
// Make newform 576.6.c.a in Magma, downloaded from the LMFDB on 12 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_576_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_576_c_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_576_6_c_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_576_6_c_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;
    Rf_num := [[1, 0], [0, 3]];
    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_576_c();"
function MakeCharacter_576_c()
    N := 576;
    order := 2;
    char_gens := [127, 325, 65];
    v := [1, 2, 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_576_c_Hecke();"
function MakeCharacter_576_c_Hecke(Kf)
    N := 576;
    order := 2;
    char_gens := [127, 325, 65];
    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]];
    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, 0], [0, 1], [0, 0], [0, 0], [-244, 0], [0, -239], [0, 0], [0, 0], [0, 1919], [0, 0], [-12242, 0], [0, -4317], [0, 0], [0, 0], [0, 7923], [0, 0], [-18950, 0], [0, 0], [0, 0], [-20144, 0], [0, 0], [0, 0], [0, 31919], [-160808, 0], [0, -13683], [0, 0], [0, 0], [28100, 0], [0, 60475], [0, 0], [0, 0], [0, 59525], [0, 0], [0, 97919], [0, 0], [-493658, 0], [0, 0], [0, 0], [0, 185525], [0, 0], [-763900, 0], [0, 0], [497294, 0], [0, 233525], [0, 0], [0, 0], [0, 0], [0, 0], [-976564, 0], [0, 152873], [0, 0], [-1739048, 0], [0, 0], [0, 227525], [0, 0], [0, -337919], [0, 0], [2451092, 0], [0, 465367], [0, 0], [0, -137525], [0, 0], [0, 0], [-2889094, 0], [0, -474239], [0, 0], [3488608, 0], [0, 0], [4445050, 0], [0, -1090077], [0, 0], [0, 0], [-5159294, 0], [0, 0], [0, 0], [0, 419031], [-259958, 0], [0, -1305367], [4976200, 0], [0, 0], [4184852, 0], [0, 0], [7784606, 0], [0, 0], [0, 0], [0, -1254969], [-7216808, 0], [0, -2146317], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, -2604969], [0, 2884317], [0, 0], [10828100, 0], [0, 0], [0, -2521189], [0, 0], [0, 1826869], [0, 0], [3306242, 0], [0, 0], [0, -3132475], [0, 0], [-15155050, 0], [0, 0], [-11150606, 0], [0, 1848239], [0, 0], [0, 0], [0, 38633], [0, 0], [0, 0], [0, 4888077], [0, 0], [-8605250, 0], [17398994, 0], [0, 4661289], [0, 0], [0, 0], [0, -1004633], [-26471236, 0], [0, 0], [0, 0], [-10512244, 0], [0, 0], [0, 0], [0, 0], [3004508, 0], [0, 7469267], [-20284750, 0], [0, 6293027], [0, 0], [0, -7926475], [0, 8733131], [0, 0], [0, -1775267], [0, 0], [0, 0], [38764100, 0], [0, 0], [-41824994, 0], [0, 10084811], [0, 0], [0, 0], [23838058, 0], [0, 9580317], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, -4231919], [-11026042, 0], [0, -8208733], [0, 0], [0, -12996475], [0, 0], [0, 0], [0, -2015289], [0, 0], [0, 0], [-36210542, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_576_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_576_6_c_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_576_6_c_a(:prec:=2)
    chi := MakeCharacter_576_c();
    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_576_6_c_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_576_6_c_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_576_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<5,R![18, 0, 1]>],Snew);
    return Vf;
end function;