
// Make newform 418.2.f.a in Magma, downloaded from the LMFDB on 20 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_a();".
// 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_a();".
// 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, 0, -2], [-3, 1, -3, 0], [-3, 3, 0, 2], [2, -4, 1, -2], [4, -4, 4, -4], [-1, 0, -1, 0], [0, 0, 1, 0], [-1, 0, 3, -3], [-6, 6, 0, 2], [4, -6, 6, -4], [-2, 2, 0, 2], [0, 4, -6, 4], [-7, 0, -1, 1], [0, -3, 4, -3], [-10, 8, -8, 10], [8, -8, 0, -2], [-3, 6, -3, 0], [-8, 0, -2, 2], [-2, 6, -2, 0], [-8, 8, 0, 6], [8, -6, 6, -8], [3, -7, 3, 0], [-8, 0, -10, 10], [-12, 4, -4, 12], [2, -11, 11, -2], [-8, 8, 0, 2], [0, -4, 4, -4], [-4, 0, 8, -8], [0, 4, -14, 4], [-6, 12, -6, 0], [12, 0, 3, -3], [9, -5, 9, 0], [5, -5, 0, -11], [-4, 10, -4, 0], [0, 2, -14, 2], [0, -7, -11, -7], [-8, -7, 7, 8], [-10, 16, -16, 10], [0, 0, 4, 0], [0, -18, 8, -18], [12, -14, 12, 0], [1, -1, 0, 14], [4, 12, 4, 0], [-6, 0, -8, 8], [26, 0, 1, -1], [-10, 2, -2, 10], [0, -14, 12, -14], [-10, 10, 0, 20], [9, -6, 6, -9], [-4, 11, -11, 4], [0, -7, -4, -7], [-22, 0, 4, -4], [13, -14, 14, -13], [-10, 10, 0, 12], [-16, 0, 0, 0], [-6, -10, -6, 0], [5, -5, 0, -25], [18, -14, 14, -18], [0, 4, 0, 0], [0, -3, -5, -3], [0, 0, 0, 24], [-4, 0, 4, -4], [0, -9, 20, -9], [17, -12, 17, 0], [6, -2, 2, -6], [-8, 0, -4, 4], [10, -10, 0, -20], [-11, 14, -11, 0], [0, -7, 21, -7], [-16, 0, -29, 29], [11, -11, 0, 9], [5, -5, 0, -6], [-16, 0, -4, 4], [-16, 18, -16, 0], [10, 4, -4, -10], [15, -15, 0, 7], [11, 0, 9, -9], [2, 16, 2, 0], [-32, 36, -36, 32], [-17, 0, -19, 19], [0, 4, 18, 4], [-18, 30, -30, 18], [16, -16, 0, -2], [6, 0, 20, -20], [0, -15, 3, -15], [-12, 6, -6, 12], [-5, -3, -5, 0], [-14, 0, -27, 27], [2, 0, -23, 23], [-15, 15, -15, 0], [5, -28, 5, 0], [0, 12, 2, 12], [-19, 19, 0, 12], [-13, 13, 0, 12], [0, 8, 12, 8], [0, 2, 20, 2], [-12, 12, 0, 16], [12, 10, 12, 0], [-13, 10, -10, 13], [0, -28, 2, -28], [-3, 3, 0, -30], [-28, 20, -20, 28], [0, 10, -2, 10], [0, 0, 23, -23], [-28, 6, -28, 0], [-4, 4, 0, -36], [-1, 0, 19, -19], [-12, -10, -12, 0], [28, -28, 0, -12], [8, -6, 6, -8], [0, -15, 22, -15], [-10, 0, 20, -20], [0, 15, 10, 15], [4, -4, 0, 16], [0, -16, 28, -16], [-7, -18, -7, 0], [16, 12, -12, -16], [1, -1, 0, -27], [22, 0, 12, -12], [-12, 0, -12, 12], [16, -2, 2, -16], [12, -2, 12, 0], [-6, 0, -6, 6], [0, -23, 23, 0], [0, 5, 34, 5], [-13, -25, -13, 0], [15, -15, 0, 18], [15, 0, 31, -31], [11, -11, 0, -14], [32, -9, 9, -32], [-12, 16, -12, 0], [0, -16, -12, -16], [-18, -2, 2, 18], [34, -38, 38, -34], [-4, 0, -15, 15], [0, 20, -18, 20], [-12, 2, -12, 0], [0, -34, 0, 0], [-5, -27, -5, 0], [0, 2, -16, 2], [1, -1, 0, -47], [-10, 29, -29, 10], [-4, 30, -30, 4], [20, -20, 0, -30], [0, 34, -32, 34], [-3, -2, -3, 0], [12, 0, 24, -24], [-1, 0, 13, -13], [-2, 22, -2, 0], [0, 14, 10, 14], [-2, 0, 12, -12], [0, 13, 7, 13], [-28, 28, 0, 18], [14, -40, 14, 0], [-10, -12, 12, 10], [-35, 17, -35, 0], [1, 48, 1, 0], [19, -36, 36, -19], [2, -24, 2, 0], [-12, 0, 17, -17], [34, -34, 0, -2], [-47, 0, -3, 3], [0, -8, -20, -8], [8, -32, 32, -8], [-12, 12, 0, -8], [8, 0, 8, -8], [29, -29, 0, -22]];
    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_a();".
// 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_a(: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_a();".
// 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_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_418_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![16, 24, 16, 4, 1]>],Snew);
    return Vf;
end function;