
// Make newform 1200.3.bg.j in Magma, downloaded from the LMFDB on 13 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1200_bg();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1200_bg_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_1200_3_bg_j();".
// 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_1200_3_bg_j();".
// 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 := [9, 0, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
    Rf_basisdens := [1, 1, 3, 3];
    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_1200_bg();"
function MakeCharacter_1200_bg()
    N := 1200;
    order := 4;
    char_gens := [751, 901, 401, 577];
    v := [4, 4, 4, 3];
    // 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_1200_bg_Hecke();"
function MakeCharacter_1200_bg_Hecke(Kf)
    N := 1200;
    order := 4;
    char_gens := [751, 901, 401, 577];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [1, 0, 0, 0], [0, 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+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, -1], [0, 0, 0, 0], [0, 6, 0, 0], [18, 0, 0, 0], [0, 0, 0, -6], [0, 4, 0, 0], [0, 0, 10, 0], [0, 0, 0, 16], [0, 0, 0, 0], [-22, 0, 0, 0], [0, -6, 0, 0], [-18, 0, 0, 0], [0, 0, 0, -24], [0, 36, 0, 0], [0, 0, 0, 4], [0, 0, 90, 0], [2, 0, 0, 0], [0, 36, 0, 0], [-72, 0, 0, 0], [0, 0, 0, -36], [0, 0, -70, 0], [0, 0, 0, -44], [0, 0, -90, 0], [0, 84, 0, 0], [-108, 0, 0, 0], [0, 0, 0, -54], [0, 36, 0, 0], [0, 0, -170, 0], [0, 0, 0, -36], [0, 6, 0, 0], [18, 0, 0, 0], [0, -116, 0, 0], [0, 0, -170, 0], [0, 0, 180, 0], [-22, 0, 0, 0], [0, 114, 0, 0], [0, 0, 0, 96], [0, 16, 0, 0], [0, 0, 0, -116], [0, 0, -90, 0], [-98, 0, 0, 0], [-252, 0, 0, 0], [0, 0, 0, -216], [0, 104, 0, 0], [0, 0, 290, 0], [-122, 0, 0, 0], [0, 0, 0, 66], [0, -44, 0, 0], [0, 0, 50, 0], [0, 0, 0, -36], [0, 0, 180, 0], [-178, 0, 0, 0], [-342, 0, 0, 0], [0, -236, 0, 0], [0, 0, 0, 216], [0, 0, 0, 0], [478, 0, 0, 0], [0, -66, 0, 0], [162, 0, 0, 0], [0, 0, 0, -204], [0, 0, 0, 304], [0, -324, 0, 0], [-252, 0, 0, 0], [0, 0, 0, -96], [0, 224, 0, 0], [418, 0, 0, 0], [0, -156, 0, 0], [0, 36, 0, 0], [0, 0, -70, 0], [0, 0, 0, -36], [0, 0, -540, 0], [0, -54, 0, 0], [0, 0, 0, 174], [0, 0, -170, 0], [0, 0, 0, -124], [0, 0, 360, 0], [0, -246, 0, 0], [-558, 0, 0, 0], [0, 0, -670, 0], [0, 0, 630, 0], [142, 0, 0, 0], [-612, 0, 0, 0], [0, 0, 0, 84], [0, 0, -430, 0], [0, 0, 0, 476], [0, 0, -90, 0], [0, -156, 0, 0], [-828, 0, 0, 0], [0, 0, 0, 6], [0, -284, 0, 0], [0, 0, -360, 0], [0, -474, 0, 0], [18, 0, 0, 0], [0, 0, 590, 0], [0, 0, 0, 76], [0, 0, -540, 0], [342, 0, 0, 0], [0, 0, 0, 216], [2, 0, 0, 0], [0, 216, 0, 0], [0, -396, 0, 0], [0, 0, 0, -324], [0, 0, -630, 0], [-302, 0, 0, 0], [0, -396, 0, 0], [0, 476, 0, 0], [0, 0, 0, -436], [0, 0, -540, 0], [-758, 0, 0, 0], [0, 546, 0, 0], [0, 0, 0, 54], [0, 484, 0, 0], [0, 0, -1030, 0], [-242, 0, 0, 0], [162, 0, 0, 0], [0, 0, 0, 216], [0, 76, 0, 0], [0, 0, 0, -56], [0, 0, 630, 0], [622, 0, 0, 0], [0, 0, 0, 84], [0, 144, 0, 0], [0, 0, 0, -364], [-682, 0, 0, 0], [-468, 0, 0, 0], [0, 0, -310, 0], [0, 0, 180, 0], [0, -414, 0, 0], [0, 0, 0, 654], [0, 0, -350, 0], [0, 0, 0, 756], [338, 0, 0, 0], [0, 234, 0, 0], [-1278, 0, 0, 0], [0, 0, -590, 0], [0, 0, 0, 104], [0, 216, 0, 0], [0, 164, 0, 0], [0, 0, 630, 0], [218, 0, 