
// Make newform 1200.3.bg.g in Magma, downloaded from the LMFDB on 13 September 2025.
// 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_g();".
// 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_g();".
// 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 := false)
    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 := ISA(Type(Kf), Fld), 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]];
    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, 5, 0, 0], [-6, 0, 0, 0], [0, 0, 0, 3], [0, -14, 0, 0], [0, 0, -23, 0], [0, 0, 0, 10], [0, 0, 6, 0], [-25, 0, 0, 0], [0, 20, 0, 0], [-60, 0, 0, 0], [0, 0, 0, -49], [0, 6, 0, 0], [0, 0, 0, -20], [0, 0, 18, 0], [-37, 0, 0, 0], [0, -21, 0, 0], [-132, 0, 0, 0], [0, 0, 0, -20], [0, 0, -10, 0], [0, 0, 0, -2], [0, 0, -132, 0], [0, -19, 0, 0], [96, 0, 0, 0], [0, 0, 0, 16], [0, -24, 0, 0], [0, 0, -167, 0], [0, 0, 0, -48], [0, -36, 0, 0], [-108, 0, 0, 0], [0, -98, 0, 0], [0, 0, 58, 0], [0, 0, -186, 0], [83, 0, 0, 0], [0, 37, 0, 0], [0, 0, 0, -81], [0, -80, 0, 0], [0, 0, 0, 142], [0, 0, -150, 0], [-215, 0, 0, 0], [234, 0, 0, 0], [0, 0, 0, -53], [0, 158, 0, 0], [0, 0, 11, 0], [85, 0, 0, 0], [0, 0, 0, -83], [0, 160, 0, 0], [0, 0, 227, 0], [0, 0, 0, 84], [0, 0, -228, 0], [191, 0, 0, 0], [-192, 0, 0, 0], [0, -116, 0, 0], [0, 0, 0, 204], [0, 0, -462, 0], [-446, 0, 0, 0], [0, 143, 0, 0], [-546, 0, 0, 0], [0, 0, 0, 157], [0, 0, 0, 226], [0, 71, 0, 0], [294, 0, 0, 0], [0, 0, 0, 321], [0, -76, 0, 0], [178, 0, 0, 0], [0, 123, 0, 0], [0, -306, 0, 0], [0, 0, -514, 0], [0, 0, 0, 198], [0, 0, 648, 0], [0, 117, 0, 0], [0, 0, 0, 29], [0, 0, -215, 0], [0, 0, 0, -16], [0, 0, 192, 0], [0, 299, 0, 0], [228, 0, 0, 0], [0, 0, 707, 0], [0, 0, -48, 0], [514, 0, 0, 0], [-30, 0, 0, 0], [0, 0, 0, -289], [0, 0, 575, 0], [0, 0, 0, 92], [0, 0, 204, 0], [0, -360, 0, 0], [132, 0, 0, 0], [0, 0, 0, 356], [0, 226, 0, 0], [0, 0, -810, 0], [0, 399, 0, 0], [-348, 0, 0, 0], [0, 0, 227, 0], [0, 0, 0, 112], [0, 0, -618, 0], [690, 0, 0, 0], [0, 0, 0, -209], [-325, 0, 0, 0], [0, -400, 0, 0], [0, 126, 0, 0], [0, 0, 0, 498], [0, 0, 198, 0], [169, 0, 0, 0], [0, -199, 0, 0], [0, 236, 0, 0], [0, 0, 0, 344], [0, 0, -144, 0], [-899, 0, 0, 0], [0, 4, 0, 0], [0, 0, 0, 212], [0, -8, 0, 0], [0, 0, -637, 0], [-287, 0, 0, 0], [-60, 0, 0, 0], [0, 0, 0, 156], [0, -536, 0, 0], [0, 0, 0, -614], [0, 0, -420, 0], [-158, 0, 0, 0], [0, 0, 0, 100], [0, 198, 0, 0], [0, 0, 0, -16], [74, 0, 0, 0], [-102, 0, 0, 0], [0, 0, -817, 0], [0, 0, 6, 0], [0, -37, 0, 0], [0, 0, 0, -396], [0, 0, 298, 0], [0, 0, 0, -204], [-1186, 0, 0, 0], [0, 607, 0, 0], [-576, 0, 0, 0], [0, 0, -491, 0], [0, 0, 0, -424], [0, 101, 0, 0], [0, -376, 0, 0], [0, 0, 900, 0], [-1477, 0, 0, 0], [-786, 0, 0, 0], [0, 0, 0, 303], [0, 644, 0, 0], [0, 0, 170, 0], [0, 0, 990, 0], [0, 0, 0, 591], [0, 604, 0, 0], [0, 0, -1510, 0], [0, 0, 0, -304], [0, -245, 0, 0], [-216, 0, 0, 0], [0, 0, 0, 691], [0, -814, 0, 0], [0, -44, 0, 0], [558, 0, 0, 0], [0, 0, -755, 