
// Make newform 1200.3.bg.i in Magma, downloaded from the LMFDB on 19 April 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_i();".
// 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_i();".
// 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, 3, 0, 0], [-6, 0, 0, 0], [0, 0, 0, 5], [0, -18, 0, 0], [0, 0, 25, 0], [0, 0, 0, 6], [0, 0, -42, 0], [-49, 0, 0, 0], [0, -4, 0, 0], [-60, 0, 0, 0], [0, 0, 0, 1], [0, 42, 0, 0], [0, 0, 0, -12], [0, 0, -78, 0], [-13, 0, 0, 0], [0, -43, 0, 0], [60, 0, 0, 0], [0, 0, 0, 52], [0, 0, -106, 0], [0, 0, 0, 66], [0, 0, 60, 0], [0, 99, 0, 0], [-192, 0, 0, 0], [0, 0, 0, 64], [0, 24, 0, 0], [0, 0, 1, 0], [0, 0, 0, -48], [0, -12, 0, 0], [-12, 0, 0, 0], [0, 66, 0, 0], [0, 0, 58, 0], [0, 0, -234, 0], [-85, 0, 0, 0], [0, -61, 0, 0], [0, 0, 0, 65], [0, 144, 0, 0], [0, 0, 0, 66], [0, 0, 42, 0], [145, 0, 0, 0], [-294, 0, 0, 0], [0, 0, 0, 21], [0, 18, 0, 0], [0, 0, 179, 0], [85, 0, 0, 0], [0, 0, 0, -85], [0, 0, 0, 0], [0, 0, -373, 0], [0, 0, 0, -180], [0, 0, -36, 0], [-97, 0, 0, 0], [-288, 0, 0, 0], [0, 276, 0, 0], [0, 0, 0, -204], [0, 0, -318, 0], [226, 0, 0, 0], [0, -311, 0, 0], [126, 0, 0, 0], [0, 0, 0, 323], [0, 0, 0, -18], [0, -71, 0, 0], [246, 0, 0, 0], [0, 0, 0, 31], [0, 12, 0, 0], [178, 0, 0, 0], [0, 341, 0, 0], [0, -366, 0, 0], [0, 0, 542, 0], [0, 0, 0, -198], [0, 0, -408, 0], [0, 163, 0, 0], [0, 0, 0, -261], [0, 0, -71, 0], [0, 0, 0, 48], [0, 0, -288, 0], [0, 61, 0, 0], [36, 0, 0, 0], [0, 0, -13, 0], [0, 0, -144, 0], [610, 0, 0, 0], [402, 0, 0, 0], [0, 0, 0, -239], [0, 0, 119, 0], [0, 0, 0, -156], [0, 0, -84, 0], [0, -120, 0, 0], [516, 0, 0, 0], [0, 0, 0, 428], [0, 414, 0, 0], [0, 0, 102, 0], [0, -199, 0, 0], [-444, 0, 0, 0], [0, 0, 707, 0], [0, 0, 0, 144], [0, 0, -666, 0], [-558, 0, 0, 0], [0, 0, 0, -447], [803, 0, 0, 0], [0, 176, 0, 0], [0, -366, 0, 0], [0, 0, 0, -210], [0, 0, 582, 0], [-215, 0, 0, 0], [0, -201, 0, 0], [0, -12, 0, 0], [0, 0, 0, 264], [0, 0, 528, 0], [493, 0, 0, 0], [0, -596, 0, 0], [0, 0, 0, 92], [0, -312, 0, 0], [0, 0, -733, 0], [-647, 0, 0, 0], [-636, 0, 0, 0], [0, 0, 0, 324], [0, 24, 0, 0], [0, 0, 0, -426], [0, 0, -36, 0], [-1022, 0, 0, 0], [0, 0, 0, 700], [0, -438, 0, 0], [0, 0, 0, 624], [842, 0, 0, 0], [-534, 0, 0, 0], [0, 0, 743, 0], [0, 0, -906, 0], [0, 61, 0, 0], [0, 0, 0, -516], [0, 0, -470, 0], [0, 0, 0, 204], [-1282, 0, 0, 0], [0, -87, 0, 0], [-768, 0, 0, 0], [0, 0, -491, 0], [0, 0, 0, 648], [0, -309, 0, 0], [0, 504, 0, 0], [0, 0, -348, 0], [-1045, 0, 0, 0], [606, 0, 0, 0], [0, 0, 0, -247], [0, -324, 0, 0], [0, 0, -406, 0], [0, 0, 462, 0], [0, 0, 0, 233], [0, -156, 0, 0], [0, 0, 410, 0], [0, 0, 0, 912], [0, -147, 0, 0], [-120, 0, 0, 0], [0, 0, 0, 477], [0, -162, 0, 0], [0, 844, 0, 0], [-1218, 0, 0, 0], [0, 0, 469, 0], [0, 0, -474, 0], [0, -579, 0, 0], [990, 0, 0, 0], [0, -846, 