
// Make newform 2205.2.d.l in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2205_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2205_d_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_2205_2_d_l();".
// 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_2205_2_d_l();".
// 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 := [2, -4, 4, 2, 2, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [-1, 6, -26, 11, 1, -3], [-2, -80, -6, -1, 2, -6], [-4, 24, 34, 21, -19, 11], [-26, 64, 60, 10, -20, 14], [56, -60, -16, -41, 36, -16]];
    Rf_basisdens := [1, 23, 23, 23, 23, 23];
    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_2205_d();"
function MakeCharacter_2205_d()
    N := 2205;
    order := 2;
    char_gens := [1226, 442, 1081];
    v := [2, 1, 2];
    // 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_2205_d_Hecke();"
function MakeCharacter_2205_d_Hecke(Kf)
    N := 2205;
    order := 2;
    char_gens := [1226, 442, 1081];
    char_values := [[1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0], [1, 0, 0, 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, 0, 0], [0, 1, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0], [-2, 0, 0, 0, 0, 0], [0, 2, 1, 0, 1, 1], [0, -2, -1, 0, 0, -1], [2, 0, -1, 2, -1, 1], [0, 0, 1, 0, -1, 1], [2, 0, -2, 0, 0, 2], [-2, 0, 1, 2, -1, -1], [0, -4, 0, 0, 0, 0], [0, 0, -1, 4, -2, 1], [0, 4, 4, 0, 0, 4], [0, 0, 2, 0, -2, 2], [0, -2, 1, 0, -3, 1], [-4, 0, -4, 4, -2, 4], [2, 0, -2, 4, -2, 2], [0, 0, 2, 0, -2, 2], [-2, 0, 0, 0, 0, 0], [0, -2, -1, 0, 3, -1], [-4, 0, 2, -4, 2, -2], [0, 4, 0, 0, 2, 0], [-4, 0, -1, 0, 0, 1], [0, -2, 3, 0, -5, 3], [-8, 0, 3, -4, 2, -3], [0, 4, 2, 0, 0, 2], [0, -4, -1, 0, 5, -1], [2, 0, 0, 4, -2, 0], [0, 6, 3, 0, -1, 3], [0, 4, 2, 0, 2, 2], [8, 0, -4, 4, -2, 4], [0, 2, -3, 0, 7, -3], [6, 0, 3, -2, 1, -3], [-6, 0, 0, 8, -4, 0], [8, 0, 2, -4, 2, -2], [0, -2, -1, 0, -3, -1], [0, 4, 2, 0, 6, 2], [0, -4, -6, 0, -2, -6], [0, -2, 3, 0, 2, 3], [10, 0, 0, 0, 0, 0], [2, 0, 4, -4, 2, -4], [2, 0, -4, 0, 0, 4], [0, -8, -6, 0, -2, -6], [0, -2, 1, 0, 1, 1], [-2, 0, -1, -6, 3, 1], [8, 0, 0, -8, 4, 0], [0, 8, 0, 0, 0, 0], [0, 4, -2, 0, 4, -2], [2, 0, 0, 4, -2, 0], [0, -6, -3, 0, -5, -3], [-6, 0, -4, 0, 0, 4], [6, 0, 2, 0, 0, -2], [20, 0, 0, -4, 2, 0], [0, 2, 1, 0, -2, 1], [0, 0, -1, 0, 5, -1], [4, 0, 3, 4, -2, -3], [2, 0, -5, -2, 1, 5], [0, 4, -4, 0, 0, -4], [-10, 0, 2, 8, -4, -2], [0, -4, 2, 0, 2, 2], [0, 2, 1, 0, 2, 1], [0, -12, -6, 0, 2, -6], [0, 0, -2, 12, -6, 2], [0, -10, -1, 0, 5, -1], [0, -6, -1, 0, 3, -1], [16, 0, -4, 0, 0, 4], [0, 0, -4, 0, -4, -4], [0, -4, -5, 0, -3, -5], [-10, 0, 2, 4, -2, -2], [0, -2, 7, 0, 0, 7], [-18, 0, 4, -8, 4, -4], [0, 8, 4, 0, -8, 4], [0, 0, 0, 0, 8, 0], [-4, 0, 2, -4, 2, -2], [0, 4, 0, 0, -8, 0], [-10, 0, 6, -8, 4, -6], [0, 2, -7, 0, -1, -7], [18, 0, -6, 8, -4, 6], [-18, 0, 2, 8, -4, -2], [-16, 0, 2, -8, 4, -2], [-26, 0, 0, 4, -2, 0], [6, 0, -4, 8, -4, 