
// Make newform 507.2.e.k in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_507_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_507_e_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_2_e_k();".
// 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_2_e_k();".
// 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, -2, 5, 0, 3, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [6, -2, 5, -9, 3, -1], [18, -6, 15, -14, 9, -3], [-2, -34, -6, -10, -1, -4], [10, -25, -9, -15, 5, -6]];
    Rf_basisdens := [1, 1, 13, 13, 13, 13];
    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_e();"
function MakeCharacter_507_e()
    N := 507;
    order := 3;
    char_gens := [170, 340];
    v := [3, 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_507_e_Hecke();"
function MakeCharacter_507_e_Hecke(Kf)
    N := 507;
    order := 3;
    char_gens := [170, 340];
    char_values := [[1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1]];
    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 := 2;
    raw_aps := [[0, 1, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 1], [-1, 0, 0, 1, 0, 0], [0, 0, 0, -1, -1, 3], [2, -3, 0, 0, -4, -2], [0, 0, 0, 0, 0, 0], [0, 1, -1, 1, 1, 3], [0, -3, 3, 1, 1, 3], [-3, 2, 0, 0, 5, 3], [3, 1, 0, 0, -2, -3], [-5, 0, 5, -2, 0, 0], [4, 1, 0, 0, 1, -4], [0, 1, 0, 0, 0, 0], [0, -4, 4, -2, -2, -1], [7, 0, -9, 3, 0, 0], [-4, 0, 1, 2, 0, 0], [0, -4, 4, -6, -6, -8], [0, 5, -5, 3, 3, 7], [4, -4, 0, 0, -3, -4], [0, 5, -5, -2, -2, -1], [-7, 0, 2, -1, 0, 0], [0, 0, -4, 5, 0, 0], [6, 0, 1, 3, 0, 0], [-2, 2, 0, 0, -1, 2], [0, -4, 4, -10, -10, -3], [-1, 6, 0, 0, -3, 1], [2, 0, 4, 8, 0, 0], [4, 3, 0, 0, -6, -4], [-6, 0, 3, -6, 0, 0], [0, 7, -7, -1, -1, 6], [-1, 6, 0, 0, 10, 1], [1, 0, 11, -4, 0, 0], [0, 8, -8, -4, -4, -1], [0, 11, -11, 4, 4, 4], [0, -12, 12, -7, -7, -12], [1, 0, -8, 5, 0, 0], [1, 0, -2, -4, 0, 0], [0, -2, 2, 3, 3, 4], [-3, 0, 0, 0, -10, 3], [0, -5, 5, 0, 0, -7], [-7, -6, 0, 0, -5, 7], [-6, 0, 14, -11, 0, 0], [0, -8, 8, 3, 3, -6], [-1, 4, 0, 0, 8, 1], [14, 3, 0, 0, 9, -14], [0, 10, -10, 9, 9, 18], [3, 1, 0, 0, -2, -3], [2, 5, 0, 0, -8, -2], [0, 10, -10, 14, 14, 3], [-3, 0, -3, 2, 0, 0], [-13, 0, 13, -13, 0, 0], [-4, 0, 7, -11, 0, 0], [0, -2, 2, 3, 3, 4], [0, 1, -1, 1, 1, 3], [14, -16, 0, 0, -11, -14], [10, 2, 0, 0, 9, -10], [0, -11, 11, -17, -17, -11], [14, -5, 0, 0, -13, -14], [0, -7, 7, 4, 4, -6], [-21, 0, 0, -5, 0, 0], [3, 1, 0, 0, 7, -3], [0, 8, -8, 9, 9, 25], [5, 0, 13, 8, 0, 0], [-4, 0, -8, -3, 0, 0], [9, 0, 1, -11, 0, 0], [-8, 0, -14, -1, 0, 0], [0, 11, -11, 12, 12, -4], [9, 0, -21, 10, 0, 0], [0, 1, -1, -15, -15, -4], [-14, 4, 0, 0, 8, 14], [-4, 6, 0, 0, 3, 4], [14, 0, -8, 5, 0, 0], [-12, 10, 0, 0, -7, 12], [0, 11, -11, 4, 4, 30], [-18, 17, 0, 0, 7, 18], [0, -12, 12, -7, -7, -12], [-21, 0, 