
// Make newform 1728.3.b.f in Magma, downloaded from the LMFDB on 24 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_f();".
// 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_f();".
// 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], [-3, 0, 6, 0], [0, 6, 0, -3]];
    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, 6, 0, 0], [0, 5, 0, 0], [6, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, -6], [0, 0, 0, -5], [0, 0, 6, 0], [0, -36, 0, 0], [0, -26, 0, 0], [0, 0, -5, 0], [0, 0, 0, 0], [0, 0, 0, -4], [0, 0, 6, 0], [0, 60, 0, 0], [18, 0, 0, 0], [0, 0, -15, 0], [0, 0, 0, -15], [0, 0, 0, 0], [25, 0, 0, 0], [0, 31, 0, 0], [120, 0, 0, 0], [0, 0, 0, 30], [-85, 0, 0, 0], [0, -156, 0, 0], [0, -125, 0, 0], [-150, 0, 0, 0], [0, 0, -20, 0], [0, 0, 0, -6], [0, 110, 0, 0], [-156, 0, 0, 0], [0, 0, 0, 30], [0, 0, 0, -35], [0, -24, 0, 0], [0, -31, 0, 0], [0, 0, 4, 0], [0, 0, 0, -59], [0, 0, 30, 0], [0, 120, 0, 0], [192, 0, 0, 0], [0, 0, -65, 0], [0, 0, 30, 0], [-325, 0, 0, 0], [0, 30, 0, 0], [0, 259, 0, 0], [0, 0, 0, -5], [0, -310, 0, 0], [-360, 0, 0, 0], [0, 0, -40, 0], [0, 0, 0, -60], [0, 0, 60, 0], [-131, 0, 0, 0], [-216, 0, 0, 0], [0, 0, 0, -36], [0, 0, 60, 0], [0, -414, 0, 0], [0, 317, 0, 0], [0, 0, -100, 0], [0, 0, 0, -60], [0, 0, 0, -20], [0, 270, 0, 0], [0, 0, 0, 84], [0, 0, -90, 0], [85, 0, 0, 0], [0, 120, 0, 0], [0, 0, 0, 5], [-325, 0, 0, 0], [300, 0, 0, 0], [0, 0, 115, 0], [0, 0, 0, 84], [0, 0, 90, 0], [0, -185, 0, 0], [0, 0, 11, 0], [0, 0, 0, 25], [0, 0, -36, 0], [0, 354, 0, 0], [0, 0, 124, 0], [0, 0, 0, 60], [-539, 0, 0, 0], [-306, 0, 0, 0], [0, 0, 5, 0], [0, 0, -30, 0], [250, 0, 0, 0], [0, 22, 0, 0], [-540, 0, 0, 0], [0, 0, 0, 30], [-230, 0, 0, 0], [0, 78, 0, 0], [0, -55, 0, 0], [-150, 0, 0, 0], [0, 0, -60, 0], [0, 785, 0, 0], [618, 0, 0, 0], [0, 0, 0, 120], [0, 0, -126, 0], [0, 366, 0, 0], [0, 0, 0, -150], [0, 0, 0, 21], [0, 0, 125, 0], [0, 0, 0, 81], [0, 690, 0, 0], [-840, 0, 0, 0], [0, 0, 0, -210], [0, 0, 0, 155], [385, 0, 0, 0], [930, 0, 0, 0], [0, 0, 0, -96], [0, 0, 60, 0], [-346, 0, 0, 0], [0, -595, 0, 0], [0, 0, 21, 0], [0, 0, 0, -114], [0, 0, 0, 135], [0, 713, 0, 0], [0, 0, 0, 0], [0, 0, 0, -84], [0, 0, 120, 0], [0, -960, 0, 0], [-132, 0, 0, 0], [0, 0, -195, 0], [445, 0, 0, 0], [0, -990, 0, 0], [-1260, 0, 0, 0], [0, 0, 0, 20], [0, -246, 0, 0], [0, 0, 25, 0], [0, 0, 240, 0], [0, 770, 0, 0], [0, 0, 220, 0], [0, 0, 0, -140], [0, 0, 114, 0], [0, 1099, 0, 0], [0, 0, 81, 0], [0, 0, 0, -30], [-1009, 0, 0, 0], [0, 360, 0, 0], [0, 0, 0, 99], [0, 660, 0, 0], [0, 0, 0, 240], [0, 0, 0, -100], [0, 1242, 0, 0], [0, -955, 0, 0], [-90, 0, 0, 0], [0, 0, 35, 0], [0, 0, -300, 0], [0, 0, -9, 0], [0, 0, 0, 120], [0, 0, 0, -25], [0, 0, -66, 0], [0, 0, 199, 0], [0, 0, 0, 90], [0, 0, 0, -55], [0, 0, -96, 0], [0, 0, 0, -211], [0, 0, 120, 0], [0, -1006, 0, 0], [0, 0, 0, -60], [565, 0, 