4], [0, -2, 3, 0, 1, 3], [-18, 0, -1, 2, -1, 1], [0, 12, 5, 0, -1, 5], [-6, 0, 8, -16, 8, -8], [0, -4, -2, 0, 6, -2], [-4, 0, 1, 0, 0, -1], [0, 4, 2, 0, 6, 2], [0, 4, -2, 0, 4, -2], [0, 0, 2, 4, -2, -2], [0, -12, -4, 0, 4, -4], [-2, 0, 0, 0, 0, 0], [-12, 0, -6, 4, -2, 6], [0, -4, 0, 0, 4, 0], [4, 0, 5, -12, 6, -5], [-28, 0, 3, -4, 2, -3], [0, 0, 0, 0, 2, 0], [2, 0, 8, -4, 2, -8], [0, 0, -2, 0, 10, -2], [0, 10, 11, 0, 3, 11], [0, 8, -2, 0, -8, -2], [-10, 0, -6, 8, -4, 6], [4, 0, 8, 0, 0, -8], [0, 10, -7, 0, 7, -7], [0, -16, 2, 0, -8, 2], [0, -6, -11, 0, 0, -11], [-10, 0, 4, 8, -4, -4], [2, 0, -4, 0, 0, 4], [0, 8, 8, 0, 0, 8], [0, -4, 2, 0, -2, 2], [0, 6, 3, 0, 13, 3], [22, 0, 3, -10, 5, -3], [4, 0, 6, -4, 2, -6], [6, 0, -10, 16, -8, 10], [0, -12, -10, 0, -10, -10], [0, -4, -6, 0, -2, -6], [0, 6, 5, 0, -17, 5], [-2, 0, 4, -16, 8, -4], [-6, 0, -6, 20, -10, 6], [0, -12, -8, 0, 0, -8], [0, -14, -7, 0, 6, -7], [0, -4, -3, 0, 7, -3], [-2, 0, -1, -2, 1, 1], [-10, 0, 10, -16, 8, -10], [-2, 0, 8, -8, 4, -8], [4, 0, -2, 4, -2, 2], [0, -12, -10, 0, 4, -10], [0, 2, -7, 0, 5, -7], [-12, 0, 8, -8, 4, -8], [0, 8, 5, 0, -5, 5], [12, 0, -12, 16, -8, 12], [0, 16, 4, 0, -12, 4], [20, 0, -9, 8, -4, 9], [6, 0, 0, -8, 4, 0], [0, -14, -11, 0, 2, -11], [0, 16, 0, 0, 2, 0], [0, 2, 5, 0, 6, 5], [6, 0, 0, -8, 4, 0], [2, 0, 13, -6, 3, -13], [26, 0, 8, -8, 4, -8], [0, 4, 10, 0, 2, 10], [0, -4, -9, 0, -11, -9], [22, 0, 0, 4, -2, 0], [-4, 0, 4, 0, 0, -4], [0, -2, -9, 0, -3, -9], [0, 18, 9, 0, -10, 9], [-2, 0, 1, -2, 1, -1], [0, -16, -11, 0, -1, -11], [0, -20, -10, 0, -6, -10], [16, 0, -5, 8, -4, 5], [0, -8, -4, 0, -8, -4], [0, 24, 10, 0, -2, 10], [0, 8, -12, 0, 8, -12], [10, 0, -4, 16, -8, 4], [-12, 0, -8, 0, 0, 8], [-12, 0, 5, 16, -8, -5], [0, 6, 7, 0, 1, 7], [16, 0, 7, -8, 4, -7], [0, -4, 5, 0, -13, 5], [0, -10, 3, 0, -17, 3], [0, 24, 10, 0, -2, 10], [4, 0, 10, -16, 8, -10], [0, 14, 11, 0, -3, 11], [0, 0, 0, 0, -12, 0], [-12, 0, -12, 8, -4, 12], [0, 10, -7, 0, 7, -7], [-2, 0, -8, 0, 0, 8], [0, 10, 13, 0, -6, 13], [34, 0, -4, 0, 0, 4], [-2, 0, -14, 12, -6, 14], [14, 0, 0, -8, 4, 0], [0, 16, 6, 0, -14, 6], [-30, 0, 9, -2, 1, -9], [-36, 0, -1, -4, 2, 1], [0, 0, -2, 12, -6, 2], [-6, 0, 6, -16, 8, -6], [0, -12, 6, 0, -8, 6], [18, 0, -14, 28, -14, 14], [0, 16, -2, 0, 6, -2], [4, 0, 2, 8, -4, -2], [0, -8, 6, 0, 2, 6], [0, 10, -3, 0, 8, -3], [0, 0, 3, 0, 13, 3], [0, 0, -9, 12, -6, 9], [0, 12, 6, 0, -2, 6], [0, 8, 4, 0, -2, 4], [10, 0, 12, -16, 8, -12], [4, 0, 6, -4, 2, -6], [0, 10, 5, 0, 17, 5], [0, -4, -7, 0, 19, -7], [-12, 0, 2, -4, 2, -2], [4, 0, 15, -16, 8, -15], [0, 28, 5, 0, -1, 5], [0, 2, 1, 0, 12, 1], [2, 0, 0, 0, 0, 0], [0, -16, -10, 0, 2, -10], [0, -6, -3, 0, 2, -3], [0, -12, 2, 0, -10, 2], [14, 0, 0, 8, -4, 0], [42, 0, 1, -10, 5, -1], [0, -14, -19, 0, -11, -19], [26, 0, -10, -8, 4, 10], [-28, 0, 2, 8, -4, -2], [0, 2, -11, 0, 14, -11], [2, 0, 5, -26, 13, -5], [0, 20, 13, 0, 3, 13], [-2, 0, 0, 