
// Make newform 432.2.y.a in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_432_y();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_432_y_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_432_2_y_a();".
// 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_432_2_y_a();".
// 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, 0, -1, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_432_y();"
function MakeCharacter_432_y()
    N := 432;
    order := 12;
    char_gens := [271, 325, 353];
    v := [12, 3, 4];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_432_y_Hecke();"
function MakeCharacter_432_y_Hecke(Kf)
    N := 432;
    order := 12;
    char_gens := [271, 325, 353];
    char_values := [[1, 0, 0, 0], [0, 0, 0, 1], [-1, 0, 1, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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, 1, 0], [0, 0, 0, 0], [0, 0, -1, 1], [-4, 1, 2, -1], [-1, 1, 0, 0], [2, -2, -3, -1], [-4, 0, 0, 0], [-3, 0, 0, 3], [-2, 3, -2, 0], [1, 2, 1, -1], [-4, 3, 4, -6], [-5, -2, -2, -5], [-3, -6, -3, 0], [-1, -5, 6, 6], [0, -3, -4, -3], [-5, 2, 2, -5], [-3, -3, 0, 3], [7, 8, 1, -1], [5, -5, -2, 3], [-4, 0, 8, 4], [-2, 0, 4, -4], [0, 1, 0, 1], [-1, 9, 10, -10], [8, 0, -16, 2], [0, 0, -1, 0], [-1, 0, 1, 1], [-4, 9, -4, 0], [-5, 12, 12, -5], [1, 2, -2, -1], [9, -2, -9, 4], [-10, -12, 0, 6], [1, -1, 6, 7], [-10, -8, 5, 8], [5, 5, -10, 5], [6, 6, 7, -13], [0, -7, 0, 7], [2, -2, -1, 1], [9, 8, -8, -9], [-2, 13, -2, 0], [7, 4, -3, 3], [-5, -8, 8, 5], [-7, 2, 2, -7], [0, -3, -8, -3], [1, -2, -1, 4], [-5, -4, -4, -5], [8, 0, -16, 8], [11, -11, -4, 7], [0, -9, 0, -9], [1, -13, -14, 14], [8, 8, -11, 3], [-4, 0, 8, 16], [6, 3, -6, -6], [0, -2, -9, -2], [9, -12, -12, 9], [-7, 10, 7, -20], [-8, 3, 4, -3], [-7, -2, 2, 7], [-8, -8, 0, 4], [-3, -6, 9, 9], [-18, -4, 9, 4], [-11, -11, 10, 1], [8, 8, -1, -7], [-3, -8, 8, 3], [-6, -21, -6, 0], [-22, 10, 11, -10], [-15, 4, 19, -19], [9, 7, -16, -16], [-9, 6, 9, -12], [-11, -11, 2, 9], [5, 4, -1, 1], [0, 2, 27, 2], [-4, 0, 8, -22], [0, 19, -2, 19], [-8, -8, 3, 5], [-9, 18, 18, -9], [-16, -5, 16, 10], [7, -4, -3, -3], [-13, -22, 22, 13], [15, -10, -15, 20], [5, -14, 5, 0], [-11, 11, 4, -7], [3, -12, 9, 9], [-14, -20, 0, 10], [14, 12, 0, -6], [-16, -7, 8, 7], [5, 5, -10, -10], [-18, 20, 0, -10], [-30, 4, 15, -4], [1, -2, -3, 3], [-2, 5, 2, -10], [11, 24, -24, -11], [0, 5, -10, 5], [12, 0, -24, -14], [1, 1, 0, -1], [5, -5, 0, 5], [8, 0, -16, 0], [6, -6, -13, -7], [-8, 0, 16, -28], [17, -14, -14, 17], [-15, 0, 0, 15], [-21, -13, 8, -8], [-17, -30, 30, 17], [1, -1, -16, -15], [-2, 30, 1, -30], [3, 3, -22, 19], [6, 36, 0, -18], [5, -9, 4, 4], [2, -4, 0, 2], [0, 35, 0, 0], [38, 2, -19, -2], [-4, -3, 4, 6], [17, -18, -18, 17], [5, -18, 5, 0], [17, 1, -18, -18], [-14, 0, 28, 8], [0, 2, -15, 2], [-17, 17, 4, -13], [-6, 0, 12, 36], [-24, 24, -5, -29], [-3, -5, -2, 2], [-6, -6, -9, 15], [0, -10, 25, -10], [23, 2, -25, -25], [-11, 12, 12, -11], [-7, -11, -4, 4], [13, 22, -22, -13], [1, 28, -29, -29], [-6, -20, 0, 10], [4, 7, -2, -7], [-14, 14, -11, -25], [-29, -2, 2, 29], [0, -29, 0, 0], [-20, 17, 20, -34], [3, 4, 4, 3], [3, 24, 3, 0], [-7, 0, 7, 0], [17, -18, -18, 17], [-7, 7, 20, 13], [24, -24, -7, 17], [-20, 0, 40, -2], [-17, 4, 4, -17], [9, -26, 17, 17], [-10, 25, -10, 0], [27, -12, -12, 27], [-17, -2, 2, 17], [24, 13, -12, -13], [-1, -6, 7, 7], [38, 16, -19, -16], [9, 9, 0, -9], [30, 4, 0, -2], [34, -34, -27, 7], [-14, 20, 0, -10], [-3, 0, 0, 3], [10, 7, 10, 0], [-1, -7, 8, 8], [0, 1, 10, 1], [-2, 0, 4, 40], [0, 16, 9, 16], [4, 0, -8, -26], [18, -18, 1, 19], [-11, 15, 26, -26], [2, 0, -4, -36], [14, -7, 14, 0], [13, -30, -30, 13], [7, -24, -7, 48], [-28, -23, 14, 23], [-34, 12, 0, -6], [-3, -4, 7, 7]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_432_y_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_432_2_y_a();".
// 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_432_2_y_a(:prec:=4)
    chi := MakeCharacter_432_y();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_432_2_y_a();".
// 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_432_2_y_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_432_y();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![1, 4, 5, 2, 1]>],Snew);
    return Vf;
end function;