
// Make newform 507.4.b.a in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_507_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_507_b_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_507_4_b_a();".
// 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_507_4_b_a();".
// 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, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [-1, 2]];
    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_507_b();"
function MakeCharacter_507_b()
    N := 507;
    order := 2;
    char_gens := [170, 340];
    v := [2, 1];
    // 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_507_b_Hecke();"
function MakeCharacter_507_b_Hecke(Kf)
    N := 507;
    order := 2;
    char_gens := [170, 340];
    char_values := [[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 := 4;
    raw_aps := [[0, 0], [-3, 0], [0, 3], [0, -6], [0, 30], [0, 0], [-117, 0], [0, 14], [18, 0], [-99, 0], [0, 112], [0, 65], [0, -21], [-82, 0], [0, -42], [-261, 0], [0, 456], [-719, 0], [0, -406], [0, 270], [0, 395], [-440, 0], [0, 690], [0, 876], [0, 668], [1575, 0], [794, 0], [450, 0], [0, 344], [-1701, 0], [-1664, 0], [-1476, 0], [0, -585], [1124, 0], [0, -1887], [0, -946], [1259, 0], [0, -1704], [0, -1812], [4266, 0], [3006, 0], [-1873, 0], [-2736, 0], [0, -1503], [0, 2148], [1198, 0], [2392, 0], [0, -1176], [0, -1242], [0, 2004], [-1854, 0], [0, -2574], [0, 241], [4104, 0], [1989, 0], [738, 0], [-2106, 0], [0, 396], [3665, 0], [0, -993], [-1826, 0], [0, 291], [0, 1126], [-3798, 0], [1378, 0], [0, 4101], [0, -5816], [9001, 0], [3294, 0], [0, -6088], [0, -1431], [0, -3126], [-9946, 0], [-7301, 0], [0, -1976], [0, 3336], [9153, 0], [0, -1168], [0, 4797], [0, 6015], [4176, 0], [0, 8355], [0, -3798], [-6605, 0], [-8542, 0], [-14328, 0], [0, -1740], [0, -1667], [0, -2079], [0, 1546], [13878, 0], [0, 636], [0, 9886], [450, 0], [0, 7632], [5346, 0], [0, -3387], [-9657, 0], [21626, 0], [0, -3101], [16946, 0], [0, 2229], [21672, 0], [-1386, 0], [1162, 0], [0, -4645], [0, -15948], [0, -159], [22356, 0], [-18083, 0], [5480, 0], [0, 10241], [0, -5697], [0, 2376], [0, 7312], [-3789, 0], [0, -9764], [-27792, 0], [-594, 0], [17748, 0], [0, 9117], [20933, 0], [3402, 0], [0, -14424], [0, 8006], [-21906, 0], [0, 7539], [-14220, 0], [-5282, 0], [0, 6593], [0, -11856], [0, -12024], [-4834, 0], [9046, 0], [0, 6948], [0, 21676], [0, 9072], [0, 2168], [-7830, 0], [-6165, 0], [0, -17228], [0, -17184], [-8920, 0], [0, -10416], [-21023, 0], [0, 12744], [0, -15363], [-12771, 0], [17134, 0], [0, -4578], [0, -5695], [29169, 0], [928, 0], [-14400, 0], [19684, 0], [24480, 0], [38608, 0], [0, -5895], [28495, 0], [0, 6192], [0, 19608], [23814, 0], [0, 29762], [-12312, 0], [0, 12435], [0, -18828], [-22330, 0], [-24931, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_507_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_507_4_b_a();".
// 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_507_4_b_a(:prec:=2)
    chi := MakeCharacter_507_b();
    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_507_4_b_a();".
// 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_507_4_b_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_507_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![0, 1]>,<5,R![27, 0, 1]>],Snew);
    return Vf;
end function;