
// Make newform 600.3.u.e in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_600_u();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_600_u_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_600_3_u_e();".
// 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_600_3_u_e();".
// 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_600_u();"
function MakeCharacter_600_u()
    N := 600;
    order := 4;
    char_gens := [151, 301, 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_600_u_Hecke();"
function MakeCharacter_600_u_Hecke(Kf)
    N := 600;
    order := 4;
    char_gens := [151, 301, 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, -1, 0, 0], [0, 0, 0, 0], [3, 0, -3, 2], [-4, 1, 0, -1], [-12, 2, -12, 0], [6, 0, -6, 14], [0, 6, -2, 6], [14, 6, 14, 0], [0, -9, -22, -9], [-20, -10, 0, 10], [-28, 0, 28, -6], [-14, -14, 0, 14], [2, 28, 2, 0], [4, 0, -4, 38], [30, 20, 30, 0], [0, 21, -4, 21], [-6, -22, 0, 22], [2, 0, -2, 68], [68, 8, 0, -8], [-27, -40, -27, 0], [0, -6, 112, -6], [-68, 22, -68, 0], [0, -4, 62, -4], [37, 0, -37, 0], [38, -23, 0, 23], [-31, 34, -31, 0], [40, 0, -40, -22], [0, -34, -14, -34], [36, -54, 36, 0], [-93, 0, 93, -22], [104, -7, 0, 7], [-24, 0, 24, -78], [0, 24, -82, 24], [0, -39, -42, -39], [-262, -12, 0, 12], [-38, 0, 38, -74], [0, -96, 0, 0], [90, 0, -90, -46], [-16, 10, -16, 0], [0, -25, -16, -25], [-50, 86, 0, -86], [100, 62, 0, -62], [-95, -64, -95, 0], [-42, 0, 42, -36], [0, 68, 138, 68], [38, 96, 0, -96], [-231, 46, -231, 0], [156, 0, -156, 64], [0, -60, 138, -60], [146, -78, 146, 0], [0, -124, 24, -124], [-64, -136, 0, 136], [120, -27, 0, 27], [-224, 0, 224, 70], [146, -14, 146, 0], [0, 45, 2, 45], [246, -32, 0, 32], [232, 0, -232, -54], [-18, -130, 0, 130], [166, -152, 166, 0], [178, 56, 178, 0], [-238, 0, 238, 104], [-76, -32, 0, 32], [63, 160, 63, 0], [20, 0, -20, 14], [-238, -106, 0, 106], [-69, 0, 69, 168], [216, 0, -216, -204], [0, -34, 158, -34], [-46, 66, -46, 0], [0, 22, 76, 22], [187, 0, -187, 22], [-90, 98, -90, 0], [0, -112, -150, -112], [-372, 38, -372, 0], [0, 67, 534, 67], [58, 0, -58, 258], [-122, -96, 0, 96], [0, 32, -12, 32], [0, 153, 40, 153], [410, 144, 0, -144], [232, -86, 0, 86], [-137, -124, -137, 0], [0, -36, -322, -36], [-8, -208, -8, 0], [0, 114, 562, 114], [161, 0, -161, 112], [-110, 63, 0, -63], [29, 134, 29, 0], [-468, 0, 468, -74], [0, -116, 172, -116], [15, 0, -15, -22], [-52, -357, 0, 357], [0, 50, -190, 50], [276, -78, 276, 0], [0, -153, -546, -153], [-270, -26, 0, 26], [258, 100, 258, 0], [230, -30, 0, 30], [42, 0, -42, 24], [-366, 0, 366, -316], [-532, 140, -532, 0], [0, 188, 26, 188], [158, 118, 0, -118], [-21, 0, 21, -220], [-364, 0, 364, 150], [52, 22, 52, 0], [0, 182, -64, 182], [-26, -84, 0, 84], [-65, 0, 65, 26], [78, -150, 78, 0], [402, 0, -402, 294], [0, 72, 258, 72], [400, 170, 0, -170], [-742, -80, 0, 80], [40, -84, 40, 0], [368, 0, -368, 230], [-72, -182, -72, 0], [0, -291, -444, -291], [610, 122, 0, -122], [-143, -564, -143, 0], [-248, 0, 248, -366], [-36, -98, -36, 0], [966, -78, 0, 78], [562, -115, 0, 115], [0, -224, -378, -224], [0, 68, 772, 68], [-177, 0, 177, 186], [64, 474, 64, 0], [0, 374, 266, 374], [240, 270, 240, 0], [-60, 98, 0, -98], [-168, 0, 168, -290], [94, 334, 0, -334], [0, -44, -94, -44], [356, -170, 356, 0], [330, 0, -330, -96], [72, 0, -72, -46], [0, -24, 758, -24], [370, -188, 0, 188], [410, -65, 0, 65], [-235, 214, -235, 0], [-732, 0, 732, -116], [0, -638, -14, -638], [0, 346, 116, 346], [150, -330, 150, 0], [6, 0, -6, -566], [0, -212, 766, -212], [882, 138, 882, 0], [222, 0, -222, 206], [1202, -184, 0, 184], [324, 280, 324, 0], [326, 0, -326, 46], [-652, 0, 652, -60], [-32, 50, 0, -50], [0, 434, 404, 434], [0, -110, -638, -110], [331, 0, -331, 400], [546, -495, 0, 495], [-460, 0, 460, 472], [240, 18, 240, 0], [-477, 0, 477, 654], [1192, 183, 0, -183], [906, 0, -906, 118], [-144, -58, -144, 0], [-398, -344, 0, 344], [-456, 0, 456, 362]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_600_u_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_600_3_u_e();".
// 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_600_3_u_e(:prec:=4)
    chi := MakeCharacter_600_u();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_600_3_u_e();".
// 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_600_3_u_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_600_u();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![36, -72, 72, -12, 1]>],Snew);
    return Vf;
end function;