
// Make newform 1728.3.q.b in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1728_q();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1728_q_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_q_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_q_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
        poly := [1, -1, 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_1728_q();"
function MakeCharacter_1728_q()
    N := 1728;
    order := 6;
    char_gens := [703, 325, 1217];
    v := [6, 6, 5];
    // 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_q_Hecke();"
function MakeCharacter_1728_q_Hecke(Kf)
    N := 1728;
    order := 6;
    char_gens := [703, 325, 1217];
    char_values := [[1, 0], [1, 0], [0, 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, 0], [4, -2], [2, -2], [1, 1], [0, -4], [-9, 18], [11, 0], [-32, 16], [26, 26], [0, 32], [34, 0], [14, -7], [61, -61], [-28, -28], [0, 0], [-58, 29], [56, -56], [0, 31], [-18, 36], [65, 0], [38, -38], [28, 28], [-72, 144], [115, -115], [26, 26], [0, -40], [81, -162], [52, 0], [104, -52], [16, 0], [-184, 92], [109, 109], [0, -5], [-176, 88], [20, -20], [0, -40], [-106, 0], [220, -110], [134, 134], [-36, 72], [232, 0], [134, 134], [0, 265], [72, -144], [-290, 0], [0, 94], [-52, 52], [109, 109], [0, 266], [-117, 234], [-464, 232], [-119, 119], [225, -450], [-202, 101], [26, 26], [-108, 216], [268, 0], [56, -56], [28, 28], [0, -374], [292, -146], [533, 0], [-284, 142], [-155, 155], [-28, -28], [-2, 2], [0, -77], [-130, 65], [416, -416], [1, 1], [342, -684], [-358, 358], [0, -580], [83, 0], [-644, 322], [-298, -298], [-362, 0], [-454, 227], [0, -221], [-904, 452], [-682, 682], [-162, 324], [-295, 0], [812, -812], [-53, -53], [369, -738], [-65, 65], [-460, -460], [0, 734], [117, -234], [350, 350], [106, 0], [266, -133], [0, 787], [360, -720], [-248, 124], [117, -234], [-250, 0], [-650, 0], [-623, 623], [306, -612], [-130, 65], [-377, -377], [0, -545], [-871, 0], [1, 1], [-108, 216], [652, -326], [-461, 461], [0, -112], [-902, 0], [410, -205], [799, -799], [-830, 0], [217, 217], [0, 13], [270, -540], [436, -218], [568, 568], [0, -382], [-578, 578], [404, 404], [603, -1206], [-182, 182], [0, 0], [-700, 700], [-342, 684], [-664, 664], [0, -670], [317, 0], [-716, 358], [0, 1310], [-218, 0], [-760, 380], [0, -1022], [684, -1368], [0, 130], [364, -182], [81, -162], [299, 0], [350, 350], [0, -814], [828, -1656], [718, 0], [-460, -460], [1424, -1424], [-404, -404], [0, -311], [-594, 1188], [0, 104], [-36, 72], [119, 0], [1372, -686], [-695, 695], [-1000, -1000], [-56, 0], [460, 460], [470, 0], [-464, 232], [1, 1], [-477, 954], [0, 1202], [108, -216], [482, -241], [674, 674], [1420, 0], [524, -524], [-1222, 0], [-1688, 844], [-1025, -1025], [0, 338], [-104, 52], [1753, -1753], [0, -1678], [1976, -988], [1339, -1339], [342, -684], [250, 0], [-358, 358], [1190, -1190], [-53, -53], [0, 1130], [-927, 1854], [-1652, 826], [836, 836], [1168, 0], [-722, 722], [0, -203], [-18, 36], [2279, 0], [-593, -593], [-115, 0], [-352, -352], [459, -918], [266, -133], [0, 787], [524, -524], [379, 379], [-468, 936], [1228, -614], [-1816, 1816], [0, -148], [79, -79], [-1359, 2718], [-900, 1800], [-182, 0], [1742, -871], [1, 1], [0, -1175], [1271, 0], [-752, 376], [110, -110], [460, 460], [-1112, 556], [-605, 605], [0, 2300], [-485, -485], [-396, 792], [2344, -1172], [0, 140], [0, -202], [-706, 353], [-1766, 0], [2138, -1069], [1442, -1442], [676, 676], [126, -252], [-2518, 2518], [28, 28], [0, -1498], [218, 0], [0, -1786], [-2290, 1145], [1366, -1366], [-730, -730], [0, 130], [774, -1548], [338, -169], [-432, 864], [-701, -701], [-2122, 0], [0, 1058], [-668, 0], [-3334, 1667], [404, 404], [-596, 0], [3290, -1645], [0, 598], [36, -72], [0, -2722], [279, -558], [472, -236], [1762, -1762], [404, 404], [-1269, 2538], [-1460, 0], [-650, 650], [1170, -2340], [1181, 0], [2612, -2612], [-377, -377], [0, 500], [-686, 0], [1112, -556], [2635, -2635], [-252, 504], [-1727, -1727], [0, 2821], [-1508, 754], [0, 1832], [-3094, 0], [-2171, 2171], [0, -1750], [0, -1175], [3076, 0], [-760, 380], [-1618, 1618], [-2095, 0], [-1889, -1889], [2308, -1154], [0, 2660], [-1702, -1702], [-988, 988], [0, 2425], [-468, 936], [2342, 0], [-932, 466], [236, -236], [1863, -3726], [-1378, -1378], [-711, 1422], [950, -475], [-1930, 965], [-1114, 1114], [2056, -1028], [3080, -3080], [26, 26], [-504, 1008], [-182, 182], [0, 1633], [1512, -3024], [2590, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1728_q_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1728_3_q_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_q_b(:prec:=2)
    chi := MakeCharacter_1728_q();
    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_q_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_q_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1728_q();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![12, -6, 1]>,<7,R![4, -2, 1]>],Snew);
    return Vf;
end function;