
// Make newform 418.2.f.e in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_418_f();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_418_f_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_f_e();".
// 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_f_e();".
// 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, -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_f();"
function MakeCharacter_418_f()
    N := 418;
    order := 5;
    char_gens := [343, 287];
    v := [2, 5];
    // 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_f_Hecke();"
function MakeCharacter_418_f_Hecke(Kf)
    N := 418;
    order := 5;
    char_gens := [343, 287];
    char_values := [[0, 0, 0, -1], [1, 0, 0, 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 := 2;
    raw_aps := [[-1, 1, -1, 1], [0, 2, -1, 2], [1, 0, 1, 0], [-3, 3, 0, 2], [-2, 1, -4, 2], [0, 1, -1, 0], [-2, 5, -2, 0], [0, 0, 1, 0], [3, 0, 2, -2], [4, -4, 0, 4], [1, -7, 7, -1], [3, -3, 0, 3], [0, -5, 1, -5], [-6, 0, -6, 6], [0, -4, 9, -4], [7, 1, -1, -7], [5, -5, 0, 5], [1, 5, 1, 0], [2, 0, 5, -5], [2, 5, 2, 0], [-7, 7, 0, 10], [-3, 3, -3, 3], [5, 2, 5, 0], [7, 0, 10, -10], [3, 9, -9, -3], [7, -3, 3, -7], [9, -9, 0, -6], [0, 0, 3, 0], [-1, 0, -7, 7], [0, -12, 9, -12], [-10, 13, -10, 0], [6, 0, 14, -14], [4, 1, 4, 0], [3, -3, 0, 0], [-2, -19, -2, 0], [0, 6, 4, 6], [0, 6, 0, 6], [-16, 3, -3, 16], [10, -9, 9, -10], [0, -11, 2, -11], [0, 2, -16, 2], [-8, -9, -8, 0], [-8, 8, 0, -3], [-9, 20, -9, 0], [17, 0, 10, -10], [4, 0, 16, -16], [21, -13, 13, -21], [0, 0, -20, 0], [-13, 13, 0, 8], [-19, 10, -10, 19], [-15, 1, -1, 15], [0, -21, 9, -21], [4, 0, 7, -7], [-16, 8, -8, 16], [-7, 7, 0, 5], [8, 0, 13, -13], [8, -23, 8, 0], [6, -6, 0, -2], [8, -11, 11, -8], [-13, -7, -13, 0], [0, -6, -9, -6], [4, -4, 0, -17], [-15, 0, 2, -2], [0, -4, 14, -4], [8, -5, 8, 0], [3, -3, 3, -3], [0, 0, -6, 6], [10, -10, 0, -1], [-22, 12, -22, 0], [0, -7, -17, -7], [14, 0, -8, 8], [3, -3, 0, 15], [9, -9, 0, -28], [10, 0, 18, -18], [3, -20, 3, 0], [13, -14, 14, -13], [-26, 26, 0, 14], [-9, 0, -24, 24], [-4, 27, -4, 0], [3, -3, 3, -3], [12, 0, -12, 12], [0, 12, -3, 12], [6, -19, 19, -6], [4, -4, 0, 26], [22, 0, -3, 3], [0, -3, 0, -3], [-3, 9, -9, 3], [0, -9, 0, 0], [13, 0, 28, -28], [-21, 0, -3, 3], [-17, 6, -17, 0], [-2, 7, -2, 0], [0, 8, 3, 8], [3, -3, 0, 24], [0, 0, 0, -31], [0, -11, 12, -11], [0, -30, 9, -30], [-4, 4, 0, -9], [-19, -6, -19, 0], [19, 8, -8, -19], [0, -11, 12, -11], [8, -8, 0, 14], [23, -1, 1, -23], [0, -4, -8, -4], [-15, 0, -16, 16], [23, -2, 23, 0], [-12, 12, 0, 27], [9, 0, -12, 12], [3, -9, 3, 0], [-15, 15, 0, -13], [-3, 3, -3, 3], [0, 18, -10, 18], [-10, 0, 1, -1], [0, -5, -4, -5], [-3, 3, 0, 7], [0, 38, -14, 38], [12, -18, 12, 0], [-1, 0, 0, 1], [-7, 7, 0, -21], [-47, 0, -4, 4], [-45, 0, 1, -1], [31, -16, 16, -31], [20, 15, 20, 0], [-14, 0, -23, 23], [-24, 26, -26, 24], [0, -3, -21, -3], [2, 33, 2, 0], [3, -3, 0, -30], [-31, 0, -24, 24], [-9, 9, 0, 20], [14, 4, -4, -14], [10, -18, 10, 0], [0, 16, -39, 16], [33, -4, 4, -33], [25, -41, 41, -25], [45, 0, 6, -6], [0, 17, 16, 17], [22, -16, 22, 0], [-11, -17, -11, 0], [8, -16, 8, 0], [0, -4, 52, -4], [9, -9, 0, -15], [20, -18, 18, -20], [-46, 35, -35, 46], [-20, 20, 0, 0], [0, -31, 8, -31], [1, -3, 1, 0], [31, 0, 5, -5], [-9, 0, -4, 4], [-10, 24, -10, 0], [0, 17, -24, 17], [-36, 0, -8, 8], [0, -30, -6, -30], [22, -22, 0, 9], [3, 53, 3, 0], [12, 8, -8, -12], [12, -9, 12, 0], [8, -25, 8, 0], [-6, 0, 0, 6], [8, -35, 8, 0], [-7, 0, -8, 8], [26, -26, 0, 4], [-1, 0, 33, -33], [0, 13, -30, 13], [14, 8, -8, -14], [30, -30, 0, -30], [39, 0, 24, -24], [-4, 4, 0, -21]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_418_f_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_418_2_f_e();".
// 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_f_e(:prec:=4)
    chi := MakeCharacter_418_f();
    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_f_e();".
// 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_f_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_418_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![25, -25, 15, -5, 1]>],Snew);
    return Vf;
end function;