
// Make newform 405.2.e.k in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_405_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_405_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_405_2_e_k();".
// 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_405_2_e_k();".
// 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 := [9, 3, 4, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-3, -4, 4, -1], [7, 0, 0, 1]];
    Rf_basisdens := [1, 1, 12, 4];
    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_405_e();"
function MakeCharacter_405_e()
    N := 405;
    order := 3;
    char_gens := [326, 82];
    v := [2, 3];
    // 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_405_e_Hecke();"
function MakeCharacter_405_e_Hecke(Kf)
    N := 405;
    order := 3;
    char_gens := [326, 82];
    char_values := [[0, 0, 1, 0], [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 := [[0, 1, 0, 0], [0, 0, 0, 0], [0, 0, 1, 0], [-2, 2, -2, 0], [0, -2, 0, 0], [-2, 2, 2, 2], [-1, 0, 0, -2], [1, 0, 0, -2], [0, 0, -3, 0], [-6, 2, -6, 0], [2, -2, -1, -2], [2, 0, 0, 0], [2, -2, 0, -2], [4, -2, 4, 0], [0, 4, 0, 0], [-1, 0, 0, -2], [-2, 2, -6, 2], [-5, 4, -5, 0], [-4, 4, -10, 4], [10, 0, 0, 2], [10, 0, 0, -2], [7, 2, 7, 0], [-3, 0, -3, 0], [-6, 0, 0, 6], [-8, 0, -8, 0], [-12, 0, -12, 0], [0, 0, -4, 0], [0, 0, 0, 0], [-7, 0, 0, 0], [4, -4, -6, -4], [-14, 0, 0, 4], [0, 0, 6, 0], [-3, -6, -3, 0], [0, 0, -4, 0], [-10, 10, 0, 10], [4, -8, 4, 0], [6, -6, -4, -6], [2, 0, 0, 0], [0, 0, -3, 0], [-3, 6, -3, 0], [-16, 0, 0, 4], [-7, 0, 0, 0], [-6, 8, -6, 0], [6, -6, 14, -6], [-15, 0, 0, 6], [4, 0, 0, 4], [6, -6, 5, -6], [10, 0, 10, 0], [21, -4, 21, 0], [-4, 4, -1, 4], [-18, 0, 0, 0], [4, -4, -6, -4], [-23, 4, -23, 0], [12, 0, 0, -12], [6, -6, 15, -6], [12, -4, 12, 0], [2, 0, 0, 4], [13, 0, 0, -14], [16, -10, 16, 0], [6, 6, 6, 0], [2, -2, 8, -2], [-6, 6, -21, 6], [-2, 0, 0, -8], [6, -6, 6, -6], [-8, 12, -8, 0], [3, 2, 3, 0], [-20, -4, -20, 0], [-2, 2, 2, 2], [-8, 8, 12, 8], [-5, 16, -5, 0], [6, -12, 6, 0], [6, 0, 0, 12], [-2, 2, -2, 0], [4, -4, -16, -4], [19, 0, 0, -2], [-12, 12, -3, 12], [-12, -6, -12, 0], [-8, 0, 0, -2], [0, 0, -30, 0], [0, 0, 5, 0], [-10, 10, 0, 10], [13, -8, 13, 0], [12, 0, 0, 6], [4, 0, 0, -14], [-23, -2, -23, 0], [-3, 12, -3, 0], [20, 0, 0, -14], [4, 4, 4, 0], [6, -12, 6, 0], [-4, 4, -10, 4], [7, 0, 0, -4], [24, 6, 24, 0], [-22, 0, 0, 6], [-4, 4, 18, 4], [-2, 2, -25, 2], [13, 0, 0, 8], [16, -16, 12, -16], [-26, 0, 0, 2], [28, 0, 0, -2], [-30, 0, 0, 8], [4, -2, 4, 0], [-6, 0, 0, -12], [0, 0, 24, 0], [-30, 6, -30, 0], [-18, 18, -13, 18], [14, 0, 0, 6], [-21, 0, -21, 0], [-11, 0, 0, 14], [2, -2, -18, -2], [-17, 12, -17, 0], [-2, 2, 20, 2], [-8, 0, 0, 16], [-18, 18, -3, 18], [-8, -12, -8, 0], [7, 0, 0, 10], [24, 0, 24, 0], [-10, 10, -10, 10], [29, 0, 0, 4], [14, -14, 9, -14], [18, -2, 18, 0], [16, -16, 2, -16], [-2, -10, -2, 0], [6, -12, 6, 0], [-9, 0, 0, 0], [-5, -2, -5, 0], [22, 0, 0, 8], [-14, 16, -14, 0], [6, 0, 0, -24], [-8, -12, -8, 0], [0, 0, 32, 0], [7, 0, 0, 10], [-8, 8, -24, 8], [10, -10, -7, -10], [22, 0, 0, 4], [-18, 18, -30, 18], [0, 0, 41, 0], [3, 0, 0, 6], [10, -10, -16, -10], [-22, 22, -9, 22], [46, 0, 0, 2], [24, 0, 0, 8], [-30, 0, -30, 0], [6, -6, -4, -6], [-43, 0, 0, 4], [-18, 0, 0, -4], [-24, -4, -24, 0], [-20, -4, -20, 0], [27, -14, 27, 0], [10, -10, -7, -10], [25, 0, 0, -4], [-22, 22, -28, 22], [-28, 0, 0, -2], [8, 0, 0, -6], [12, -12, -3, -12], [-20, 8, -20, 0], [-18, 10, -18, 0], [16, 0, 0, -8], [-24, 2, -24, 0], [36, 0, 0, -4], [8, -8, 18, -8], [-39, 0, -39, 0], [-10, 0, 0, 16], [0, 0, 50, 0], [-18, 0, 0, 0], [4, -4, -6, -4], [15, 12, 15, 0], [-37, 0, 0, -6], [-20, 26, -20, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_405_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_405_2_e_k();".
// 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_405_2_e_k(:prec:=4)
    chi := MakeCharacter_405_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_405_2_e_k();".
// 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_405_2_e_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_405_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![9, 3, 4, -1, 1]>,<7,R![144, -24, 16, 2, 1]>,<11,R![144, -24, 16, 2, 1]>],Snew);
    return Vf;
end function;