3, 1, 0, 3], [7, -1, 0, 1, 1], [2, 2, 0, 1, 2], [0, 1, 1, -1, 0], [3, -2, -1, 0, 1], [1, -1, -1, -1, -1], [6, -1, 1, -1, 0], [-3, -1, 0, 0, 3], [3, -4, 1, 0, 1], [5, -1, -1, 0, 1], [-3, 1, 0, 2, -1], [-5, -3, 1, -1, -3], [5, -1, -1, 1, 1], [8, -2, 1, 0, -2], [11, 0, 0, -1, 1], [-4, -1, -3, 1, 0], [-10, -1, -1, 0, -2], [-2, 2, -1, 0, 2], [8, -2, 3, 0, -2], [-5, 2, -1, 0, -3], [12, -2, 1, 0, 0], [-5, -3, 1, 1, 1], [-4, 3, -1, 1, 0], [8, 2, 2, -1, 2], [-5, 3, 1, 2, -1], [-5, 1, -1, -1, -5], [4, -3, -1, 1, 4], [2, 2, 2, -1, 0], [-7, 0, 2, 0, -3], [7, -1, 0, -1, -1], [-4, -2, 0, 0, 4], [0, 3, -1, 1, 2], [-4, 0, 2, 2, 0], [7, 1, -2, 1, -5], [-3, 3, 3, -1, 1], [2, 1, 3, -1, 0], [-3, 0, -4, 2, -1], [-11, -2, -1, -1, -5], [-5, 3, 3, -1, 1], [10, -4, -1, 0, 4], [-11, 7, -1, 1, -1], [2, -3, -2, -1, -4], [1, -2, 2, -3, 1], [-3, 0, 0, -1, -1], [10, 0, -2, 2, 0], [-4, 0, 2, 0, -2], [8, 0, -2, 2, 2], [10, 2, 1, 1, 6], [-21, 4, -2, 0, -1], [-2, 6, 0, 0, 6], [2, 5, -1, 1, 0], [1, 1, 1, 0, -5], [4, -6, -2, 0, 0], [-5, -5, -2, -1, -5], [2, 7, -1, 1, 2], [14, 0, 1, -1, -2], [6, -1, -2, -1, -2], [8, 0, -1, 2, 4], [1, 4, 1, 0, 1], [-2, 2, -2, -2, -2], [2, -1, -1, 1, 8], [-10, 0, 2, -1, 8], [-14, -2, 0, 2, 2], [-6, 4, 3, 1, -2], [-3, -4, 1, 0, 1], [-3, -5, 1, 1, 3], [2, -2, 3, 1, 2], [-2, 2, 1, 2, 2], [-9, -4, 2, 0, -5], [4, 0, 3, 1, 8], [-12, -1, -1, -3, -2], [0, -1, 4, 0, 6], [-2, 6, 2, 0, 2], [-1, -11, 1, 0, -1], [0, -1, 1, -3, 2], [-8, 2, -2, -1, -2], [18, 0, 0, 0, 4], [-3, -6, 1, -1, -1], [9, 1, -1, -1, -1], [-1, 5, 2, -1, 1], [-13, 5, 2, 2, 1], [4, -1, 1, 2, -2], [16, 0, 0, 0, 2], [20, -1, -3, -1, 0], [-15, -3, -1, -3, -5], [-7, -2, 1, 0, 7], [4, 0, 2, 0, -2], [9, -1, -4, 0, -1], [11, 5, 1, 1, 1], [2, -2, 1, 3, 2], [-3, 4, 2, 1, -1], [16, 0, 3, 3, 4], [-4, -5, -1, 1, -6], [4, -2, 3, -1, -6], [5, 3, -2, 3, 3], [-9, -5, -2, 1, 3], [-2, -2, -2, 0, 0], [9, 1, 0, -3, 1], [-9, -3, 0, 3, 1], [-9, 1, 4, 0, 11], [-3, 0, -3, 0, 1], [3, 3, -3, 4, -3], [-2, 1, 3, 2, -2], [12, 5, 1, -3, 4], [-11, -4, -3, 0, -1], [-20, -4, 4, -2, 4], [17, 5, 3, 1, 7], [22, -2, 3, -2, 0], [-10, -7, 3, 1, 6], [-12, -1, -5, 3, -4], [6, 4, 5, -2, -4], [3, 2, 4, 0, -3], [-15, -6, 1, -2, -3], [5, -1, 