
// Make newform 1728.3.e.b in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1728_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1728_e_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_1728_3_e_b();".
// 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_1728_3_e_b();".
// 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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_1728_e();"
function MakeCharacter_1728_e()
    N := 1728;
    order := 2;
    char_gens := [703, 325, 1217];
    v := [2, 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_1728_e_Hecke();"
function MakeCharacter_1728_e_Hecke(Kf)
    N := 1728;
    order := 2;
    char_gens := [703, 325, 1217];
    char_values := [[1], [1], [-1]];
    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 := 3;
    raw_aps := [[0], [0], [0], [-11], [0], [-23], [0], [-37], [0], [0], [46], [73], [0], [-22], [0], [0], [0], [-47], [-13], [0], [143], [-11], [0], [0], [-169], [0], [157], [0], [214], [0], [-146], [0], [0], [-229], [0], [-227], [118], [-37], [0], [0], [0], [1], [0], [239], [0], [109], [419], [-338], [0], [-26], [0], [0], [479], [0], [0], [0], [0], [541], [-122], [0], [458], [0], [-358], [0], [-457], [0], [299], [167], [0], [169], [0], [0], [-227], [121], [611], [0], [0], [-362], [0], [-769], [0], [-839], [0], [-862], [94], [0], [0], [-814], [0], [-923], [0], [0], [613], [0], [26], [0], [0], [0], [179], [793], [587], [0], [0], [0], [1067], [71], [0], [0], [0], [-526], [373], [-143], [0], [1163], [1261], [0], [314], [0], [0], [0], [1201], [-1177], [0], [0], [-1318], [0], [457], [0], [-482], [-1034], [-1222], [0], [1381], [-1511], [0], [671], [0], [-949], [0], [0], [1514], [0], [-563], [0], [-1151], [0], [1177], [0], [131], [0], [1129], [0], [1259], [0], [1667], [0], [-866], [0], [-649], [0], [0], [0], [-1787], [0], [0], [0], [1693], [1894], [-1657], [0], [0], [1846], [0], [-2041], [2029], [0], [1994], [0], [757], [-1706], [-1307], [0], [1702], [0], [0], [0], [1873], [2243], [-1417], [0], [719], [0], [467], [0], [0], [0], [1319], [-2063], [0], [0], [0], [61], [1633], [1991], [0], [0], [-2483], [0], [0], [-2461], [2447], [0], [2437], [0], [0], [-241], [2638], [0], [0], [0], [913], [-1826], [0], [2446], [0], [2689], [0], [0], [1213], [0], [-1319], [218], [-2834], [0], [2963], [0], [-1129], [0], [0], [0], [0], [1979], [2461], [-2231], [0], [0], [1909], [0], [-2134], [0], [694], [0], [0], [2711], [0], [0], [433], [3146], [0], [1586], [3301], [0], [3289], [2161], [0], [698], [0], [0], [-1846], [0], [-3407], [-3133], [-3406], [-3371], [-3358], [541], [0], [-3146], [1874], [0], [0], [-962], [0], [1321], [-2893], [0], [-3457], [0], [-2171], [0], [0], [0], [0], [0], [22], [0], [-2027], [0], [0], [-1573], [2903], [0], [1294]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1728_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1728_3_e_b();".
// 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_1728_3_e_b(:prec:=1)
    chi := MakeCharacter_1728_e();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_1728_3_e_b();".
// 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_1728_3_e_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1728_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![0, 1]>,<7,R![11, 1]>],Snew);
    return Vf;
end function;