
// Make newform 3600.3.l.d in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3600_l();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_3600_l_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_3600_3_l_d();".
// 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_3600_3_l_d();".
// 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 := [2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [0, 3]];
    Rf_basisdens := [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_3600_l();"
function MakeCharacter_3600_l()
    N := 3600;
    order := 2;
    char_gens := [3151, 901, 2801, 577];
    v := [2, 2, 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_3600_l_Hecke();"
function MakeCharacter_3600_l_Hecke(Kf)
    N := 3600;
    order := 2;
    char_gens := [3151, 901, 2801, 577];
    char_values := [[1, 0], [1, 0], [-1, 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], [-4, 0], [0, 4], [-8, 0], [0, -3], [16, 0], [0, 4], [0, -1], [-44, 0], [34, 0], [0, -11], [-40, 0], [0, 20], [0, 9], [0, 8], [50, 0], [8, 0], [0, -12], [16, 0], [76, 0], [0, -28], [0, -3], [-176, 0], [0, -7], [-28, 0], [0, 0], [56, 0], [0, -37], [92, 0], [0, -40], [0, 37], [-152, 0], [0, -65], [148, 0], [82, 0], [56, 0], [0, -8], [0, -41], [0, -48], [-232, 0], [0, -8], [-206, 0], [0, 39], [-20, 0], [-296, 0], [-436, 0], [0, -4], [8, 0], [0, -3], [0, 32], [32, 0], [0, 12], [0, -43], [0, -88], [0, 81], [-380, 0], [328, 0], [0, -67], [-208, 0], [0, -103], [-520, 0], [0, -88], [94, 0], [0, 79], [-536, 0], [208, 0], [0, -68], [-238, 0], [0, -53], [0, -132], [284, 0], [190, 0], [160, 0], [0, 64], [0, 95], [-146, 0], [0, -77], [368, 0], [0, 92], [-40, 0], [0, -36], [-542, 0], [4, 0], [0, -76], [0, -51], [400, 0], [0, 71], [-604, 0], [0, -84], [0, 124], [596, 0], [0, -64], [-224, 0], [0, -204], [0, -113], [0, 123], [-736, 0], [-808, 0], [536, 0], [0, -39], [0, 76], [0, -37], [-368, 0], [142, 0], [0, -88], [0, 261], [0, 188], [158, 0], [332, 0], [-578, 0], [0, -13], [-896, 0], [-20, 0], [0, -61], [728, 0], [0, 108], [0, 71], [0, -248], [62, 0], [670, 0], [0, -305], [0, 132], [40, 0], [0, -225], [968, 0], [0, 276], [-508, 0], [1144, 0], [304, 0], [0, -200], [-188, 0], [1240, 0], [0, 37], [-910, 0], [0, -327], [-1360, 0], [0, -25], [0, 261], [160, 0], [0, -103], [332, 0], [0, -24], [632, 0], [0, 172], [-446, 0], [0, -101], [-728, 0], [0, 216], [910, 0], [0, 219], [1064, 0], [0, 328], [-1768, 0], [0, -56], [-380, 0], [0, -157], [178, 0], [0, 103], [0, -424], [0, 309], [1700, 0], [0, 108], [0, 179], [0, 248], [772, 0], [-194, 0], [560, 0], [0, 335], [0, -272], [-946, 0], [0, -64], [544, 0], [-1412, 0], [0, 445], [832, 0], [0, 303], [-196, 0], [878, 0], [1508, 0], [0, 4], [-1736, 0], [0, 51], [0, -176], [0, -271], [-776, 0], [-1552, 0], [512, 0], [0, 24], [-1280, 0], [0, 368], [1168, 0], [0, 551], [0, -216], [0, -19], [-46, 0], [1048, 0], [0, 91], [0, 240], [0, 313], [148, 0], [-728, 0], [1616, 0], [0, -228], [0, -33], [1708, 0], [0, 196], [0, -349], [1000, 0], [430, 0], [0, -287], [428, 0], [0, 32], [0, -148], [-1426, 0], [-1900, 0], [0, 77], [0, 192], [0, 311], [1880, 0], [-620, 0], [0, 19], [-1852, 0], [0, 220], [-2722, 0], [0, -35], [0, 516], [2012, 0], [0, -344], [1000, 0], [-1784, 0], [748, 0], [0, -653], [1544, 0], [0, 620], [-496, 0], [0, -159], [0, -412], [0, 468], [0, 284], [-632, 0], [-1468, 0], [2126, 0], [0, 533], [0, 340], [164, 0], [0, 440], [-440, 0], [0, -456], [-1880, 0], [0, 555], [0, 124], [-1120, 0], [0, -65], [0, -72], [-808, 0], [-544, 0], [0, 393], [2194, 0], [1364, 0], [0, -304], [-136, 0], [-686, 0], [0, -301], [184, 0], [0, 105], [0, -781], [944, 0], [0, 17], [-1774, 0], [2576, 0], [-896, 0], [-908, 0], [-1376, 0], [-1780, 0], [0, -404], [-1678, 0], [-2878, 0], [0, 208], [0, 556], [-1484, 0], [0, 572], [-3262, 0], [-496, 0], [0, -384], [-128, 0], [0, 209], [-3308, 0], [0, -171], [0, -103], [0, -744], [0, -157], [0, -592], [-2606, 0], [0, -233], [1732, 0], [0, -521], [0, 84], [-1552, 0], [514, 0], [0, -279], [700, 0], [0, -584], [3016, 0], [-1280, 0], [0, 116], [-3736, 0], [0, 452], [-2, 0], [0, -688], [0, 489], [0, 413], [1664, 0], [0, -696], [128, 0], [0, -344], [0, -388], [-752, 0], [0, -149], [-32, 0], [-3230, 0], [0, -129], [-2356, 0], [0, -875], [3728, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3600_l_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3600_3_l_d();".
// 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_3600_3_l_d(:prec:=2)
    chi := MakeCharacter_3600_l();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2161) - 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_3600_3_l_d();".
// 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_3600_3_l_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3600_l();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![4, 1]>,<11,R![288, 0, 1]>,<13,R![8, 1]>],Snew);
    return Vf;
end function;