3, 4, 5], [-8, 0, -1, 0, -2], [3, -3, -2, 3, -1], [-7, 3, -2, 0, -1], [-20, 0, 1, 0, 2], [-1, -1, -3, -1, 1], [-10, -7, -1, 1, -2], [9, 3, -5, 1, -7], [-18, 0, 0, 4, 0], [-8, -6, 4, -2, 8], [-16, -4, 4, -4, -6], [16, -6, -2, -3, 2], [-6, -2, 0, 0, 2], [4, 4, 0, 2, 4], [2, 10, -2, 0, 4], [-34, -1, -1, 1, -4], [-6, 0, -2, -4, 6], [13, 3, 2, 3, 9], [-8, 4, 0, 2, -8], [-9, 1, 2, 1, 5], [-11, 5, 6, 2, 3], [13, -3, -1, 2, 1], [1, -9, 2, 1, -5], [-7, -1, 1, 1, 7], [15, 5, 2, 1, 3], [-2, 0, 4, -1, 8], [5, 8, -2, 2, 9], [4, -2, 1, -2, 0], [-5, -1, -1, -2, -9], [14, 8, 0, 0, -6], [-10, 3, 3, -1, -2], [14, -5, 1, -1, -4], [-26, -5, -2, -1, -2], [9, -5, 0, 1, 5], [-28, -2, -2, 0, 4], [26, -3, 1, 3, 6], [3, 9, 0, 1, 3], [-13, 8, 1, 2, 7], [-6, -1, -3, 1, 10], [-6, 5, -3, -1, 0], [-10, 12, -1, 0, -4], [-3, 4, -2, -1, 9], [-10, -4, 6, -3, 4], [20, -1, 3, -1, -2], [17, 3, 1, 0, -3], [9, 11, -4, 1, -5], [0, 2, -3, -4, -2], [-27, 4, -6, 0, -1], [-1, -9, 4, -1, 7], [-17, -7, 2, -3, -5], [3, 3, -2, 1, -3], [-19, -2, -6, 3, -1], [7, 4, -3, 2, 11], [10, -1, 3, 3, -6], [-14, -4, -3, -2, -4], [14, 8, -4, 0, -10], [-12, 0, -4, 4, 0], [17, -10, 2, -1, 1], [7, 5, 6, -5, -3], [10, -4, 4, -4, 0], [25, -3, 4, -5, 1], [-18, -4, 4, 2, 8], [-13, 6, -2, -4, -7], [9, -3, 0, -3, -3], [-27, 1, 1, -3, -9], [27, 3, -2, 0, 9], [1, -4, -1, 4, -7], [3, -9, 4, 3, 9], [1, 10, -3, 0, 1], [15, -13, 2, -1, 3], [-26, 0, 3, -1, -6], [13, -2, -3, 2, 7], [21, 1, -3, 1, 7], [16, 2, 6, 2, 4], [-25, 3, -4, -3, 5], [5, -13, 2, 1, 5], [-6, 13, -1, 1, -2], [12, 0, 2, 0, -8], [3, -7, -5, -1, 3], [-3, -13, 2, -1, -9], [-8, -2, 2, 1, -6], [12, -14, 4, 1, -2], [9, -7, -6, -1, -7], [-10, -2, 1, 7, -2], [-8, -3, -11, 3, -2], [27, 6, 0, 0, -9], [-8, -6, 1, 1, -4], [13, -3, -7, 0, -11], [-8, 7, -1, -1, 2], [0, -6, 0, 0, -6], [6, -4, -6, 2, -4], [-11, -6, 1, -4, 3], [-30, -2, 4, -1, 8], [16, -4, 2, 2, -12], [14, 0, 4, 4, 4], [34, -1, -1, 4, 12], [-11, 0, 4, -4, -3], [42, -6, -4, -1, 0], [10, 2, 0, -6, 6], [8, 8, 4, 0, 0], [-34, 0, -2, -2, 0], [-25, 3, 1, -1, -3], [-14, -4, -7, 2, -6], [11, 9, 1, 3, 5], [22, -1, -7, -1, -6], [7, -12, -5, -3, -7], [26, -11, 1, -1, -2], [3, -1, -2, -1, -13], [-5, -8, -8, -2, -15], [11, 