
// Make newform 507.2.a.i in Magma, downloaded from the LMFDB on 24 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_i();".
// 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_i();".
// 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 := [[-1, 1, 1], [-1, 0, 0], [-1, -2, 1], [-1, 0, -1], [-3, 1, -3], [0, 0, 0], [-2, 3, -2], [2, 1, 0], [3, -4, 5], [-2, 1, -3], [8, -3, 5], [-7, -1, 0], [-5, 1, -3], [-7, 4, -2], [-2, -1, 0], [-5, 1, 3], [0, 4, -2], [-2, -1, 6], [0, -4, 7], [4, -7, 5], [-7, 6, -9], [0, 0, -3], [-3, -1, 2], [-2, 6, 1], [-3, 12, -2], [1, -4, -3], [-6, 0, -4], [-1, -7, 3], [-5, -5, -5], [9, -1, 2], [3, 6, 0], [-2, 9, -9], [-5, -10, 0], [-9, 9, -5], [-14, 0, 3], [7, 8, -1], [11, -10, 14], [-6, 6, 5], [-7, -2, 2], [8, -11, -5], [3, -12, -1], [-8, -6, 7], [-12, 12, -13], [3, -16, 12], [-3, -9, 8], [8, -6, 1], [8, 5, 5], [7, -5, -7], [-3, 0, 8], [6, -3, 13], [8, -1, 2], [-3, 13, 2], [2, -14, 1], [6, -1, 18], [0, 8, 7], [6, -6, -11], [16, -9, 8], [21, 1, 0], [-9, -9, 1], [-1, -2, 7], [0, 1, -14], [-5, 2, -17], [6, 1, -1], [-18, 4, -1], [2, -13, -10], [-12, 14, -7], [9, -5, 9], [4, 1, 1], [-7, 1, 2], [-14, 8, 0], [24, -10, 21], [8, -12, -7], [2, -10, 11], [9, -15, -1], [-5, 11, -6], [6, -16, 3], [-19, 0, -4], [4, 3, -4], [6, -1, 13], [27, -2, -1], [7, -4, 12], [-10, 5, -6], [-20, 16, -9], [-20, 15, -14], [-4, -11, -7], [10, -3, 4], [18, 3, 4], [-19, 14, -9], [-13, 12, -8], [20, -12, 4], [-15, 26, -20], [3, -5, 2], [17, 6, -12], [9, -5, 23], [10, 2, 16], [17, -12, 21], [10, -2, 2], [3, -2, 16], [5, 2, -17], [-1, -19, 5], [-1, 27, -16], [-24, 13, -5], [-23, 9, -10], [-11, 7, 5], [7, -6, 27], [-13, 5, -4], [-10, 2, -14], [-12, 18, -26], [14, 16, -11], [-5, 22, 2], [6, -5, -9], [-13, 7, -24], [10, 0, 12], [-28, 11, -22], [17, -13, -14], [19, -18, -9], [0, -21, -8], [-9, 11, -13], [-16, 22, -23], [-39, 9, -17], [-6, 28, -5], [-5, -3, 3], [3, 1, 0], [-15, 0, 3], [-3, 5, 15], [-1, 17, -3], [-12, -1, 11], [-4, 20, -4], [-1, -27, 2], [-24, 18, 1], [-19, 7, -24], [20, 1, 16], [-19, -8, 9], [17, -5, 1], [24, -14, 7], [4, 6, -16], [25, -14, 7], [15, -13, 13], [20, -2, 4], [7, -4, 2], [2, 7, -26], [13, -35, 29], [-1, -18, 21], [-2, -31, 3], [29, -20, 1], [-13, 6, 6], [-9, 16, -5], [10, -34, 22], [15, -6, -25], [0, 15, 0], [-37, 2, -6], [-34, 5, 2], [14, -17, 10], [-8, -15, 7], [-13, -27, 3], [18, 3, -19], [-7, -21, 11], [-19, 16, -14], [0, 10, -19], [-3, -29, 11], [39, -3, 11], [-32, 16, -25], [-19, 3, -28], [1, 10, 18], [-23, 25, -16], [-24, 18, -27], [9, 1, 1], [-16, 24, -23]];
    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_i();".
// 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_i(: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_i();".
// 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_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_507_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-13, -4, 3, 1]>,<5,R![-13, 5, 6, 1]>],Snew);
    return Vf;
end function;