
// Make newform 55.2.g.a in Magma, downloaded from the LMFDB on 24 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_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_55_2_g_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 := [25, 25, 10, -5, 21, -11, 6, -1, 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], [375, 43855, -84434, 40069, -18164, 7489, -2159, 555], [35160, -68120, 27673, -14943, 9063, -6608, -1002, -970], [-13875, -13500, 38305, -81659, 28414, -12059, 4159, -1604], [-90925, -19395, 35785, -82042, 21007, -19912, 2252, -2052], [-12035, -4035, 72554, -82441, 50856, -17466, 6691, -2667], [66675, 54640, 22635, 59219, -26434, 21519, -1464, 4024]];
    Rf_basisdens := [1, 1, 94655, 94655, 94655, 94655, 94655, 94655];
    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], [-1, 0, 1, 1, 0, 0, 1, 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, -1, 0, 1, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0], [-2, 0, 3, 3, 0, -1, 2, 0], [0, 2, -1, -1, -1, 0, -2, 0], [0, -1, -1, 2, 1, 0, 2, 0], [0, -1, 0, 0, -1, 1, -1, 0], [1, 1, -1, -4, 0, 0, 0, 0], [-5, -1, 3, 0, -3, 3, 3, 1], [3, 0, 0, 0, 0, 0, -3, -1], [0, -3, -1, 2, 1, 2, 2, 2], [3, 0, -1, -1, 0, -3, -3, 1], [0, -4, 0, 3, 3, 0, 0, 3], [5, 1, -2, 0, -2, 2, -2, -1], [1, 2, -1, -1, 1, 0, 0, 1], [0, -2, 2, 0, 1, 1, 0, 1], [2, 0, -5, -5, 0, -3, -2, 2], [1, 2, 0, -1, -2, -2, -3, 0], [0, 3, -2, 0, 0, 0, -2, -3], [8, 3, 0, -8, -2, -3, -2, 0], [-2, 0, -4, -4, 0, 1, 2, -2], [0, 1, 2, 5, 2, -3, 5, -3], [-3, 1, 0, 3, 1, -1, -2, 0], [-3, 4, 6, 0, 2, -2, 6, -4], [0, 4, 1, 2, -2, -2, 2, -2], [0, 0, 8, 6, -1, 1, 6, 1], [3, 0, -5, -5, 0, 2, -3, 4], [-6, -5, 6, 1, -2, 0, 0, -2], [2, -3, 0, 0, -2, 2, 0, 3], [-5, 6, 5, 1, -7, 0, 0, -7], [-12, -2, 0, 12, 1, 2, 0, 0], [1, 1, -6, 0, 4, -4, -6, -1], [9, 1, 0, -9, -2, -1, 0, 0], [4, 0, -2, -2, 0, -3, -4, 9], [0, 0, 0, 0, 5, 0, 10, 0], [-4, 1, 4, -1, 0, 0, 0, 0], [-4, 3, 4, -1, -1, 0, 0, -1], [0, -3, 7, 1, 2, 1, 1, 1], [0, -3, -8, 3, 3, 0, 3, 0], [13, -5, -13, -8, 5, 0, 0, 5], [2, -6, -2, -5, 0, 0, 0, 0], [2, -3, 0, -2, -2, 3, -11, 0], [8, 0, 4, 4, 0, -1, -8, -4], [1, -1, 0, -1, -3, 1, -12, 0], [-2, -7, -2, 0, -7, 7, -2, 7], [6, -4, -7, 0, 3, -3, -7, 4], [0, -5, -14, 0, 1, 4, 0, 4], [0, -4, 0, -6, 2, 0, 0, 2], [-19, 0, 11, 11, 0, 0, 19, -7], [0, 0, 1, -13, 5, -5, -13, -5], [0, 1, 5, 7, -1, 0, 7, 0], [-17, 2, 17, 8, -3, 0, 0, -3], [-9, 6, 13, 0, -4, 4, 13, -6], [0, 9, 0, 4, -8, -1, 4, -1], [-2, 0, -11, -11, 0, 7, 2, 1], [6, 1, -4, 0, 8, -8, -4, -1], [0, -1, 0, 0, 0, 1, 3, 0], [-3, 0, 2, 2, 0, 1, 3, -10], [0, 14, 4, -7, -12, -2, -7, -2], [-13, -2, 0, 13, 3, 2, 16, 0], [9, -5, -9, -16, 5, 0, 0, 5], [-2, 0, 13, 13, 0, 7, 2, 0], [6, 1, 8, 0, 5, -5, 8, -1], [3, -2, -3, 5, -3, 0, 0, -3], [-5, 8, 0, 5, -5, -8, -8, 0], [0, -4, 0, -1, 5, -1, -1, -1], [-2, -4, -8, 0, 0, 0, -8, 4], [10, 0, -3, -3, 0, -2, -10, -4], [-1, -3, 0, 1, -2, 3, -3, 0], [10, 9, -10, -5, -3, 0, 0, -3], [-12, -3, -5, 0, 0, 0, -5, 3], [9, 0, -5, -5, 0, 2, -9, 8], [-19, 0, 24, 24, 0, 4, 19, 1], [8, -2, -10, 0, -7, 7, -10, 2], [-7, -2, 0, 7, 4, 2, 11, 0], [0, 2, -4, 8, 3, -5, 8, -5], [-3, 0, -1, -1, 0, 2, 3, -10], [4, -2, 7, 0, 6, -6, 7, 2], [6, 1, 0, -6, 13, -1, -5, 