
// Make newform 169.4.c.h in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_169_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_169_c_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_169_4_c_h();".
// 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_169_4_c_h();".
// 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, 0, 1, 0], [0, 1, 0, 1], [0, 2, 0, -1]];
    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_169_c();"
function MakeCharacter_169_c()
    N := 169;
    order := 3;
    char_gens := [2];
    v := [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_169_c_Hecke();"
function MakeCharacter_169_c_Hecke(Kf)
    N := 169;
    order := 3;
    char_gens := [2];
    char_values := [[-1, 1, 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+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 := 4;
    raw_aps := [[0, 0, -1, 0], [0, -2, 0, 0], [0, 0, 0, 1], [0, 0, -8, 8], [0, 0, 8, 0], [0, 0, 0, 0], [117, -117, 0, 0], [0, 0, 66, -66], [0, -78, 0, 0], [0, 141, 0, 0], [0, 0, 0, -90], [0, 0, -83, 0], [0, 0, -157, 0], [104, -104, 0, 0], [0, 0, 0, 174], [93, 0, 0, 0], [0, 0, -164, 164], [-145, 145, 0, 0], [0, 0, 454, 0], [0, 0, -610, 610], [0, 0, 0, 265], [1276, 0, 0, 0], [0, 0, 0, 456], [0, 0, -564, 0], [0, 0, -116, 116], [0, 429, 0, 0], [-182, 0, 0, 0], [0, 1506, 0, 0], [0, 0, 0, -896], [687, -687, 0, 0], [0, 286, 0, 0], [-1974, 0, 0, 0], [0, 0, -489, 489], [-236, 236, 0, 0], [0, 0, 27, -27], [0, 0, 0, -1022], [1211, 0, 0, 0], [0, 0, -580, 580], [0, 0, 528, 0], [2574, -2574, 0, 0], [0, 3744, 0, 0], [637, 0, 0, 0], [2598, -2598, 0, 0], [0, 0, 645, 0], [0, 0, -1184, 0], [-2522, 2522, 0, 0], [0, -1042, 0, 0], [0, 0, 1390, 0], [0, 0, -1390, 1390], [0, 0, 0, 1448], [5850, 0, 0, 0], [0, 0, 0, 3108], [0, 0, 2839, -2839], [-3978, 3978, 0, 0], [0, 2067, 0, 0], [0, 2052, 0, 0], [-3330, 3330, 0, 0], [0, 0, 1620, 0], [377, -377, 0, 0], [0, 0, 0, -21], [0, -7124, 0, 0], [0, 0, -4805, 4805], [0, 0, 0, -1282], [-4914, 0, 0, 0], [-518, 0, 0, 0], [0, 0, 0, -2261], [0, 0, 4304, -4304], [3575, 0, 0, 0], [6966, -6966, 0, 0], [0, 0, -3840, 0], [0, 0, -3251, 0], [0, 0, 0, -4116], [0, -2, 0, 0], [-3499, 3499, 0, 0], [0, 0, -3186, 0], [0, 0, 4252, -4252], [1209, 0, 0, 0], [0, 0, 6752, -6752], [0, 0, -1721, 0], [0, 0, -25, 25], [0, 9462, 0, 0], [0, 0, 0, -4081], [0, 0, 5732, 0], [-6617, 6617, 0, 0], [0, 13988, 0, 0], [2004, 0, 0, 0], [0, 0, 5244, -5244], [0, 0, -1457, 0], [0, 0, -11309, 11309], [0, 0, 0, 4984], [-5460, 0, 0, 0], [0, 0, -1474, 0], [0, 0, -6252, 6252], [0, 11388, 0, 0], [0, 0, 0, 10206], [-3876, 3876, 0, 0], [0, 0, 9853, 0], [2121, 0, 0, 0], [0, 11464, 0, 0], [0, 0, 0, -2751], [6554, 0, 0, 0], [0, 0, -10457, 0], [-12168, 12168, 0, 0], [0, -7722, 0, 0], [-11440, 0, 0, 0], [0, 0, 0, -8917], [0, 0, 8124, 0], [0, 0, 0, 15553], [-10554, 0, 0, 0], [0, 14831, 0, 0], [7954, -7954, 0, 0], [0, 0, -14561, 0], [0, 0, 10037, -10037], [0, 0, 0, -4740], [0, 0, 7428, -7428], [6201, -6201, 0, 0], [0, 0, 9712, -9712], [0, -13494, 0, 0], [0, 11334, 0, 0], [-13236, 13236, 0, 0], [0, 0, -6843, 0], [0, 8021, 0, 0], [-21630, 0, 0, 0], [0, 0, -15322, 15322], [0, 0, 480, 0], [-30186, 0, 0, 0], [0, 0, -6859, 6859], [18408, -18408, 0, 0], [-21112, 0, 0, 0], [0, 0, 0, 13833], [0, 0, 1828, 0], [0, 0, 17380, 0], [0, 28496, 0, 0], [0, -17422, 0, 0], [0, 0, 23860, -23860], [0, 0, 8124, 0], [0, 0, -116, 116], [0, 0, -3986, 3986], [-31278, 31278, 0, 0], [0, -8049, 0, 0], [0, 0, 0, -8098], [0, 0, 4640, 0], [-40300, 40300, 0, 0], [0, 0, 0, 22820], [0, -12311, 0, 0], [0, 0, 12394, -12394], [0, 0, 0, 447], [-13923, 0, 0, 0], [-22358, 0, 0, 0], [0, 0, 0, 1288], [0, 0, 9673, -9673], [0, 17355, 0, 0], [46982, 0, 0, 0], [0, 8916, 0, 0], [0, -30836, 0, 0], [-27480, 0, 0, 0], [28442, -28442, 0, 0], [0, 0, -4029, 4029], [-38465, 0, 0, 0], [0, 0, 0, -2820], [0, 0, 12566, 0], [6474, -6474, 0, 0], [0, 0, 0, -4354], [-34998, 34998, 0, 0], [0, 0, 14559, 0], [0, 0, 0, -32586], [0, -59282, 0, 0], [37711, -37711, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_169_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_169_4_c_h();".
// 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_169_4_c_h(:prec:=4)
    chi := MakeCharacter_169_c();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_169_4_c_h();".
// 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_169_4_c_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_169_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![9, 0, 3, 0, 1]>],Snew);
    return Vf;
end function;