
// Make newform 55.2.g.b in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_55_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_55_g_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_55_2_g_b();".
// 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_55_2_g_b();".
// 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, 3, 5, 3, 4, -3, 5, -3, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [-7, -12, -7, -4, 0, -3, 2, -1], [9, 6, 19, -16, 20, -7, 0, 1], [-1, -10, 13, -16, 12, -9, 4, -1], [-5, -8, -13, -4, 8, -17, 10, -3], [-1, 6, 1, 16, -20, 23, -12, 3], [-7, -12, -11, -28, 32, -35, 18, -5]];
    Rf_basisdens := [1, 1, 8, 8, 8, 8, 8, 8];
    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_55_g();"
function MakeCharacter_55_g()
    N := 55;
    order := 5;
    char_gens := [12, 46];
    v := [5, 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_55_g_Hecke();"
function MakeCharacter_55_g_Hecke(Kf)
    N := 55;
    order := 5;
    char_gens := [12, 46];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 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 := [[0, 0, 0, 0, 0, 0, -1, 0], [-1, 1, 0, 1, -1, -1, -1, 0], [1, 0, 0, -1, 1, 0, 0, 1], [0, 2, 1, -1, 1, 0, -2, 1], [1, -1, 2, -2, 1, 1, 0, 2], [-2, 0, 0, 0, -2, 0, 3, -2], [-1, 0, 1, 1, 2, 0, -1, 2], [0, 3, 0, 5, 0, -1, -1, 0], [2, -1, -1, 0, 0, 0, 0, 0], [0, -1, -2, -2, 2, 0, 1, -3], [0, 0, 0, 0, 0, 0, -5, 0], [0, -3, -3, 7, -7, 0, 3, -4], [1, -4, 0, 1, 1, 3, 3, 0], [-6, 2, 2, 0, 0, -1, 0, 0], [0, -2, 0, -3, 0, 3, 3, 0], [1, 0, -5, 0, -5, 5, 1, 1], [0, 4, 1, -5, 5, 0, -4, 7], [2, 4, 2, -2, 3, -4, -2, 3], [-5, -4, -4, 6, 0, 1, 0, -6], [-3, -3, -3, 3, -1, 3, 3, -1], [0, 2, 3, 0, 0, 0, -2, -1], [6, 0, 3, 0, -3, -3, -4, 6], [-1, 4, 1, 1, -7, -4, -1, -7], [1, -6, -6, 0, 0, 6, 0, 0], [8, 0, 2, 0, 13, -2, 0, 8], [-1, 0, -7, 0, -3, 7, 3, -1], [0, -4, 0, -6, 6, 0, 4, 5], [-7, 5, 0, -2, -7, 3, 3, 0], [-5, 4, 4, -6, 0, -3, 0, 6], [2, 2, 0, -1, 2, -1, -1, 0], [5, -5, -2, -5, 2, 5, 2, 2], [-6, 6, 6, -2, 0, -5, 0, 2], [8, 5, 9, -8, 11, -5, -9, 11], [0, -5, -5, 7, -7, 0, 5, -15], [1, -1, -6, -1, -6, 1, 6, -6], [-5, 7, 0, 12, -5, -9, -9, 0], [10, 3, 0, -6, 10, -6, -6, 0], [-4, 0, 9, 0, 0, -9, 2, -4], [-7, 0, -6, 0, 0, 6, 5, -7], [3, 1, 0, -11, 3, -2, -2, 0], [0, -10, 0, 5, 0, -2, -2, 0], [0, 11, 3, 0, -4, -11, -3, -4], [0, -6, -5, 0, 0, 0, 6, -1], [7, -14, -5, -7, 5, 14, 5, 5], [-2, 1, 1, 7, 0, -6, 0, -7], [-4, 3, 3, -2, 0, -5, 0, 2], [12, 0, 2, 0, -5, -2, -1, 12], [-10, 4, 0, 6, -10, -2, -2, 0], [0, -7, 2, -7, 7, 0, 7, -3], [2, 0, -13, 0, 3, 13, 1, 2], [-3, 0, 6, 0, 9, -6, -13, -3], [16, -14, 0, -4, 16, 7, 7, 0], [-2, 0, 0, -9, 0, 10, 0, 9], [-9, 0, 1, 0, 0, -1, 2, -9], [0, 3, 1, 14, -14, 0, -3, -7], [1, 0, 0, 4, 0, 7, 0, -4], [4, -1, 5, -4, 0, 1, -5, 0], [0, 0, -7, 5, -5, 0, 0, -12], [1, 0, 0, 0, -5, 0, -2, 1], [-6, 11, 8, 6, 10, -11, -8, 10], [3, 1, 0, 17, 3, -10, -10, 0], [0, -2, 1, -7, 7, 0, 2, 17], [0, -7, -7, -7, 0, 2, 0, 7], [-16, 6, 0, 9, -16, 7, 7, 0], [6, 13, 8, -6, 8, -13, -8, 8], [4, 0, 7, 0, 10, -7, -13, 4], [-2, 4, 4, -12, 0, 16, 0, 12], [0, -6, 6, 5, -5, 0, 6, -16], [5, 3, 3, -5, 6, -3, -3, 6], [-16, 7, 0, 10, -16, 0, 0, 0], [-4, 2, 2, 7, 0, -19, 0, -7], [0, 8, 6, 0, 0, 0, -8, 5], [0, 11, 2, -8, 8, 0, -11, 5], [-1, 5, 5, 11, 0, 1, 0, -11], [-2, -6, -2, 2, -11, 6, 2, -11], [7, 0, 5, 0, -9, -5, -5, 7], [0, -6, -22, 18, -18, 0, 6, -7], [-11, 0, 0, -3, 0, -2, 