11, 0, -1, -5], [20, -8, -1, 1, 4], [9, -6, 1, 0, 1], [14, 14, 3, -2, 8], [55, -3, 2, 1, 3], [22, -12, -6, -2, 0], [11, 3, -1, -7, -3], [3, -5, 2, 1, -9], [-26, 1, -2, -1, 8], [-11, 4, 3, 1, 5], [13, -3, -1, -1, 1], [-2, 6, 5, 2, 4], [8, 2, 3, -6, -2], [27, 1, -2, 5, 15], [-15, 6, 3, -4, -7], [-5, 3, 4, 2, 13], [24, -3, 1, 1, 2], [-12, 8, 4, -2, 2], [-16, 1, 1, -3, -6], [3, -13, -5, 0, -9], [20, 2, 6, 2, 8], [8, -5, 8, -4, 4], [28, -6, -1, -1, -8], [16, -2, -1, -2, -6], [-5, 9, 0, -2, -13], [28, 4, 2, -2, 6], [6, 4, 1, 1, -12], [-2, 8, 6, 2, -4], [35, 3, 2, 0, 1], [2, 2, -2, 0, -4], [-8, 0, -1, 7, -4], [29, 6, 3, -2, 1], [-5, -3, 5, 5, 1], [-18, 11, 4, 3, 2], [2, 3, 1, -1, 4], [25, -11, -4, -5, -3], [12, 6, 9, -5, 0], [-17, 7, -2, -5, -9], [35, 1, -5, 1, 5], [30, 2, -1, 4, 4], [21, -6, -1, 2, -1], [2, 4, -4, 4, 4], [-13, 1, 3, 5, 1], [-6, 0, 9, -1, 2], [-16, 6, 4, 3, 6], [5, 9, -2, 3, -7], [52, -10, 1, -2, 0], [3, -4, -6, 0, 5], [-13, 3, -1, -3, 3], [11, 0, -7, 2, -3], [24, 9, -5, -2, -12], [-2, -1, 5, -1, 10], [-7, -13, 2, -1, -15], [0, -21, 3, 1, -2], [24, 8, -1, 0, 2], [-4, 6, -6, -2, -2], [7, 13, 3, -1, -3], [-19, -3, -6, 4, -1], [26, 8, 7, -4, 2], [-2, 8, -4, 4, 10], [-29, 11, 2, 1, 9], [-6, -4, 3, -2, 14], [20, 10, 0, 0, 6], [10, 10, 2, -2, 6], [-4, 5, 7, -3, -6], [12, -8, -8, 2, -4], [-12, 3, 0, -6, -4], [3, -1, -2, -2, 9], [8, 10, -4, 0, 0], [26, -2, -6, 4, 14], [2, -6, -4, -4, -12], [16, 7, -3, 1, 16], [-41, -4, 2, 4, 3], [4, -4, 6, -2, 2], [-24, -10, -2, -1, -10], [-23, 5, 2, 0, -5], [16, 12, -2, -4, -10], [-13, -4, -3, 3, -9], [2, 14, -9, 5, -6], [-29, 6, 6, 6, 5], [10, -5, -4, -1, -8], [16, -13, 4, -4, 6], [19, 3, -5, 2, 13], [4, -6, 0, -4, -12], [-23, 7, -6, 0, 9], [-11, 3, -3, -5, -9], [35, -7, 4, 3, 3], [-31, 7, -6, 2, 3], [-1, 9, 7, -1, -1], [-22, 4, -9, -2, 4], [-30, -2, 2, 0, -6], [22, 7, -1, 9, 6], [-1, 10, -6, 4, 1], [-32, 2, 4, 4, -6], [3, -9, -8, 2, -3], [-4, -3, -1, 3, -4], [20, 1, 1, 1, 6], [-21, 5, -3, -5, -3], [-30, 2, -10, 0, 6], [37, 0, 2, -3, -7], [-17, -4, 7, -6, 9], [36, -6, -4, -2, -2], [-13, 6, -4, 1, 9], [1, 8, 0, 4, 13], [11, 1, 0, 3, -1], [-20, 2, -1, -2, -2], [-48, -2, -4, -4, -10], [-6, 2, 3, -5, 2], [-2, 