
// Make newform 432.2.y.c in Magma, downloaded from the LMFDB on 24 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_c();".
// 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_c();".
// 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 := [[0, -1, 1, 1], [0, 0, 0, 0], [2, 2, 0, -2], [-2, 1, 1, -1], [3, -2, -1, -1], [-1, 1, -3, -4], [4, -2, 0, 1], [-1, 1, -1, 1], [-1, 3, -1, 0], [2, 1, -1, 1], [-4, -2, 4, 4], [4, -2, -2, 4], [0, 3, 0, 0], [-3, 5, -2, -2], [0, 5, 1, 5], [6, -4, -4, 6], [-4, -4, 5, -1], [-6, 0, 6, -6], [3, -3, 2, 5], [4, 0, -8, -4], [-1, 0, 2, 8], [0, 0, -12, 0], [0, 1, 1, -1], [0, 0, 0, -2], [0, 1, -10, 1], [4, 0, -4, -4], [-7, 3, -7, 0], [-7, 15, 15, -7], [8, -2, 2, -8], [0, -8, 0, 16], [-1, -6, 0, 3], [5, -5, -3, 2], [-4, -13, 2, 13], [-7, -7, 8, -1], [-3, -3, 5, -2], [-2, -1, 1, 1], [-2, 2, -2, -4], [-7, 0, 0, 7], [2, -4, 2, 0], [6, 7, 1, -1], [-9, -8, 8, 9], [-3, 12, 12, -3], [0, -11, 5, -11], [-20, 1, 20, -2], [-10, 10, 10, -10], [-7, 0, 14, -13], [3, -3, -3, 0], [0, 11, 3, 11], [7, 8, 1, -1], [5, 5, -5, 0], [-3, 0, 6, 2], [21, 3, -21, -6], [0, 3, -18, 3], [4, -5, -5, 4], [-5, 8, 5, -16], [14, 15, -7, -15], [-4, -2, 2, 4], [10, 12, 0, -6], [4, -17, 13, 13], [0, 10, 0, -10], [13, 13, 7, -20], [-7, -7, -1, 8], [-12, -11, 11, 12], [-11, 3, -11, 0], [4, 25, -2, -25], [-22, -15, 7, -7], [8, 3, -11, -11], [-3, -12, 3, 24], [17, 17, -14, -3], [-10, -2, 8, -8], [0, 10, 3, 10], [-1, 0, 2, 13], [0, -14, 4, -14], [1, 1, -15, 14], [-1, -13, -13, -1], [10, -2, -10, 4], [-4, 13, -9, -9], [-12, -2, 2, 12], [-12, -9, 12, 18], [-18, 1, -18, 0], [-1, 1, 17, 16], [-12, 18, -6, -6], [11, -6, 0, 3], [-4, 2, 0, -1], [12, -16, -6, 16], [1, 13, -14, -14], [12, 10, 0, -5], [12, 13, -6, -13], [14, -7, -21, 21], [-8, -6, 8, 12], [-4, -1, 1, 4], [0, -5, -17, -5], [3, 0, -6, -11], [17, 17, -12, -5], [6, -6, -5, 1], [-16, 0, 32, 0], [4, -4, 8, 12], [0, 0, 0, 13], [-9, -4, -4, -9], [-6, -6, 6, 6], [-27, -13, 14, -14], [23, 6, -6, -23], [-15, 15, 2, -13], [-28, -3, 14, 3], [-12, -12, 5, 7], [15, 24, 0, -12], [13, -15, 2, 2], [-14, 4, 0, -2], [-4, -20, -4, 0], [2, -22, -1, 22], [14, 18, -14, -36], [-12, 10, 10, -12], [-13, 18, -13, 0], [-23, 3, 20, 20], [6, 0, -12, -28], [0, -9, 24, -9], [-2, 2, 25, 23], [1, 0, -2, 15], [15, -15, -1, 14], [0, 11, 11, -11], [-6, -6, 6, 0], [0, -6, 28, -6], [4, -2, -2, -2], [17, -9, -9, 17], [6, 15, 9, -9], [-7, -10, 10, 7], [24, -18, -6, -6], [6, 12, 0, -6], [-2, -35, 1, 35], [-11, 11, 19, 8], [15, 17, -17, -15], [9, -13, 9, 0], [10, 16, -10, -32], [-17, 8, 8, -17], [6, -42, 6, 0], [-4, 14, 4, -28], [-29, 18, 18, -29], [31, -31, -35, -4], [26, -26, -36, -10], [-10, 0, 20, 11], [-10, 11, 11, -10], [18, -1, -17, -17], [-6, -2, -6, 0], [-19, -4, -4, -19], [15, 26, -26, -15], [-8, -34, 4, 34], [6, 1, -7, -7], [-16, 38, 8, -38], [20, 20, -13, -7], [12, -4, 0, 2], [18, -18, -4, 14], [-10, 20, 0, -10], [-14, -1, 1, 14], [14, 14, 14, 0], [9, 0, -9, -9], [0, 4, 2, 4], [-15, 0, 30, 7], [0, 4, 30, 4], [18, 0, -36, -20], [-2, 2, -6, -8], [11, -15, -26, 26], [22, 0, -44, -21], [-9, 5, -9, 0], [-17, -4, -4, -17], [32, -7, -32, 14], [0, -16, 0, 16], [2, -40, 0, 20], [-26, 12, 14, 14]];
    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_c();".
// 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_c(: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_c();".
// 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_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_432_y();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![16, -16, 20, -8, 1]>],Snew);
    return Vf;
end function;