
// Make newform 32.6.a.d in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_32_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_32_6_a_d();".
// 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_32_6_a_d();".
// 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 := [-3, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [0, 16]];
    Rf_basisdens := [1, 1];
    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_32_a();"
function MakeCharacter_32_a()
    N := 32;
    order := 1;
    char_gens := [31, 5];
    v := [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_32_a_Hecke(Kf)
    return MakeCharacter_32_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+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 := 6;
    raw_aps := [[0, 0], [0, 1], [46, 0], [0, -6], [0, 3], [-42, 0], [962, 0], [0, -75], [0, -114], [-2554, 0], [0, 72], [11950, 0], [-5078, 0], [0, -453], [0, 444], [-19714, 0], [0, -321], [29318, 0], [0, 609], [0, 2922], [37914, 0], [0, 3204], [0, -1419], [13930, 0], [163602, 0], [-148562, 0], [0, -3630], [0, 7035], [123222, 0], [-51278, 0], [0, -5376], [0, -7983], [241034, 0], [0, 6099], [-103810, 0], [0, -3666], [-234458, 0], [0, 14025], [0, -2430], [-206282, 0], [0, 7149], [-19746, 0], [0, 25296], [-628622, 0], [-334034, 0], [0, -11478], [0, -26571], [0, 47352], [0, -12423], [-664626, 0], [1094330, 0], [0, 21516], [-64734, 0], [0, -46737], [52226, 0], [0, -42342], [-359498, 0], [0, -36204], [-638370, 0], [-112790, 0], [0, 17607], [990350, 0], [0, -12291], [0, 51030], [940154, 0], [-2428250, 0], [0, 27507], [2975058, 0], [0, 105063], [1606406, 0], [-4228190, 0], [0, -74862], [0, 84108], [1506302, 0], [0, 30447], [0, -129312], [786446, 0], [1168886, 0], [-4583998, 0], [4313562, 0], [0, -131079], [-5016882, 0], [0, -6852], [5216866, 0], [0, -121482], [0, -159729], [443506, 0], [590538, 0], [-4794634, 0], [0, 50532], [0, 119685], [0, -180552], [0, 212946], [0, -294261], [0, -16995], [0, 333222], [-2952826, 0], [10456970, 0], [0, -208413], [-3864090, 0], [0, 306585], [6332534, 0], [0, 305853], [1692794, 0], [0, -162705], [-4075774, 0], [0, 294291], [-528622, 0], [0, -174786], [-2118038, 0], [0, 250104], [-3215986, 0], [14632570, 0], [0, 87579], [0, 585414], [-12239950, 0], [0, 86769], [0, -503430], [-15939146, 0], [0, 121413], [-8511938, 0], [11516882, 0], [-23269202, 0], [0, -687045], [0, 258285], [8776838, 0], [8084814, 0], [0, -515484], [0, 423006], [18166086, 0], [0, -459399], [0, -66030], [0, 203724], [139230, 0], [13625146, 0], [-10683982, 0], [22236686, 0], [0, 1015173], [-24345050, 0], [23081002, 0], [0, 1203723], [-20313410, 0], [0, 249174], [0, 1127631], [23201478, 0], [0, -1052406], [-9231458, 0], [-7340774, 0], [0, -567513], [0, -206760], [35712278, 0], [-1769998, 0], [0, -975699], [0, 1111014], [0, -702669], [0, 883572], [0, -742098], [-45017134, 0], [23115130, 0], [-664042, 0], [0, -1866915], [28903034, 0], [0, -853590], [0, -1137405], [4827778, 0], [0, 1016286], [0, -1065576], [-51630610, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_32_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_32_6_a_d();".
// 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_32_6_a_d(:prec:=2)
    chi := MakeCharacter_32_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 6));
    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_32_6_a_d();".
// 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_32_6_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_32_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<3,R![-768, 0, 1]>],Snew);
    return Vf;
end function;