0, 3], [15, -8, -11, -15, 2, 8, 11, 2], [-3, 0, -2, 0, 6, 2, 8, -3], [-11, 9, 9, 5, 0, -18, 0, -5], [17, -15, 0, -5, 17, 0, 0, 0], [15, 0, 13, 0, 13, -13, -11, 15], [0, 7, -18, 17, -17, 0, -7, -9], [0, -5, -5, 8, 0, 5, 0, -8], [3, 19, 0, 3, 3, -2, -2, 0], [-7, 0, 13, 0, 9, -13, 0, -7], [9, -14, -8, -9, 4, 14, 8, 4], [8, -3, -3, 2, 0, -2, 0, -2], [19, -9, -9, -17, 0, 7, 0, 17], [-18, 10, 4, 18, -27, -10, -4, -27], [-8, -12, -13, 8, -10, 12, 13, -10], [1, -15, 0, 12, 1, 3, 3, 0], [0, 4, 5, 0, 0, 0, -4, -15], [0, 11, 12, 5, -5, 0, -11, 5], [0, 6, 0, -38, 0, 5, 5, 0], [9, 5, 0, -3, 9, -6, -6, 0], [0, 0, 13, 3, -3, 0, 0, -8], [-12, 3, -2, 12, 0, -3, 2, 0], [-1, 0, -4, 0, 3, 4, 2, -1], [14, -3, 0, 2, 14, 1, 1, 0], [0, 5, -20, 22, -22, 0, -5, -17], [2, 0, 9, 0, 11, -9, 2, 2], [-20, 18, 0, 3, -20, -13, -13, 0], [-6, 0, 0, -16, 0, -1, 0, 16], [-26, -8, -13, 26, -18, 8, 13, -18], [0, -15, 10, -16, 16, 0, 15, 5], [36, -1, -1, -6, 0, -1, 0, 6], [5, 0, 3, -5, 17, 0, -3, 17], [0, 0, 9, 16, -16, 0, 0, 8], [6, 0, 13, 0, 8, -13, 11, 6], [-4, 4, 0, -11, -4, 11, 11, 0], [0, 7, 7, -15, 0, -2, 0, 15], [5, -7, 0, -21, 5, 17, 17, 0], [0, 12, 16, -20, 20, 0, -12, 5], [-12, 18, 0, 21, -12, -6, -6, 0], [-9, 9, -8, 9, -5, -9, 8, -5], [-29, 0, -2, 0, -28, 2, 4, -29], [0, -3, 25, -6, 6, 0, 3, 15], [-10, 2, 2, -9, 0, 5, 0, 9], [-18, 9, 9, 8, 0, -23, 0, -8], [7, 0, -13, 0, 3, 13, -20, 7], [19, -18, 4, -19, 25, 18, -4, 25], [-15, -1, -1, -8, 0, 12, 0, 8], [15, 0, 5, 0, 25, -5, -2, 15], [1, -14, 0, 21, 1, -9, -9, 0], [15, 3, 10, -15, 13, -3, -10, 13], [0, -15, -9, 6, -6, 0, 15, -25], [19, -18, -18, -6, 0, 18, 0, 6], [0, 14, 4, -11, 11, 0, -14, 29], [-8, 0, 17, 0, -12, -17, 0, -8], [2, -10, -4, -2, -28, 10, 4, -28], [-13, 16, 0, 5, -13, -12, -12, 0], [6, 0, -6, 0, -18, 6, 0, 6], [-16, 0, -9, 0, -23, 9, 7, -16], [5, 10, 10, 20, 0, -28, 0, -20], [7, -31, 0, -9, 7, 12, 12, 0], [24, -2, 6, -24, 13, 2, -6, 13], [17, -11, -6, -17, 10, 11, 6, 10], [3, -8, 2, -3, 11, 8, -2, 11], [-5, -20, 0, 4, -5, 3, 3, 0], [0, 0, 7, -31, 31, 0, 0, 1], [1, 0, 4, 0, -3, -4, 24, 1], [35, 0, -3, 0, 9, 3, -1, 35], [0, -5, 5, -5, 5, 0, 5, -2], [-8, 17, 0, -12, -8, 3, 3, 0], [-11, -7, 10, 11, 1, 7, -10, 1], [-29, 7, 7, 7, 0, 13, 0, -7], [7, -11, -11, -2, 0, 8, 0, 2], [-5, -6, 2, 5, 8, 6, -2, 8], [-14, 10, 0, 13, -14, 4, 4, 0], [14, 17, 17, -15, 0, -7, 0, 15], [-19, -21, 0, 0, -19, 8, 8, 0], [0, -7, -16, -17, 17, 0, 7, 6], [-2, 25, 23, 2, 6, -25, -23, 6], [-20, 0, 2, 0, -12, -2, 8, -20], [-24, -2, -8, 24, -2, 2, 8, -2], [8, 25, 14, -8, -5, -25, -14, -5], [-12, 0, -13, 0, -24, 13, 7, -12], [38, -13, -8, -38, 34, 13, 8, 34], [9, -6, -6, 19, 0, -5, 0, -19], [0, 9, -10, 13, -13, 0, -9, 4], [-12, 4, 4, 7, 0, -15, 0, -7], [-3, 9, 0, 17, -3, -2, -2, 0], [1, 0, -17, 0, -5, 17, 15, 1], [0, 14, -8, -17, 17, 0, -14, 40], [15, 1, 1, 0, 0, 6, 0, 0], [0, -2, -19, 17, -17, 0, 2, -9]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_55_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_55_2_g_b();".
// 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_55_2_g_b(:prec:=8)
    chi := MakeCharacter_55_g();
    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_55_2_g_b();".
// 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_55_2_g_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_55_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, -2, 5, -2, -1, 2, 5, 2, 1]>],Snew);
    return Vf;
end function;