
// Make newform 1728.3.b.e in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1728_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1728_b_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_b_e();".
// 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_b_e();".
// 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 := [1, 0, -1, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 0, 0, 1], [-1, 0, 2, 0], [0, 2, 0, -1]];
    Rf_basisdens := [1, 1, 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_1728_b();"
function MakeCharacter_1728_b()
    N := 1728;
    order := 2;
    char_gens := [703, 325, 1217];
    v := [1, 1, 2];
    // 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_b_Hecke();"
function MakeCharacter_1728_b_Hecke(Kf)
    N := 1728;
    order := 2;
    char_gens := [703, 325, 1217];
    char_values := [[-1, 0, 0, 0], [-1, 0, 0, 0], [1, 0, 0, 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 := 3;
    raw_aps := [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -2, 0], [0, -1, 0, 0], [0, 0, 0, 10], [0, 0, 1, 0], [6, 0, 0, 0], [0, 0, 0, 1], [0, 30, 0, 0], [0, 0, 12, 0], [0, -14, 0, 0], [0, 0, 11, 0], [-48, 0, 0, 0], [0, 0, 0, 12], [0, -66, 0, 0], [0, 0, 28, 0], [0, 0, 0, -18], [0, 0, 25, 0], [0, 0, 0, 35], [0, 48, 0, 0], [49, 0, 0, 0], [0, -83, 0, 0], [0, 0, 0, 8], [66, 0, 0, 0], [107, 0, 0, 0], [0, 0, -92, 0], [0, -95, 0, 0], [0, 0, 0, -10], [0, 0, 108, 0], [150, 0, 0, 0], [0, -190, 0, 0], [0, 0, 0, 28], [-222, 0, 0, 0], [0, 0, 0, 95], [0, 0, 56, 0], [0, -109, 0, 0], [0, 0, 36, 0], [0, 0, 0, -49], [0, -234, 0, 0], [0, 0, 104, 0], [0, 0, 0, -112], [0, 0, 95, 0], [0, 198, 0, 0], [227, 0, 0, 0], [0, 0, 134, 0], [0, -239, 0, 0], [0, 0, 0, 169], [0, -130, 0, 0], [0, 0, 0, 24], [0, 0, -192, 0], [-276, 0, 0, 0], [0, 204, 0, 0], [133, 0, 0, 0], [0, 0, 0, 264], [-300, 0, 0, 0], [0, -84, 0, 0], [0, 0, -54, 0], [0, -265, 0, 0], [0, 0, -180, 0], [540, 0, 0, 0], [0, 0, 0, 204], [0, 0, -90, 0], [0, 0, 0, -204], [0, 462, 0, 0], [-35, 0, 0, 0], [0, 0, 152, 0], [0, 0, 0, 23], [59, 0, 0, 0], [0, 0, 0, -364], [0, 0, 107, 0], [12, 0, 0, 0], [0, -318, 0, 0], [0, 109, 0, 0], [0, 0, -421, 0], [0, 0, 0, -253], [0, -516, 0, 0], [0, 0, 314, 0], [0, 0, 396, 0], [-252, 0, 0, 0], [-11, 0, 0, 0], [0, 0, 0, -78], [0, 0, -347, 0], [0, -486, 0, 0], [-494, 0, 0, 0], [0, 610, 0, 0], [0, 0, 0, -84], [-366, 0, 0, 0], [-398, 0, 0, 0], [0, 0, -410, 0], [0, 875, 0, 0], [0, 0, 0, -202], [0, 228, 0, 0], [0, 251, 0, 0], [0, 0, 0, 406], [0, 0, 0, 216], [0, 762, 0, 0], [0, 0, 118, 0], [726, 0, 0, 0], [0, 0, 0, -481], [0, 0, 373, 0], [0, 0, 0, -181], [0, 0, 394, 0], [0, 0, 0, 376], [-846, 0, 0, 0], [0, 0, 0, 217], [-791, 0, 0, 0], [0, 0, 0, 158], [-768, 0, 0, 0], [0, -564, 0, 0], [-178, 0, 0, 0], [0, 1103, 0, 0], [0, 0, -131, 0], [162, 0, 0, 0], [0, 0, 0, -203], [0, -85, 0, 0], [-720, 0, 0, 0], [0, 0, 0, -228], [0, -312, 0, 0], [0, 0, 416, 0], [0, 0, 0, 116], [0, 0, -539, 0], [-323, 0, 0, 0], [0, 0, 426, 0], [0, 0, 0, -612], [0, 0, 0, 324], [0, 0, 322, 0], [0, 0, -527, 0], [0, 432, 0, 0], [0, 758, 0, 0], [0, 0, 156, 0], [0, 0, 0, -492], [0, -150, 0, 0], [0, 457, 0, 0], [0, 0, -359, 0], [-1458, 0, 0, 0], [47, 0, 0, 0], [0, 0, -408, 0], [0, 0, 0, 409], [0, 0, -652, 0], [288, 0, 0, 0], [0, 0, 0, 108], [0, 0, 66, 