
// Make newform 100.4.a.d in Magma, downloaded from the LMFDB on 04 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_100_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_100_4_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_100_4_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 := [-19, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [0, 2]];
    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_100_a();"
function MakeCharacter_100_a()
    N := 100;
    order := 1;
    char_gens := [51, 77];
    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_100_a_Hecke(Kf)
    return MakeCharacter_100_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 := 4;
    raw_aps := [[0, 0], [0, 1], [0, 0], [0, 1], [20, 0], [0, -6], [0, -8], [84, 0], [0, 7], [-6, 0], [-224, 0], [0, -14], [266, 0], [0, -35], [0, -43], [0, -42], [28, 0], [182, 0], [0, -49], [408, 0], [0, 124], [-48, 0], [0, -23], [1526, 0], [0, -64], [1246, 0], [0, -97], [0, 147], [-902, 0], [0, 168], [0, 301], [2940, 0], [0, 84], [364, 0], [-254, 0], [-2360, 0], [0, 254], [0, -175], [0, -351], [0, 218], [1972, 0], [-1330, 0], [1728, 0], [0, 112], [0, 434], [1512, 0], [1644, 0], [0, -125], [0, -113], [-3934, 0], [0, -532], [-3856, 0], [994, 0], [-6300, 0], [0, -768], [0, -161], [5082, 0], [2800, 0], [0, -182], [-1254, 0], [0, 533], [0, 750], [0, -729], [-5208, 0], [0, -524], [0, -98], [7828, 0], [0, 616], [0, 315], [8890, 0], [0, 176], [-9144, 0], [0, 1189], [0, -378], [2588, 0], [0, 467], [5426, 0], [0, 1126], [370, 0], [-4186, 0], [-2716, 0], [1966, 0], [-11824, 0], [0, -296], [13272, 0], [0, 357], [2094, 0], [0, -812], [-9450, 0], [0, -357], [0, 615], [-13216, 0], [0, 273], [-12300, 0], [-15756, 0], [0, 2199], [2394, 0], [6874, 0], [0, 1853], [7174, 0], [0, -497], [0, -1050], [0, -1591], [16070, 0], [-3932, 0], [0, -1856], [0, -1293], [0, -1224], [6840, 0], [-10150, 0], [0, -211], [0, 574], [0, -2828], [27020, 0], [-10648, 0], [1682, 0], [0, -2127], [0, -3519], [0, 154], [-21356, 0], [-13762, 0], [0, -672], [0, 3394], [0, 1981], [-1652, 0], [28790, 0], [7554, 0], [-28336, 0], [0, 2697], [0, 386], [-14300, 0], [0, -3633], [-11824, 0], [0, 154], [6762, 0], [-29442, 0], [0, 4110], [0, 2743], [0, 1966], [8646, 0], [-18956, 0], [13726, 0], [0, 5271], [0, -245], [8778, 0], [18088, 0], [0, 2630], [0, -516], [-40516, 0], [0, -3213], [0, 3878], [19362, 0], [0, -567], [0, 5049], [0, -4893], [-15888, 0], [-7944, 0], [20622, 0], [0, 4724], [40502, 0], [0, -441], [0, 3668], [0, -2191], [17556, 0], [0, 504], [0, 1079], [28576, 0], [0, 3618]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_100_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_100_4_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_100_4_a_d(:prec:=2)
    chi := MakeCharacter_100_a();
    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_100_4_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_100_4_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_100_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![-76, 0, 1]>],Snew);
    return Vf;
end function;