0], [0, -18, -19, -5, 8, 10, -5, 10], [-9, 5, 2, 0, 7, -7, 2, -5], [7, 7, -7, -2, -1, 0, 0, -1], [0, -2, 6, -6, 1, 1, -6, 1], [3, 0, -1, -1, 0, -12, -3, -5], [-2, -14, 3, 0, -5, 5, 3, 14], [-3, 5, 3, -24, 3, 0, 0, 3], [0, 11, 11, 6, -6, -5, 6, -5], [11, 0, 0, -11, 4, 0, 8, 0], [-17, -3, 1, 0, -5, 5, 1, 3], [8, 2, -8, 0, -9, 9, -8, -2], [-5, -6, 0, 5, 0, 6, 5, 0], [-23, -5, 0, 23, -9, 5, 25, 0], [2, 3, -2, 11, -6, 0, 0, -6], [0, 0, -6, -6, 0, -3, 0, 6], [1, 0, 9, 9, 0, 6, -1, -1], [-5, 4, 5, -4, -3, 0, 0, -3], [-3, -13, 3, 20, 9, 0, 0, 9], [-3, 0, 20, 20, 0, -3, 3, -6], [1, 6, 0, -1, -13, -6, -18, 0], [0, 10, -6, -5, 2, -12, -5, -12], [9, 15, -9, -19, -2, 0, 0, -2], [25, 0, -20, -20, 0, 6, -25, 11], [0, -9, -14, -21, 4, 5, -21, 5], [-1, 6, 1, 7, 5, 0, 0, 5], [5, -1, 2, 0, 6, -6, 2, 1], [-3, 9, 0, 3, 7, -9, 1, 0], [21, 0, -22, -22, 0, -12, -21, 11], [-8, 2, 5, 0, 9, -9, 5, -2], [-1, -1, 0, 1, 3, 1, 24, 0], [5, 0, -11, -11, 0, 13, -5, 14], [0, -2, 10, 11, 9, -7, 11, -7], [9, -12, -9, 9, -1, 0, 0, -1], [-20, 11, 14, 0, 7, -7, 14, -11], [4, 11, -4, -12, -12, 0, 0, -12], [27, 0, -3, -3, 0, -4, -27, 0], [2, -14, -2, -5, 12, 0, 0, 12], [-7, -2, 0, 7, -1, 2, 27, 0], [0, 16, -7, -19, -16, 0, -19, 0], [-16, 0, -11, -11, 0, 3, 16, 3], [0, 3, -23, 0, 0, 0, -23, -3], [-14, 12, 13, 0, -11, 11, 13, -12], [0, 13, -13, -18, -6, -7, -18, -7], [-8, -2, 0, 8, 10, 2, 17, 0], [2, 9, -1, 0, -3, 3, -1, -9], [0, -15, 11, 16, 20, -5, 16, -5], [-12, -2, 12, 9, -11, 0, 0, -11], [11, 4, 0, -11, -3, -4, -3, 0], [6, 0, 1, 1, 0, -7, -6, -1], [9, -10, -24, 0, 14, -14, -24, 10], [-17, 0, -8, -8, 0, 2, 17, 8], [0, -9, -21, 1, 10, -1, 1, -1], [4, -8, 0, -4, -10, 8, -12, 0], [19, 0, -19, 17, 8, 0, 0, 8], [0, 14, 2, 16, -4, -10, 16, -10], [0, 14, 11, 17, -9, -5, 17, -5], [3, 18, 10, 0, 2, -2, 10, -18], [27, 11, -27, -10, 3, 0, 0, 3], [19, 12, 0, -19, 2, -12, -29, 0], [-6, -10, 0, 6, -5, 10, -20, 0], [30, 10, 0, -30, 2, -10, -23, 0], [-10, -8, 10, -18, 3, 0, 0, 3], [10, 0, -11, -11, 0, -13, -10, 6], [0, -8, 12, 17, 6, 2, 17, 2], [0, -14, -14, -4, -5, 19, -4, 19], [13, 0, 9, 9, 0, 11, -13, 5], [5, 3, -5, -3, -10, 0, 0, -10], [-13, -8, 0, 13, 13, 8, 7, 0], [18, 12, -12, 0, 15, -15, -12, -12], [-12, 3, 5, 0, 7, -7, 5, -3], [-21, -6, 0, 21, 2, 6, -6, 0], [-10, 20, 10, 7, -20, 0, 0, -20], [-1, -4, 8, 0, 9, -9, 8, 4], [-11, -1, 11, 9, 7, 0, 0, 7], [14, 0, -4, -4, 0, 4, -14, 9], [-2, 9, 0, 2, -10, -9, -19, 0], [0, 2, -26, -6, 4, -6, -6, -6], [-4, -20, 0, 4, 6, 20, 18, 0], [4, 8, 0, -4, -17, -8, -13, 0], [0, -20, 22, 11, 9, 11, 11, 11], [-9, -18, 0, 9, -5, 18, 4, 0], [7, -3, 11, 0, 8, -8, 11, 3], [-18, 0, 18, 18, 0, -10, 18, 1], [-18, -1, 37, 0, 2, -2, 37, 1], [5, 3, -5, -36, -3, 0, 0, -3], [0, -4, -12, 16, -1, 5, 16, 5], [12, 0, -11, -11, 0, 0, -12, 8], [0, 5, -29, 0, -9, 9, -29, -5], [-4, 0, -1, -1, 0, 3, 4, -12]];
    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_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_55_2_g_a(: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_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_55_2_g_a( : 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, 18, 127, 90, 71, 30, 13, 4, 1]>],Snew);
    return Vf;
end function;