0, 0], [432, 0, 0, 0], [0, 0, 0, -294], [0, 716, 0, 0], [0, 0, -70, 0], [0, 0, -540, 0], [0, 0, 0, -666], [0, 244, 0, 0], [0, 0, -310, 0], [0, 0, 0, -184], [0, 174, 0, 0], [1062, 0, 0, 0], [0, 0, 0, 96], [0, 416, 0, 0], [0, 336, 0, 0], [-792, 0, 0, 0], [0, 0, 430, 0], [0, 0, -1530, 0], [0, 264, 0, 0], [1152, 0, 0, 0], [0, -604, 0, 0], [0, 0, 0, -36], [0, -594, 0, 0], [1278, 0, 0, 0], [0, 324, 0, 0], [0, 0, 0, 676], [118, 0, 0, 0], [0, -666, 0, 0], [0, 0, 730, 0], [0, 0, 0, -196], [0, 0, -90, 0], [-958, 0, 0, 0], [-1512, 0, 0, 0], [0, 0, 0, 984], [0, 0, 1030, 0], [0, 0, -450, 0], [-502, 0, 0, 0], [2052, 0, 0, 0], [0, 0, 0, -174], [0, 0, -410, 0], [0, -114, 0, 0], [1818, 0, 0, 0], [0, 0, 0, 774], [0, -756, 0, 0], [0, 0, 0, -1204], [0, 0, 360, 0], [0, 234, 0, 0], [0, 0, 0, 876], [0, 0, 1310, 0], [-252, 0, 0, 0], [0, 0, 0, 384], [0, 0, 0, -604], [458, 0, 0, 0], [-1548, 0, 0, 0], [0, -364, 0, 0], [0, 0, 0, 1244], [482, 0, 0, 0], [0, 0, 0, -1326], [0, 324, 0, 0], [0, 0, 0, -324], [0, 0, -1080, 0], [-2182, 0, 0, 0], [0, -786, 0, 0], [0, 0, -970, 0], [0, 0, 1530, 0], [0, 144, 0, 0], [0, 0, 2450, 0], [0, 0, 0, -164], [0, 0, 90, 0], [-442, 0, 0, 0], [0, 1344, 0, 0], [1692, 0, 0, 0], [0, 0, 0, 1206], [0, -1044, 0, 0], [0, 0, -1440, 0], [-578, 0, 0, 0], [0, -354, 0, 0], [1242, 0, 0, 0], [0, -724, 0, 0], [0, 0, 0, -916], [1682, 0, 0, 0], [0, 0, -1850, 0], [0, 0, 1350, 0], [0, 0, 0, -894], [0, -1244, 0, 0], [0, 0, 1150, 0], [0, 0, 0, 1484], [0, 0, -540, 0], [0, -174, 0, 0], [-162, 0, 0, 0], [0, 0, 0, 1254], [0, 0, 170, 0], [-242, 0, 0, 0], [1602, 0, 0, 0], [0, 0, 0, -384], [0, 296, 0, 0], [0, 0, 670, 0], [0, 0, 0, -896], [0, 0, 990, 0], [-1872, 0, 0, 0], [0, 0, 0, 756], [-1522, 0, 0, 0], [0, 0, 0, 1206], [0, 0, 1270, 0], [0, 0, 0, -36], [0, 0, 1440, 0], [0, 1446, 0, 0], [-162, 0, 0, 0], [0, 0, -1810, 0], [0, 0, 0, 896], [0, -126, 0, 0], [-1098, 0, 0, 0], [0, -1064, 0, 0], [0, 0, -2450, 0], [0, 0, 0, 404], [0, 0, 450, 0], [-1898, 0, 0, 0], [0, 1716, 0, 0], [0, -76, 0, 0], [0, 564, 0, 0], [0, 0, 0, 6], [0, 1036, 0, 0], [0, 0, 1850, 0], [0, 0, 0, -1146], [0, 884, 0, 0], [0, 0, -1570, 0], [0, 0, -720, 0], [1422, 0, 0, 0], [0, 0, 0, -624], [0, 0, 0, -936], [1462, 0, 0, 0], [0, 516, 0, 0], [0, 0, 0, 984], [0, 0, 2830, 0], [0, -1536, 0, 0], [0, 0, 0, -54], [0, -484, 0, 0], [0, 0, -2590, 0], [962, 0, 0, 0], [18, 0, 0, 0], [0, 0, 0, -144], [2498, 0, 0, 0], [0, -244, 0, 0], [-1658, 0, 0, 0], [0, 576, 0, 0], [1908, 0, 0, 0], [0, 0, 0, 1104], [0, 1224, 0, 0], [0, 0, -2950, 0], [0, 0, 1710, 0], [1872, 0, 0, 0], [0, -284, 0, 0], [0, 0, 0, -1116], [378, 0, 0, 0], [0, 0, 0, 234], [0, 0, 2700, 0], [-2, 0, 0, 0], [0, 0, 0, -396], [0, 0, 270, 0], [0, -1704, 0, 0], [0, 0, 0, -696], [0, 824, 0, 0], [0, 0, -650, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1200_bg_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1200_3_bg_j();".
// 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_1200_3_bg_j(:prec:=4)
    chi := MakeCharacter_1200_bg();
    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_1200_3_bg_j();".
// 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_1200_3_bg_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1200_bg();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![11664, 0, 0, 0, 1]>],Snew);
    return Vf;
end function;