0], [0, -601, 0, 0], [0, 0, 0, 474], [0, 0, 371, 0], [0, -972, 0, 0], [0, 0, -193, 0], [24, 0, 0, 0], [0, 0, 0, 61], [0, 150, 0, 0], [0, 0, 23, 0], [1014, 0, 0, 0], [0, 0, 0, 275], [0, 768, 0, 0], [0, 0, 0, -121], [0, -600, 0, 0], [0, 422, 0, 0], [1116, 0, 0, 0], [-275, 0, 0, 0], [0, 0, 190, 0], [0, 0, 0, -94], [-1506, 0, 0, 0], [0, -1307, 0, 0], [0, 0, 0, -362], [-840, 0, 0, 0], [0, 1134, 0, 0], [0, -1355, 0, 0], [0, 0, 228, 0], [527, 0, 0, 0], [0, 0, 942, 0], [0, 0, 0, -416], [0, 0, 852, 0], [0, 210, 0, 0], [1055, 0, 0, 0], [0, 421, 0, 0], [696, 0, 0, 0], [0, 0, 0, 372], [0, 0, 1078, 0], [0, -95, 0, 0], [0, 0, 168, 0], [0, -683, 0, 0], [0, 0, 0, -760], [0, 0, 60, 0], [-402, 0, 0, 0], [0, -2052, 0, 0], [0, 0, -220, 0], [0, 0, -49, 0], [0, 0, 0, -335], [-493, 0, 0, 0], [0, -1962, 0, 0], [901, 0, 0, 0], [0, 0, 0, 628], [0, 0, 0, 1093], [0, 0, 1178, 0], [0, 0, 0, 930], [162, 0, 0, 0], [1465, 0, 0, 0], [0, 0, 659, 0], [474, 0, 0, 0], [0, 2244, 0, 0], [0, 0, 118, 0], [0, -1945, 0, 0], [0, 0, 851, 0], [827, 0, 0, 0], [0, 0, 0, -148], [0, 0, -1044, 0], [0, -397, 0, 0], [0, 0, 0, 674], [234, 0, 0, 0], [0, 0, 0, -781], [599, 0, 0, 0], [0, 0, -402, 0], [0, 1045, 0, 0], [0, 0, 0, 106], [0, -36, 0, 0], [1619, 0, 0, 0], [0, 946, 0, 0], [-750, 0, 0, 0], [0, -2040, 0, 0], [0, 0, 120, 0], [0, 0, 863, 0], [0, -1934, 0, 0], [-1860, 0, 0, 0], [0, 758, 0, 0], [0, 0, 0, -586], [0, 0, 817, 0], [2502, 0, 0, 0], [0, 918, 0, 0], [0, 1717, 0, 0], [0, 0, 0, -846], [0, 0, 625, 0], [0, 0, 0, 1608], [0, 530, 0, 0], [-1278, 0, 0, 0], [0, 0, 0, -1415], [0, -1290, 0, 0], [-625, 0, 0, 0], [0, 0, 666, 0], [0, 0, 0, 140], [0, -1056, 0, 0], [0, 0, 0, -804], [0, 0, 0, 817], [0, -2149, 0, 0], [0, 0, -1309, 0], [342, 0, 0, 0], [0, -2730, 0, 0], [0, 2209, 0, 0], [0, 0, 0, -1700], [0, 0, 0, -384], [0, 876, 0, 0], [0, 0, -276, 0], [-1170, 0, 0, 0], [0, 1098, 0, 0], [-397, 0, 0, 0], [0, 0, 82, 0], [0, 0, 0, 286], [0, 0, -457, 0], [0, 0, 0, 564], [0, 0, 16, 0], [-10, 0, 0, 0], [0, -1225, 0, 0], [0, 0, 0, 482], [0, 0, 1657, 0], [0, 0, -1223, 0], [-744, 0, 0, 0], [0, 0, 0, -1692], [0, 0, -486, 0], [432, 0, 0, 0], [0, 0, 0, -1200], [0, 0, -1566, 0], [0, 0, -23, 0], [0, 0, 0, 913], [-2734, 0, 0, 0], [0, 3193, 0, 0], [-3326, 0, 0, 0], [0, 2425, 0, 0], [0, 0, 0, 1366], [0, 0, -1332, 0], [-1090, 0, 0, 0], [0, 0, 0, 1608], [0, -1446, 0, 0], [0, -2930, 0, 0], [0, -156, 0, 0], [0, 0, 1139, 0], [0, 0, 0, -1571], [0, 30, 0, 0], [13, 0, 0, 0], [0, 0, 1996, 0], [0, 2099, 0, 0], [-438, 0, 0, 0], [0, 0, -1254, 0], [0, 0, 0, -248], [1620, 0, 0, 0], [0, 0, 0, -1664], [0, 0, -816, 0], [0, 0, 1776, 0], [0, -1333, 0, 0], [0, 0, 826, 0], [0, 0, 0, -1220], [0, 0, 0, -1763], [-1751, 0, 0, 0], [0, 0, 1748, 0], [0, 2314, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1728_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1728_3_b_e();".
// 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_b_e(:prec:=4)
    chi := MakeCharacter_1728_b();
    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_b_e();".
// 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_b_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1728_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![12, 0, 1]>,<7,R![1, 0, 1]>,<11,R![-300, 0, 1]>,<17,R![-6, 1]>],Snew);
    return Vf;
end function;