
// Make newform 54.5.b.a in Magma, downloaded from the LMFDB on 07 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_54_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_54_b_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_54_5_b_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_54_5_b_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 := [2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [0, 2]];
    Rf_basisdens := [1, 1];
    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_54_b();"
function MakeCharacter_54_b()
    N := 54;
    order := 2;
    char_gens := [29];
    v := [1];
    // 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_54_b_Hecke();"
function MakeCharacter_54_b_Hecke(Kf)
    N := 54;
    order := 2;
    char_gens := [29];
    char_values := [[-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]];
    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 := 5;
    raw_aps := [[0, 1], [0, 0], [0, -12], [-73, 0], [0, -60], [95, 0], [0, 36], [-313, 0], [0, -228], [0, 552], [-958, 0], [-385, 0], [0, -840], [2546, 0], [0, -60], [0, -936], [0, -948], [5615, 0], [23, 0], [0, 144], [6527, 0], [-6121, 0], [0, -888], [0, -972], [9935, 0], [0, 3360], [6215, 0], [0, -2772], [-15982, 0], [0, -1020], [-10942, 0], [0, -264], [0, -7332], [-10777, 0], [0, -13008], [-505, 0], [-15790, 0], [-12409, 0], [0, 5244], [0, -2472], [0, 18576], [9263, 0], [0, 3612], [-5089, 0], [0, 4212], [44903, 0], [66551, 0], [-7102, 0], [0, -1824], [-94990, 0], [0, -14256], [0, 33936], [24191, 0], [0, 6984], [0, 5928], [0, -28176], [0, -21996], [-81241, 0], [60722, 0], [0, 36552], [145202, 0], [0, 32388], [-8494, 0], [0, 16548], [62063, 0], [0, 42024], [38087, 0], [75119, 0], [0, -70608], [-63073, 0], [0, -76560], [0, -85932], [19367, 0], [-130657, 0], [-123433, 0], [0, -79392], [0, 29676], [261650, 0], [0, -2208], [296159, 0], [0, 51612], [-315937, 0], [0, -252], [-29374, 0], [-11422, 0], [0, -37032], [0, 29556], [174626, 0], [0, -23316], [6647, 0], [0, -82116], [0, 103512], [53399, 0], [0, 28212], [51026, 0], [0, 48132], [0, 18636], [0, 26028], [477047, 0], [159743, 0], [82727, 0], [0, -197388], [0, 150792], [0, 169932], [-284665, 0], [85871, 0], [0, 27012], [0, 216504], [0, -172776], [472418, 0], [477527, 0], [-242737, 0], [0, 56004], [520199, 0], [313895, 0], [0, -239568], [155282, 0], [0, 237816], [0, -136056], [0, -38472], [-411889, 0], [-678097, 0], [0, -165300], [0, 221328], [214226, 0], [0, 99828], [-67969, 0], [0, -257400], [369890, 0], [242258, 0], [-798190, 0], [0, 201300], [831479, 0], [-954529, 0], [0, -49404], [1086143, 0], [0, 395352], [-676633, 0], [0, 295152], [0, -256752], [-664846, 0], [0, -335148], [-293017, 0], [0, -227412], [-227761, 0], [0, 333480], [766943, 0], [0, -296736], [-1184233, 0], [0, -136332], [1065695, 0], [0, -178740], [211367, 0], [0, -120792], [1486295, 0], [0, -408840], [-203038, 0], [0, -414816], [-1694545, 0], [0, -436548], [0, 538572], [0, 218628], [-1174441, 0], [0, 362772], [0, -104952], [0, 384780], [149735, 0], [402290, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_54_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_54_5_b_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_54_5_b_a(:prec:=2)
    chi := MakeCharacter_54_b();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 5));
    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_54_5_b_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_54_5_b_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_54_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,5,sign)));
    Vf := Kernel([<5,R![1152, 0, 1]>],Snew);
    return Vf;
end function;