
// Make newform 1600.3.g.e in Magma, downloaded from the LMFDB on 03 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1600_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1600_g_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_1600_3_g_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_1600_3_g_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 := [1, 0, -1, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 2, 0, -1], [-2, 0, 4, 0], [0, 0, 0, 2]];
    Rf_basisdens := [1, 1, 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_1600_g();"
function MakeCharacter_1600_g()
    N := 1600;
    order := 2;
    char_gens := [1151, 901, 577];
    v := [1, 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_1600_g_Hecke();"
function MakeCharacter_1600_g_Hecke(Kf)
    N := 1600;
    order := 2;
    char_gens := [1151, 901, 577];
    char_values := [[-1, 0, 0, 0], [-1, 0, 0, 0], [1, 0, 0, 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]];
    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, 3, 0, 0], [0, 0, 0, 0], [0, 0, 0, 1], [0, -3, 0, 0], [0, 0, -7, 0], [21, 0, 0, 0], [0, 11, 0, 0], [0, 0, 0, 18], [0, 0, 9, 0], [0, 0, 0, -16], [0, 0, -3, 0], [51, 0, 0, 0], [0, -28, 0, 0], [0, 0, 0, -39], [0, 0, 6, 0], [0, -12, 0, 0], [0, 0, -29, 0], [0, 41, 0, 0], [0, 0, 0, 45], [-83, 0, 0, 0], [0, 0, 0, -7], [0, 21, 0, 0], [33, 0, 0, 0], [74, 0, 0, 0], [0, 0, 33, 0], [0, 0, 0, 5], [0, 81, 0, 0], [0, 0, -25, 0], [159, 0, 0, 0], [0, 0, 0, 82], [0, -72, 0, 0], [-45, 0, 0, 0], [0, -51, 0, 0], [0, 0, -27, 0], [0, 0, 0, 91], [0, 0, 8, 0], [0, 17, 0, 0], [0, 0, 0, 3], [0, 0, -30, 0], [0, -177, 0, 0], [0, 0, 22, 0], [0, 0, 0, 120], [-103, 0, 0, 0], [0, 0, -72, 0], [0, 0, 0, 104], [0, 13, 0, 0], [0, 0, 0, 64], [0, -36, 0, 0], [0, 0, 98, 0], [-306, 0, 0, 0], [0, 0, 0, 69], [-155, 0, 0, 0], [0, -129, 0, 0], [222, 0, 0, 0], [0, 0, 0, -24], [0, 0, -72, 0], [0, 0, 0, 64], [0, 0, -13, 0], [-510, 0, 0, 0], [0, 11, 0, 0], [0, 0, -117, 0], [0, 103, 0, 0], [0, 0, 0, -87], [214, 0, 0, 0], [0, 0, 78, 0], [0, -195, 0, 0], [167, 0, 0, 0], [0, 105, 0, 0], [0, 0, -39, 0], [-402, 0, 0, 0], [0, 0, 0, -159], [0, 0, 0, -298], [0, 0, 14, 0], [0, -397, 0, 0], [0, 0, 0, 327], [0, 0, -12, 0], [0, 0, 196, 0], [-507, 0, 0, 0], [499, 0, 0, 0], [0, -129, 0, 0], [0, 0, -186, 0], [0, 0, 0, -237], [-791, 0, 0, 0], [0, 0, 0, -217], [0, -33, 0, 0], [-87, 0, 0, 0], [-401, 0, 0, 0], [0, 0, 126, 0], [0, 0, 0, -50], [0, -12, 0, 0], [0, 0, 0, 330], [0, 0, 0, 130], [0, 204, 0, 0], [0, 232, 0, 0], [0, 0, 0, 60], [0, 0, -216, 0], [-801, 0, 0, 0], [0, -191, 0, 0], [0, 0, 10, 0], [0, 75, 0, 0], [0, 0, -285, 0], [0, 396, 0, 0], [-657, 0, 0, 0], [0, -172, 0, 0], [343, 0, 0, 0], [0, 327, 0, 0], [243, 0, 0, 0], [0, 0, 0, 447], [-547, 0, 0, 0], [0, 0, 0, 346], [0, 0, -60, 0], [186, 0, 0, 0], [0, 500, 0, 0], [0, 0, 0, 274], [138, 0, 0, 0], [0, -36, 0, 0], [0, 0, 0, -192], [0, 0, -51, 0], [0, 213, 0, 0], [0, 0, -123, 0], [490, 0, 0, 0], [0, 0, 138, 0], [0, -135, 0, 0], [0, 423, 0, 0], [0, 0, 177, 0], [0, 0, -110, 0], [0, 0, 0, 261], [0, 0, 0, -593], [0, 0, 369, 0], [0, -612, 0, 0], [0, 0, 0, 438], [0, 0, 0, 401], [0, 0, 154, 0], [711, 0, 0, 0], [-823, 0, 0, 0], [0, 0, -21, 0], [0, -424, 0, 0], [0, 0, -108, 0], [798, 0, 0, 