0, 0], [0, 0, 0, 516], [0, 460, 0, 0], [-1338, 0, 0, 0], [0, -246, 0, 0], [0, 0, 0, 348], [-83, 0, 0, 0], [0, 468, 0, 0], [0, 0, -983, 0], [0, 0, 0, 474], [0, 0, 174, 0], [347, 0, 0, 0], [-42, 0, 0, 0], [0, 0, 0, 393], [0, 0, 877, 0], [0, 0, -1104, 0], [-169, 0, 0, 0], [1290, 0, 0, 0], [0, 0, 0, -1025], [0, 0, -227, 0], [0, 787, 0, 0], [-1836, 0, 0, 0], [0, 0, 0, -553], [0, -18, 0, 0], [0, 0, 0, 210], [0, 0, 330, 0], [0, -257, 0, 0], [0, 0, 0, -797], [0, 0, -289, 0], [1926, 0, 0, 0], [0, 0, 0, 9], [0, 0, 0, 84], [-61, 0, 0, 0], [-1806, 0, 0, 0], [0, -1146, 0, 0], [0, 0, 0, 1014], [302, 0, 0, 0], [0, 0, 0, -1060], [0, 522, 0, 0], [0, 0, 0, -930], [0, 0, 1392, 0], [-1294, 0, 0, 0], [0, 824, 0, 0], [0, 0, -538, 0], [0, 0, 1188, 0], [0, 1002, 0, 0], [0, 0, -313, 0], [0, 0, 0, -54], [0, 0, 1368, 0], [1427, 0, 0, 0], [0, -389, 0, 0], [-1950, 0, 0, 0], [0, 0, 0, -379], [0, -912, 0, 0], [0, 0, 78, 0], [610, 0, 0, 0], [0, -889, 0, 0], [246, 0, 0, 0], [0, 516, 0, 0], [0, 0, 0, -786], [1727, 0, 0, 0], [0, 0, -1103, 0], [0, 0, 1950, 0], [0, 0, 0, 564], [0, -300, 0, 0], [0, 0, -587, 0], [0, 0, 0, 798], [0, 0, 846, 0], [0, 581, 0, 0], [-1890, 0, 0, 0], [0, 0, 0, 429], [0, 0, -250, 0], [-995, 0, 0, 0], [-702, 0, 0, 0], [0, 0, 0, 188], [0, 972, 0, 0], [0, 0, -230, 0], [0, 0, 0, 84], [0, 0, 306, 0], [1608, 0, 0, 0], [0, 0, 0, -366], [-1873, 0, 0, 0], [0, 0, 0, -1383], [0, 0, 1621, 0], [0, 0, 0, -1554], [0, 0, 1374, 0], [0, 225, 0, 0], [1854, 0, 0, 0], [0, 0, 1535, 0], [0, 0, 0, 438], [0, -684, 0, 0], [-1578, 0, 0, 0], [0, -24, 0, 0], [0, 0, -1199, 0], [0, 0, 0, -78], [0, 0, -1404, 0], [-1343, 0, 0, 0], [0, 1068, 0, 0], [0, -360, 0, 0], [0, -12, 0, 0], [0, 0, 0, 441], [0, 726, 0, 0], [0, 0, -1066, 0], [0, 0, 0, -105], [0, -258, 0, 0], [0, 0, 2714, 0], [0, 0, -486, 0], [1608, 0, 0, 0], [0, 0, 0, 636], [0, 0, 0, 858], [-719, 0, 0, 0], [0, 1112, 0, 0], [0, 0, 0, 1236], [0, 0, -1427, 0], [0, -865, 0, 0], [0, 0, 0, -453], [0, 1122, 0, 0], [0, 0, 2135, 0], [863, 0, 0, 0], [1188, 0, 0, 0], [0, 0, 0, 924], [-442, 0, 0, 0], [0, 882, 0, 0], [-443, 0, 0, 0], [0, 727, 0, 0], [-2478, 0, 0, 0], [0, 0, 0, 631], [0, 732, 0, 0], [0, 0, 1478, 0], [0, 0, -2052, 0], [-2694, 0, 0, 0], [0, -2076, 0, 0], [0, 0, 0, 840], [-1956, 0, 0, 0], [0, 0, 0, 1357], [0, 0, -2748, 0], [2965, 0, 0, 0], [0, 0, 0, -1920], [0, 0, -3678, 0], [0, -412, 0, 0], [0, 0, 0, 932], [0, 144, 0, 0], [0, 0, 109, 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_i();".
// 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_i(: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_i();".
// 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_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1200_bg();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![729, 0, 0, 0, 1]>],Snew);
    return Vf;
end function;