
// Make newform 432.4.i.a in Magma, downloaded from the LMFDB on 25 November 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_432_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_432_i_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_4_i_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_4_i_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, -1, 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_i();"
function MakeCharacter_432_i()
    N := 432;
    order := 3;
    char_gens := [271, 325, 353];
    v := [3, 3, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_432_i_Hecke();"
function MakeCharacter_432_i_Hecke( : Kf := false)
    N := 432;
    order := 3;
    char_gens := [271, 325, 353];
    char_values := [[1, 0], [1, 0], [0, -1]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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]];
    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 := 4;
    raw_aps := [[0, 0], [0, 0], [0, -9], [-31, 31], [15, -15], [0, 37], [42, 0], [28, 0], [0, -195], [111, -111], [0, -205], [-166, 0], [0, -261], [-43, 43], [-177, 177], [-114, 0], [0, -159], [-191, 191], [0, -421], [156, 0], [182, 0], [1133, -1133], [1083, -1083], [1050, 0], [901, -901], [387, -387], [0, 551], [-12, 0], [-502, 0], [0, -1401], [880, 0], [0, -1503], [-2661, 2661], [0, -121], [0, -2829], [461, -461], [0, 2977], [3316, 0], [0, 681], [-3981, 3981], [2004, 0], [1274, 0], [-1161, 1161], [0, -3611], [-2046, 0], [-2996, 0], [0, 755], [-3463, 3463], [-6225, 6225], [0, 1465], [-2634, 0], [0, -6915], [1489, -1489], [-4620, 0], [0, 3351], [603, -603], [1470, 0], [-2072, 0], [-7139, 7139], [8427, -8427], [0, -457], [0, -5889], [1204, 0], [0, 3285], [10057, -10057], [2295, -2295], [-6679, 6679], [0, -2183], [0, -3891], [-2795, 2795], [4755, -4755], [4608, 0], [3845, -3845], [0, 8317], [-12560, 0], [0, -12087], [-8541, 8541], [-13174, 0], [0, 9603], [0, -11471], [0, 5973], [8905, -8905], [1416, 0], [10766, 0], [4349, -4349], [14547, -14547], [3330, 0], [-8147, 8147], [8031, -8031], [0, 4283], [5460, 0], [-429, 429], [11296, 0], [0, 14673], [0, 13439], [-17388, 0], [0, -3789], [-9786, 0], [8008, 0], [-2938, 0], [-10375, 10375], [-3306, 0], [0, 21093], [1287, -1287], [0, 15035], [1190, 0], [17883, -17883], [-20118, 0], [0, 1065], [20725, -20725], [0, -15745], [5042, 0], [0, -10053], [-5983, 5983], [19696, 0], [-10977, 10977], [0, -15829], [28224, 0], [0, 28167], [-10737, 10737], [0, -10127], [-251, 251], [8451, -8451], [-25884, 0], [6365, -6365], [-1122, 0], [-4283, 4283], [4032, 0], [24005, -24005], [0, 37501], [880, 0], [0, -1623], [0, -6889], [-12850, 0], [0, 4611], [0, 2305], [34902, 0], [0, 26255], [0, -1437], [-25734, 0], [3220, 0], [44631, -44631], [0, 24947], [-14964, 0], [-7462, 0], [-2697, 2697], [39625, -39625], [-2973, 2973], [0, -45229], [1416, 0], [0, -17555], [-30030, 0], [35740, 0], [0, -747], [-20539, 20539], [24123, -24123], [-23312, 0], [18699, -18699], [-31234, 0], [0, 11751], [25887, -25887], [41130, 0], [0, -28645], [-32256, 0], [0, 9015], [-15885, 15885], [-34904, 0], [35341, -35341]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_432_i_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_432_4_i_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_4_i_a(:prec:=2)
    chi := MakeCharacter_432_i();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_4_i_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_4_i_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_432_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![81, 9, 1]>],Snew);
    return Vf;
end function;