0], [0, 508, 0, 0], [0, 0, -30, 0], [0, 0, 0, -470], [0, 483, 0, 0], [0, 0, 213, 0], [0, 0, 0, -204], [0, 0, -206, 0], [-1005, 0, 0, 0], [0, -723, 0, 0], [0, 0, 0, 459], [0, 0, 50, 0], [-42, 0, 0, 0], [0, -281, 0, 0], [0, 0, 0, 228], [0, 948, 0, 0], [0, 0, 0, -282], [0, 0, 0, 202], [1554, 0, 0, 0], [-71, 0, 0, 0], [0, 0, 423, 0], [0, 1020, 0, 0], [-573, 0, 0, 0], [0, 0, 0, -22], [0, -495, 0, 0], [1005, 0, 0, 0], [0, 0, 0, 729], [0, 0, 0, -67], [0, 0, 468, 0], [1241, 0, 0, 0], [0, 0, -267, 0], [0, -33, 0, 0], [0, 0, -37, 0], [0, 0, 0, 207], [677, 0, 0, 0], [0, 0, 0, -97], [-303, 0, 0, 0], [0, 215, 0, 0], [0, 0, 411, 0], [0, 0, 0, -670], [0, 0, -220, 0], [0, 0, 0, 98], [0, 447, 0, 0], [0, 0, -89, 0], [921, 0, 0, 0], [0, 0, 0, -465], [0, 0, 108, 0], [0, 0, -59, 0], [0, -569, 0, 0], [-769, 0, 0, 0], [0, 0, 0, -1119], [1171, 0, 0, 0], [0, 591, 0, 0], [0, -1123, 0, 0], [0, 0, -522, 0], [0, -141, 0, 0], [186, 0, 0, 0], [739, 0, 0, 0], [0, 0, 546, 0], [1935, 0, 0, 0], [0, 0, 0, -651], [0, 0, -504, 0], [0, 0, 0, 754], [0, 0, -570, 0], [-1525, 0, 0, 0], [0, -240, 0, 0], [0, 0, 258, 0], [0, 0, 0, 544], [0, 576, 0, 0], [642, 0, 0, 0], [0, 284, 0, 0], [350, 0, 0, 0], [0, 0, 219, 0], [0, 0, 0, -37], [0, -1263, 0, 0], [0, 0, 0, 780], [635, 0, 0, 0], [0, 0, 0, -16], [-327, 0, 0, 0], [0, 0, 0, 1143], [0, 0, -627, 0], [0, 0, 266, 0], [0, 0, 0, 497], [-1815, 0, 0, 0], [0, 0, 0, -737], [0, 735, 0, 0], [0, 0, -1, 0], [1629, 0, 0, 0], [0, 0, 0, 849], [0, 0, 0, 35], [0, -633, 0, 0], [0, 0, 175, 0], [0, 297, 0, 0], [0, 0, 0, 223], [1305, 0, 0, 0], [0, -1236, 0, 0], [0, 0, 0, 609], [1709, 0, 0, 0], [0, 0, -765, 0], [0, -573, 0, 0], [0, 0, 0, -618], [0, 1563, 0, 0], [0, 1567, 0, 0], [0, 0, 0, 34], [0, 0, -266, 0], [-1077, 0, 0, 0], [0, 0, 0, -1026], [0, 0, 0, -319], [0, -1068, 0, 0], [0, 403, 0, 0], [0, 0, 0, 615], [0, 0, -72, 0], [2667, 0, 0, 0], [0, 0, 0, 675], [-835, 0, 0, 0], [0, 0, 927, 0], [0, 1593, 0, 0], [0, 0, 917, 0], [0, 243, 0, 0], [0, 0, 195, 0], [11, 0, 0, 0], [0, 0, 0, -1187], [0, -195, 0, 0], [0, 0, 579, 0], [0, 0, 343, 0], [-435, 0, 0, 0], [0, -333, 0, 0], [0, 0, 24, 0], [306, 0, 0, 0], [0, -1215, 0, 0], [0, 0, -267, 0], [0, 0, -275, 0], [0, -651, 0, 0], [1229, 0, 0, 0], [0, 0, 0, -655], [670, 0, 0, 0], [0, 0, 0, -940], [0, 621, 0, 0], [0, 0, 22, 0], [-1666, 0, 0, 0], [0, -513, 0, 0], [0, 0, 0, -798], [0, 0, 0, -1192], [0, 0, 0, -1098], [0, 0, -8, 0], [0, 535, 0, 0], [0, 0, 0, 1236], [802, 0, 0, 0], [0, 0, 225, 0], [0, 0, 0, 124], [1062, 0, 0, 0], [0, 0, 771, 0], [0, -396, 0, 0], [3633, 0, 0, 0], [0, -1881, 0, 0], [0, 0, -338, 0], [0, 0, 114, 0], [0, 0, 0, -1889], [0, 0, -297, 0], [0, 1884, 0, 0], [0, -579, 0, 0], [-773, 0, 0, 0], [0, 0, -993, 0], [0, 0, 0, -1822]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1600_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1600_3_g_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_1600_3_g_e(:prec:=4)
    chi := MakeCharacter_1600_g();
    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_1600_3_g_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_1600_3_g_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1600_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![-27, 0, 1]>,<17,R![-21, 1]>],Snew);
    return Vf;
end function;