0, 0, -1, -16], [11, 6, -1, 0, -7], [-15, 3, -6, 0, -13], [7, 1, -2, 5, -5], [49, 9, -5, 7, 9], [9, -19, 4, 2, 7], [-25, 11, -2, 3, 3], [-22, -6, 0, 6, -12], [-2, -6, 8, 1, 4], [21, -8, -9, -2, -5], [5, -9, 1, 1, -3], [10, -4, 10, 0, 6], [-48, -5, -4, -1, -2], [-9, 7, 5, -1, 9], [-9, -8, 2, 2, -17], [17, 15, 2, -3, 3], [-14, -4, -4, 2, -10], [-6, 2, 4, -2, -10], [5, 8, -4, 0, 17], [-20, 4, -2, 5, 8], [-42, -6, 4, -7, -16], [14, -6, 2, -10, 4], [-31, 7, -8, -1, -3], [-8, 0, 2, -2, 4], [6, 0, 1, 6, 6], [-6, -16, 6, 1, 4], [9, 3, 8, 7, 11], [16, 3, -4, -8, 0], [12, 4, 3, -5, -4], [1, 16, 2, 0, 9], [7, -2, 4, 2, 1], [12, 6, -3, 0, -10], [35, 7, 4, -3, -5], [21, 1, -5, -4, 9], [2, -16, 4, -2, -8], [14, 2, 11, -6, 0], [-27, 15, 5, 3, -5], [-17, 9, 2, -1, 5], [-13, 2, 0, -3, -1], [17, -5, 6, -7, 1], [-2, 0, 6, 4, 18], [16, 6, 0, 4, -16], [15, -3, -4, 1, -15], [-11, -12, 2, 0, 9], [2, -11, -3, -1, -8], [-10, 0, 10, -4, 2], [33, -3, 3, 3, -9], [16, -8, 0, -4, -6], [-2, 11, -3, 5, 0], [5, 4, 12, -4, 5], [31, -15, 1, 2, 3], [-3, 3, -5, 1, 7], [-23, 1, -4, -4, -5], [-56, -9, -1, 3, -4], [-9, -10, 2, 2, -23], [54, -9, -2, -2, 0], [-31, -6, 1, 2, 13], [2, 2, -2, -2, 12], [-20, 0, -4, 7, 4], [-18, 2, -2, 0, -6], [8, 5, 3, 2, 6], [-34, -2, 2, 0, -8], [25, 9, -6, 3, 9], [-41, -2, 2, -2, -5], [-42, 14, 0, 0, -6], [16, 12, -5, 5, 4], [19, -11, -10, -5, -5], [-15, 9, -8, 7, -3], [10, -2, -6, 4, 0], [19, -9, 2, -3, -1], [-19, -14, 1, 4, -13], [36, -2, 10, 6, 6], [68, -9, -2, -2, 0], [4, 14, -10, 6, -4], [-17, -13, 3, 1, -19], [-5, 20, -10, 7, 5], [-11, 7, 2, -7, -7]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3432_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3432_2_a_x();".
// 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_3432_2_a_x(:prec:=5)
    chi := MakeCharacter_3432_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_3432_2_a_x();".
// 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_3432_2_a_x( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3432_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![28, 70, 8, -20, -1, 1]>,<7,R![-16, 52, -24, -18, 3, 1]>,<17,R![-448, -56, 202, -20, -8, 1]>],Snew);
    return Vf;
end function;