16, -8, 0], [10, 0, -9, -14, 7, 9], [0, -20, -2, 0, 10, -2], [20, 0, -9, -12, 6, 9], [0, 8, 12, 0, 0, 12], [0, -20, -12, 0, -10, -12], [-4, 0, -8, 16, -8, 8], [-14, 0, 12, 0, 0, -12], [0, 4, 0, 0, -12, 0], [-8, 0, 5, 0, 0, -5], [0, 8, -11, 0, 7, -11], [0, 22, -3, 0, 3, -3], [18, 0, 0, -8, 4, 0], [-22, 0, 3, -18, 9, -3], [-10, 0, 4, 8, -4, -4], [0, 0, 4, 0, -4, 4], [0, 0, -4, 0, 6, -4], [22, 0, -4, -24, 12, 4], [0, -2, -9, 0, -2, -9], [-10, 0, -12, 16, -8, 12], [0, 12, -2, 0, 24, -2], [-10, 0, -20, 24, -12, 20], [0, 16, 4, 0, 12, 4], [-14, 0, -17, 18, -9, 17], [0, 0, -18, 20, -10, 18], [-38, 0, 0, 8, -4, 0], [0, -4, 6, 0, -10, 6], [0, 16, 16, 0, 12, 16], [-2, 0, 10, -24, 12, -10], [0, -30, -13, 0, 11, -13], [-14, 0, -4, 24, -12, 4], [-36, 0, -6, 12, -6, 6], [0, -4, -5, 0, 13, -5], [2, 0, 13, -6, 3, -13], [0, 0, 4, 0, -20, 4], [-30, 0, 8, 16, -8, -8], [0, 14, -1, 0, 10, -1], [52, 0, 4, -8, 4, -4], [0, 12, 10, 0, 8, 10], [32, 0, -6, 8, -4, 6], [-32, 0, -2, 12, -6, 2], [0, -8, 1, 0, 11, 1], [0, -20, -2, 0, -10, -2], [-24, 0, -7, 24, -12, 7], [0, 8, 1, 0, 11, 1], [-18, 0, 10, -16, 8, -10], [0, 10, 1, 0, -10, 1], [-6, 0, -16, 16, -8, 16], [-46, 0, -8, 8, -4, 8], [0, 36, 14, 0, -10, 14], [0, 6, -9, 0, 22, -9], [0, -6, 1, 0, -13, 1], [0, 28, -6, 0, 14, -6], [0, -12, -1, 0, -23, -1], [-30, 0, 12, 4, -2, -12], [0, 10, -3, 0, 31, -3], [0, -18, -9, 0, -12, -9], [34, 0, -7, -6, 3, 7], [-2, 0, -4, 8, -4, 4], [28, 0, 3, -12, 6, -3], [0, -24, -18, 0, -2, -18], [0, 6, 9, 0, -9, 9], [-42, 0, 6, 12, -6, -6], [0, 4, 16, 0, 0, 16], [0, 2, -3, 0, -7, -3], [44, 0, -4, 0, 0, 4], [0, -2, -1, 0, -13, -1], [0, -4, -14, 0, 0, -14], [0, 12, 11, 0, 5, 11], [50, 0, -8, 4, -2, 8], [10, 0, 12, -20, 10, -12], [-60, 0, 8, 12, -6, -8], [0, -16, 4, 0, -20, 4], [-28, 0, -2, -4, 2, 2], [0, -16, -24, 0, 0, -24], [22, 0, -2, 4, -2, 2], [0, -24, -12, 0, 18, -12], [-14, 0, 0, 16, -8, 0], [0, -4, 10, 0, -6, 10], [0, 2, -9, 0, -5, -9], [2, 0, -17, 30, -15, 17], [12, 0, -7, 0, 0, 7], [-54, 0, 4, -16, 8, -4], [0, 12, 12, 0, 10, 12], [0, -14, -11, 0, 11, -11], [-8, 0, 2, 24, -12, -2], [0, -24, -8, 0, 8, -8], [-12, 0, 3, -8, 4, -3], [14, 0, -9, -6, 3, 9], [0, 22, 3, 0, -4, 3], [28, 0, 16, -12, 6, -16], [0, 0, -8, 0, -10, -8], [0, -22, -7, 0, 11, -7], [0, 10, 19, 0, -11, 19], [24, 0, -4, 0, 0, 4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2205_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2205_2_d_l();".
// 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_2205_2_d_l(:prec:=6)
    chi := MakeCharacter_2205_d();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_2205_2_d_l();".
// 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_2205_2_d_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2205_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 0, 27, 0, 11, 0, 1]>,<11,R![2, 1]>,<13,R![64, 0, 112, 0, 44, 0, 1]>,<19,R![40, -4, -6, 1]>,<29,R![40, -52, -2, 1]>],Snew);
    return Vf;
end function;