10, -6, 0, 0], [0, 7, -7, 11, 11, 19], [-13, -7, 0, 0, 4, 13], [0, -6, 6, -19, -19, -15], [-9, -10, 0, 0, -6, 9], [-11, 0, -7, 13, 0, 0], [-2, -16, 0, 0, -5, 2], [0, -11, 11, -1, -1, 9], [3, 1, 0, 0, -6, -3], [-9, 0, -9, -5, 0, 0], [0, 7, -7, -15, -15, -7], [-17, 6, 0, 0, 3, 17], [0, -2, 2, -14, -14, 9], [-4, 0, 8, -28, 0, 0], [13, 0, 0, 0, 0, 0], [-16, -13, 0, 0, 3, 16], [0, 10, -10, 14, 14, 3], [-8, 5, 0, 0, 12, 8], [20, 0, -10, -6, 0, 0], [0, 10, -10, 19, 19, 11], [-8, -2, 0, 0, -20, 8], [3, 0, -24, 4, 0, 0], [-19, 22, 0, 0, 21, 19], [4, 0, -15, 30, 0, 0], [-10, 0, -11, -9, 0, 0], [27, -5, 0, 0, 0, -27], [0, -19, 19, -11, -11, 0], [8, -15, 0, 0, -26, -8], [1, 0, 2, 17, 0, 0], [4, 0, -3, -5, 0, 0], [-12, -10, 0, 0, -4, 12], [6, 0, 2, 12, 0, 0], [8, 0, 12, 11, 0, 0], [-5, -2, 0, 0, 4, 5], [0, 13, -13, 26, 26, 3], [26, 1, 0, 0, 13, -26], [0, 24, -24, 12, 12, 22], [-11, 0, 7, -17, 0, 0], [0, -5, 5, 9, 9, 12], [0, 0, 0, 3, 3, 23], [0, -5, 5, -21, -21, -9], [-10, 1, 0, 0, 20, 10], [-8, 18, 0, 0, 29, 8], [0, -3, 3, 6, 6, -2], [-10, 12, 0, 0, 5, 10], [-8, 5, 0, 0, 12, 8], [-18, 0, 25, -15, 0, 0], [0, -2, 2, 3, 3, 17], [2, -21, 0, 0, -8, -2], [40, 0, -11, 4, 0, 0], [0, 23, -23, 18, 18, 11], [0, 20, -20, 28, 28, 0], [-12, 0, 15, -9, 0, 0], [-26, 0, 2, 5, 0, 0], [36, -39, 0, 0, -23, -36], [1, 3, 0, 0, 9, -1], [-13, 8, 0, 0, -7, 13], [-4, -13, 0, 0, 0, 4], [0, 6, -6, -13, -13, 0], [2, 18, 0, 0, 18, -2], [0, 8, -8, 5, 5, 43], [0, -1, 1, -12, -12, 10], [0, 6, -6, 30, 30, -6], [-17, 26, 0, 0, 0, 17], [-5, 0, -21, 11, 0, 0], [18, -15, 0, 0, -20, -18], [0, 18, -18, -7, -7, -25], [-13, 0, -23, -12, 0, 0], [7, -4, 0, 0, 17, -7], [0, 16, -16, 6, 6, -5], [-15, 0, 12, -7, 0, 0], [28, 0, -22, 8, 0, 0], [17, 0, -22, 21, 0, 0], [33, 0, 3, 7, 0, 0], [0, 14, -14, -20, -20, 1], [9, 13, 0, 0, 9, -9], [-3, 0, -19, 1, 0, 0], [21, -15, 0, 0, 4, -21], [36, -11, 0, 0, -8, -36], [-13, 0, 9, -8, 0, 0], [0, -19, 19, -14, -14, -20], [0, -14, 14, -16, -16, -37], [26, 0, -30, 31, 0, 0], [20, 0, -23, 20, 0, 0], [4, 1, 0, 0, 14, -4], [0, 4, -4, -5, -5, -18], [-20, 0, 27, -23, 0, 0], [0, -12, 12, -2, -2, 35], [24, -3, 0, 0, -5, -24], [8, 0, 6, 7, 0, 0], [14, -3, 0, 0, 6, -14], [0, 12, -12, 15, 15, 36]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_507_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_507_2_e_k();".
// 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_2_e_k(:prec:=6)
    chi := MakeCharacter_507_e();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    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_2_e_k();".
// 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_2_e_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_507_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, -2, 5, 0, 3, -1, 1]>,<5,R![-1, 3, 4, 1]>],Snew);
    return Vf;
end function;