0, 0], [0, 294, 0, 0], [30, 0, 0, 0], [0, 0, 0, -174], [0, -305, 0, 0], [1194, 0, 0, 0], [0, 0, 0, -24], [0, 0, 246, 0], [0, 511, 0, 0], [0, 0, -204, 0], [119, 0, 0, 0], [0, -1530, 0, 0], [96, 0, 0, 0], [0, 0, -300, 0], [0, 0, 330, 0], [95, 0, 0, 0], [0, 1711, 0, 0], [0, 0, 0, 120], [0, 0, 0, -140], [0, 222, 0, 0], [0, 475, 0, 0], [0, 0, -80, 0], [0, -1625, 0, 0], [-1032, 0, 0, 0], [0, 0, -276, 0], [0, 0, 0, -174], [0, 0, 60, 0], [0, 1668, 0, 0], [0, 0, 39, 0], [0, 0, 0, -301], [683, 0, 0, 0], [0, 0, 270, 0], [-1955, 0, 0, 0], [-1380, 0, 0, 0], [0, 0, 0, -25], [0, 1074, 0, 0], [-450, 0, 0, 0], [0, 0, 0, -210], [577, 0, 0, 0], [0, 0, -109, 0], [0, 0, 0, -90], [0, 0, 180, 0], [0, -2226, 0, 0], [0, -787, 0, 0], [0, 0, 155, 0], [-373, 0, 0, 0], [804, 0, 0, 0], [0, -900, 0, 0], [0, -871, 0, 0], [750, 0, 0, 0], [0, 0, 0, -330], [0, 0, 0, -455], [-1465, 0, 0, 0], [0, -1674, 0, 0], [0, 1615, 0, 0], [-2010, 0, 0, 0], [0, 0, -420, 0], [851, 0, 0, 0], [0, 1990, 0, 0], [0, 0, 0, 270], [0, 0, 216, 0], [0, 360, 0, 0], [0, 0, 95, 0], [0, 1414, 0, 0], [0, 0, 0, 180], [0, -1150, 0, 0], [-150, 0, 0, 0], [0, 0, 145, 0], [0, 0, 0, 186], [0, 0, -210, 0], [0, 535, 0, 0], [414, 0, 0, 0], [0, 0, 65, 0], [0, 0, 0, -40], [0, 1214, 0, 0], [0, 0, 0, 30], [0, 0, 0, 35], [0, 0, 30, 0], [359, 0, 0, 0], [0, 1890, 0, 0], [-348, 0, 0, 0], [0, 0, -480, 0], [900, 0, 0, 0], [0, 0, 0, -285], [0, -2215, 0, 0], [0, 0, 75, 0], [0, 0, 0, 570], [0, 0, 510, 0], [0, -2045, 0, 0], [2436, 0, 0, 0], [0, 0, 0, 320], [0, 0, -324, 0], [0, 0, -84, 0], [0, 0, 0, 210], [0, 0, -510, 0], [2507, 0, 0, 0], [0, 1050, 0, 0], [-2334, 0, 0, 0], [0, 0, -105, 0], [0, 0, 0, 436], [0, -1200, 0, 0], [-610, 0, 0, 0], [0, -1435, 0, 0], [-1410, 0, 0, 0], [0, 0, 225, 0], [0, 0, -79, 0], [0, 0, 0, 504], [0, 0, 0, -380], [0, 882, 0, 0], [0, 0, 0, 240], [0, 0, 0, 544], [0, 90, 0, 0], [0, 0, -175, 0], [0, 0, 0, 611], [2930, 0, 0, 0], [0, 2971, 0, 0], [1450, 0, 0, 0], [0, -3125, 0, 0], [-390, 0, 0, 0], [0, 0, -260, 0], [-1834, 0, 0, 0], [-216, 0, 0, 0], [0, 0, -174, 0], [0, 1666, 0, 0], [0, 0, -204, 0], [0, 0, 435, 0], [0, 0, 0, 39], [0, 0, 630, 0], [-875, 0, 0, 0], [0, -1380, 0, 0], [0, 761, 0, 0], [0, 0, 0, -90], [0, -1854, 0, 0], [1320, 0, 0, 0], [0, 0, 0, -36], [-1488, 0, 0, 0], [0, 0, -120, 0], [0, 1152, 0, 0], [0, -3127, 0, 0], [0, 2130, 0, 0], [-876, 0, 0, 0], [0, 0, 0, 295], [1225, 0, 0, 0], [0, 660, 0, 0], [0, -3746, 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_f();".
// 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_f(: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_f();".
// 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_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1728_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![36, 0, 1]>,<7,R![25, 0, 1]>,<11,R![-6, 1]>,<17,R![-972, 0, 1]>],Snew);
    return Vf;
end function;