
// Make newform 507.2.a.j in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_507_a();"
// 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_a_j();".
// 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_a_j();".
// 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, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-2, 0, 1]];
    Rf_basisdens := [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_507_a();"
function MakeCharacter_507_a()
    N := 507;
    order := 1;
    char_gens := [170, 340];
    v := [1, 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;

function MakeCharacter_507_a_Hecke(Kf)
    return MakeCharacter_507_a();
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], [1, 0, 0], [-2, 1, -1], [-4, 1, -1], [2, -1, 4], [0, 0, 0], [-2, 0, 1], [-2, -4, 1], [-2, 3, -5], [-1, -3, 2], [-3, 3, 2], [-5, 0, -1], [0, -1, 0], [-1, -2, -2], [4, -6, -3], [-6, 3, -2], [2, 2, -6], [-4, 2, 3], [-1, 1, 3], [-1, 7, -2], [-6, 1, 1], [-5, 1, -5], [3, 4, -3], [3, -3, 1], [-7, 6, -10], [4, -9, 3], [-6, 12, -8], [2, -9, 6], [0, -3, 6], [-7, 8, -1], [-9, 4, -10], [5, 7, 4], [-3, 12, -4], [0, 7, 4], [5, -5, -7], [-4, -3, -5], [5, -6, 4], [-1, -5, 3], [13, -10, 10], [7, -5, 0], [12, 1, 5], [5, 3, 11], [9, -11, 3], [-7, 4, -8], [-23, 6, -9], [-9, 1, 9], [-1, -3, 2], [6, -13, 8], [11, -4, 14], [-5, -1, -2], [0, 0, 13], [7, -4, 11], [-1, -5, 3], [-2, 0, 1], [-3, 5, 11], [-19, 7, -9], [-6, 6, -17], [-1, -8, 13], [10, -11, 4], [-16, -5, 5], [-10, 6, -7], [-16, -1, 9], [-3, 21, -8], [-1, -11, 3], [20, -10, 11], [-7, -15, 1], [16, -1, 12], [-1, -11, -10], [-11, 16, -15], [6, 4, -8], [1, -3, -3], [9, -3, -5], [19, -17, 7], [-26, 7, 4], [11, -10, -7], [5, -5, -7], [-15, 4, 6], [-8, -4, 11], [9, 11, -4], [-4, 13, -19], [15, 4, 6], [-24, 6, -13], [7, 11, 5], [-10, -10, -1], [3, -7, 6], [-4, -14, 5], [-8, 22, -15], [14, -3, -3], [-23, 12, -14], [24, -20, 28], [13, 0, 0], [13, 16, -3], [11, -4, 14], [-4, 7, -12], [26, -16, 6], [8, -9, 19], [28, -18, 20], [-1, -20, -4], [-2, -1, -21], [-26, 15, -30], [-1, -20, 9], [-27, 5, 0], [-11, -8, -11], [18, -11, 26], [-16, 19, -17], [9, -8, 5], [16, 6, 4], [-6, 14, -12], [-3, 23, -11], [1, 6, -4], [23, -13, 26], [-39, 12, -13], [-10, 12, 12], [6, -10, 17], [-3, -14, 9], [-20, -3, 3], [-12, 16, -21], [-10, 19, -20], [-21, 11, -29], [8, -9, 6], [5, -7, -5], [-4, 7, -12], [-3, 10, 15], [-14, -5, 3], [6, 13, 8], [36, -7, -4], [7, 5, 18], [28, -8, 28], [-3, 6, 9], [-31, 7, -5], [-13, 16, 23], [-10, 6, -9], [20, -15, 7], [4, 13, 0], [-13, 19, -13], [-20, 0, -18], [-38, 3, 5], [-22, 11, -12], [36, -24, 30], [17, -26, 0], [-16, -10, -11], [2, -5, 20], [18, 25, -7], [-1, -35, 12], [-24, 21, -17], [11, 10, 6], [-8, 5, 7], [20, -14, -8], [-4, -1, -21], [26, 10, -7], [-21, 34, -20], [-18, -4, -9], [-4, -18, -1], [-25, 19, -4], [-28, 3, 8], [-5, 1, 8], [6, -5, -14], [21, 2, -16], [-5, 1, -31], [0, -3, -20], [-18, 13, -14], [13, 9, -5], [3, 4, 23], [-37, -10, -2], [-19, -2, 5], [1, 13, -7], [-20, 9, -6], [-21, -3, 15]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_507_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_507_2_a_j();".
// 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_a_j(:prec:=3)
    chi := MakeCharacter_507_a();
    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_a_j();".
// 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_a_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_507_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, -2, 1, 1]>,<5,R![-1, 3, 4, 1]>],Snew);
    return Vf;
end function;