0], [0, 0, -762, 0], [0, -317, 0, 0], [-690, 0, 0, 0], [0, -82, 0, 0], [0, 0, 0, -612], [0, 4, 0, 0], [-90, 0, 0, 0], [0, 246, 0, 0], [0, 0, 0, -380], [-1067, 0, 0, 0], [0, -228, 0, 0], [0, 0, -455, 0], [0, 0, 0, 662], [0, 0, 942, 0], [-1357, 0, 0, 0], [1446, 0, 0, 0], [0, 0, 0, -809], [0, 0, -1211, 0], [0, 0, 1968, 0], [1655, 0, 0, 0], [-294, 0, 0, 0], [0, 0, 0, 921], [0, 0, -155, 0], [0, 405, 0, 0], [1044, 0, 0, 0], [0, 0, 0, -863], [0, -366, 0, 0], [0, 0, 0, -994], [0, 0, 1050, 0], [0, -279, 0, 0], [0, 0, 0, -3], [0, 0, -1249, 0], [-234, 0, 0, 0], [0, 0, 0, 919], [0, 0, 0, 428], [2147, 0, 0, 0], [1986, 0, 0, 0], [0, 1274, 0, 0], [0, 0, 0, -598], [686, 0, 0, 0], [0, 0, 0, -268], [0, 1014, 0, 0], [0, 0, 0, 114], [0, 0, 624, 0], [-718, 0, 0, 0], [0, -1192, 0, 0], [0, 0, 998, 0], [0, 0, -924, 0], [0, 390, 0, 0], [0, 0, 287, 0], [0, 0, 0, 214], [0, 0, -1416, 0], [-829, 0, 0, 0], [0, 837, 0, 0], [402, 0, 0, 0], [0, 0, 0, 1155], [0, 144, 0, 0], [0, 0, 1278, 0], [1378, 0, 0, 0], [0, -767, 0, 0], [1398, 0, 0, 0], [0, 92, 0, 0], [0, 0, 0, 1154], [-2281, 0, 0, 0], [0, 0, 889, 0], [0, 0, -1890, 0], [0, 0, 0, 828], [0, 1516, 0, 0], [0, 0, -1235, 0], [0, 0, 0, -94], [0, 0, -1218, 0], [0, -1389, 0, 0], [-834, 0, 0, 0], [0, 0, 0, 1195], [0, 0, -1018, 0], [-1211, 0, 0, 0], [1698, 0, 0, 0], [0, 0, 0, -1372], [0, -1036, 0, 0], [0, 0, -614, 0], [0, 0, 0, -404], [0, 0, -1518, 0], [2472, 0, 0, 0], [0, 0, 0, 1518], [-1009, 0, 0, 0], [0, 0, 0, -401], [0, 0, 2989, 0], [0, 0, 0, -1134], [0, 0, 1518, 0], [0, 967, 0, 0], [1950, 0, 0, 0], [0, 0, -1057, 0], [0, 0, 0, -1606], [0, -900, 0, 0], [-618, 0, 0, 0], [0, 280, 0, 0], [0, 0, -623, 0], [0, 0, 0, 926], [0, 0, 2724, 0], [1417, 0, 0, 0], [0, 84, 0, 0], [0, -952, 0, 0], [0, 1068, 0, 0], [0, 0, 0, -1649], [0, 490, 0, 0], [0, 0, 2486, 0], [0, 0, 0, -703], [0, -1054, 0, 0], [0, 0, -1894, 0], [0, 0, 138, 0], [1128, 0, 0, 0], [0, 0, 0, -540], [0, 0, 0, -1098], [-1271, 0, 0, 0], [0, 296, 0, 0], [0, 0, 0, -84], [0, 0, -491, 0], [0, 1729, 0, 0], [0, 0, 0, 509], [0, 62, 0, 0], [0, 0, -2305, 0], [-1489, 0, 0, 0], [-2940, 0, 0, 0], [0, 0, 0, -1020], [806, 0, 0, 0], [0, -1090, 0, 0], [781, 0, 0, 0], [0, -999, 0, 0], [-30, 0, 0, 0], [0, 0, 0, -279], [0, 132, 0, 0], [0, 0, -1306, 0], [0, 0, 540, 0], [-2454, 0, 0, 0], [0, -644, 0, 0], [0, 0, 0, 696], [-804, 0, 0, 0], [0, 0, 0, -1669], [0, 0, -3420, 0], [-467, 0, 0, 0], [0, 0, 0, -672], [0, 0, -1950, 0], [0, -1732, 0, 0], [0, 0, 0, -868], [0, -208, 0, 0], [0, 0, 3349, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1200_bg_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1200_3_bg_g();".
// 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_g(: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_g();".
// 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_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1200_bg();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![5625, 0, 0, 0, 1]>],Snew);
    return Vf;
end function;