
// Make newform 24.8.f.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_24_f();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_24_f_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_24_8_f_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_24_8_f_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 := [2, 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_24_f();"
function MakeCharacter_24_f()
    N := 24;
    order := 2;
    char_gens := [7, 13, 17];
    v := [1, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_24_f_Hecke();"
function MakeCharacter_24_f_Hecke(Kf)
    N := 24;
    order := 2;
    char_gens := [7, 13, 17];
    char_values := [[-1, 0], [-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 := 8;
    raw_aps := [[0, 8], [-43, 13], [0, 0], [0, 0], [0, -362], [0, 0], [0, -23972], [-59722, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, -601208], [-220510, 0], [0, 0], [0, 0], [0, 2108530], [0, 0], [-3851302, 0], [0, 0], [-4865614, 0], [0, 0], [0, -6535226], [0, 7965436], [-9938890, 0], [0, 0], [0, 0], [0, -8417318], [0, 0], [0, 15353416], [0, 0], [0, -36341630], [0, 10266352], [53491262, 0], [0, 0], [0, 0], [0, 0], [-57691114, 0], [0, 0], [0, 0], [0, 47445034], [0, 0], [0, 0], [22530794, 0], [0, 0], [0, 0], [100760954, 0], [0, 0], [0, -152419034], [0, 0], [0, 263445052], [0, 0], [-267976774, 0], [0, -83266898], [0, -222720056], [0, 0], [0, 0], [0, 0], [0, 0], [0, -356661740], [-733601506, 0], [0, 0], [142695722, 0], [0, 0], [984034790, 0], [0, 0], [861506126, 0], [-551176402, 0], [0, 467362558], [0, 0], [0, -1147974032], [0, 0], [0, 0], [0, 0], [69823442, 0], [0, 0], [0, 0], [0, 0], [0, -904400252], [-2140061638, 0], [0, -2127268754], [0, 0], [0, 0], [-3310269946, 0], [0, 0], [0, -1173572426], [0, 2186001436], [4042701962, 0], [0, 0], [0, 0], [0, -1903373186], [0, 0], [0, 0], [0, 2830485970], [-4384230214, 0], [0, 0], [0, 0], [0, 2755805728], [-3590008846, 0], [0, 0], [-2217845242, 0], [0, 0], [0, 467244622], [0, -1940431376], [-1828785442, 0], [-6486044578, 0], [0, -4474134902], [0, -6330757856], [0, 0], [8358875234, 0], [0, 0], [0, 0], [0, 1427593636], [7665784334, 0], [0, 0], [0, 7971283060], [5176552550, 0], [0, 0], [0, 0], [0, 8369691226], [0, 0], [15457579190, 0], [0, 0], [0, -11411455154], [16000466534, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-21939405046, 0], [0, 0], [0, 0], [0, 0], [0, -17058985208], [7124793962, 0], [0, 0], [-27021941050, 0], [0, 0], [0, -15732115880], [27976547378, 0], [0, 0], [0, 0], [0, -14492621318], [0, 0], [0, 0], [0, 0], [0, 9277011952], [-19544098702, 0], [0, 0], [0, 0], [0, 15866083480], [-9551387914, 0], [0, 0], [41337799010, 0], [0, 0], [0, 0], [0, 8075671540], [-41827759378, 0], [0, 0], [0, -33711338126], [0, 18919642144], [0, 0], [0, -19953517298], [0, 32237958412], [0, 0], [0, 0], [0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_24_f_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_24_8_f_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_24_8_f_a(:prec:=2)
    chi := MakeCharacter_24_f();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 8));
    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_24_8_f_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_24_8_f_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_24_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,8,sign)));
    Vf := Kernel([<5,R![0, 1]>],Snew);
    return Vf;
end function;