
// Make newform 418.2.e.f in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_418_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_418_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_418_2_e_f();".
// 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_418_2_e_f();".
// 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;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_418_e();"
function MakeCharacter_418_e()
    N := 418;
    order := 3;
    char_gens := [343, 287];
    v := [3, 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_418_e_Hecke();"
function MakeCharacter_418_e_Hecke(Kf)
    N := 418;
    order := 3;
    char_gens := [343, 287];
    char_values := [[1, 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 := [[1, -1], [2, -2], [3, -3], [-1, 0], [1, 0], [0, 4], [0, 0], [-5, 3], [0, 0], [0, 6], [-4, 0], [-1, 0], [6, -6], [4, -4], [0, -6], [0, -3], [0, 0], [0, 4], [0, 4], [6, -6], [-2, 2], [-11, 11], [-3, 0], [0, 18], [19, -19], [0, 12], [-4, 0], [-9, 0], [16, -16], [-6, 0], [0, 16], [0, 0], [0, 15], [0, -17], [-18, 18], [-19, 0], [-17, 17], [-4, 0], [0, -3], [0, 0], [-24, 0], [0, -11], [-6, 0], [-2, 2], [-12, 0], [0, 10], [25, -25], [28, -28], [27, 0], [-25, 0], [0, 0], [-15, 0], [0, -2], [0, -24], [0, 9], [-27, 27], [-3, 3], [7, -7], [-28, 0], [0, 0], [13, -13], [30, 0], [-11, 11], [-18, 0], [0, 1], [0, 18], [-16, 0], [16, -16], [-3, 3], [-22, 0], [27, 0], [3, -3], [0, 28], [32, 0], [32, 0], [-24, 24], [0, -9], [-5, 5], [30, -30], [0, 4], [18, 0], [-17, 17], [0, -15], [0, -14], [-35, 35], [0, 12], [-9, 0], [-22, 0], [6, -6], [-4, 0], [30, 0], [0, 24], [2, 0], [9, -9], [4, -4], [0, 0], [0, 9], [33, 0], [0, 1], [0, -38], [0, 19], [0, -12], [9, 0], [-36, 0], [-19, 0], [-7, 0], [0, 0], [0, -36], [0, -36], [2, 0], [8, 0], [16, -16], [0, -33], [-28, 0], [0, 16], [33, -33], [-14, 14], [6, 0], [-6, 0], [0, -15], [0, -41], [-10, 0], [-18, 0], [42, 0], [-28, 0], [-24, 24], [0, 25], [36, -36], [-50, 50], [-4, 0], [52, -52], [21, -21], [0, -26], [22, -22], [30, 0], [0, 10], [0, -6], [-31, 0], [33, 0], [-12, 0], [0, 43], [0, -24], [0, -14], [0, 12], [-10, 0], [36, -36], [-26, 26], [-48, 48], [0, -2], [6, 0], [-20, 20], [18, 0], [0, -14], [0, 33], [10, -10], [42, 0], [23, 0], [-30, 30], [0, -50], [0, 0], [54, -54], [-36, 36], [7, -7], [30, -30], [57, 0], [42, -42], [-38, 38], [0, -44]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_418_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_418_2_e_f();".
// 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_418_2_e_f(:prec:=2)
    chi := MakeCharacter_418_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_418_2_e_f();".
// 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_418_2_e_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_418_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![4, -2, 1]>,<5,R![9, -3, 1]>],Snew);
    return Vf;
end function;