
// Make newform 23.5.b.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_23_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_23_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_23_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_23_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 := [-3, -6, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-4, -1, 1]];
    Rf_basisdens := [1, 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_23_b();"
function MakeCharacter_23_b()
    N := 23;
    order := 2;
    char_gens := [5];
    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_23_b_Hecke();"
function MakeCharacter_23_b_Hecke(Kf)
    N := 23;
    order := 2;
    char_gens := [5];
    char_values := [[-1, 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 := 5;
    raw_aps := [[0, 1, -2], [0, 6, -1], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, -122, -17], [0, 0, 0], [0, 0, 0], [529, 0, 0], [0, -506, -353], [0, 694, 127], [0, 0, 0], [0, 214, 991], [0, 0, 0], [0, -842, 943], [0, 0, 0], [-6286, 0, 0], [0, 0, 0], [0, 0, 0], [0, -2858, 1327], [0, -554, -3137], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [7154, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, -1034, 9103], [0, 10054, 7471], [0, 0, 0], [0, -3098, -11393], [0, 0, 0], [0, -10538, -11873], [0, 0, 0], [0, -2234, 14863], [2786, 0, 0], [-59374, 0, 0], [0, -21626, -10193], [0, 0, 0], [0, 0, 0], [0, 20086, -10961], [0, 27046, 10399], [0, 0, 0], [75794, 0, 0], [46466, 0, 0], [0, 0, 0], [0, 0, 0], [0, 16918, -25457], [0, -11786, -33569], [0, 0, 0], [0, 0, 0], [0, -38474, -28961], [0, 0, 0], [0, 50182, 20047], [-65086, 0, 0], [0, -20954, 37519], [0, 0, 0], [0, 0, 0], [0, 0, 0], [-142702, 0, 0], [0, 44374, 50527], [0, 0, 0], [-130222, 0, 0], [0, -48026, 42271], [0, 0, 0], [121586, 0, 0], [0, 48262, -50753], [0, -75914, 25999], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 97606, 63103], [0, 0, 0], [0, 2134, -96209], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 108694, 87631], [0, -35162, 103903], [-73726, 0, 0], [0, 0, 0], [0, -87098, -115553], [-419134, 0, 0], [0, 0, 0], [0, 0, 0], [0, 27478, -130577], [0, 166822, -12113], [0, 109894, -95489], [0, 0, 0], [0, -34106, 141391], [0, 0, 0], [0, 0, 0], [0, 210694, 48991], [0, 132358, 158623], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, -230474, -93281], [0, -117722, -194897], [-621502, 0, 0], [505634, 0, 0], [0, -162026, 136447], [-587902, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 88726, -216257], [0, 208006, -146897], [0, 0, 0], [0, 0, 0], [0, -92234, 235519], [0, 0, 0], [0, -183194, -256577], [-648046, 0, 0], [0, 0, 0], [0, 0, 0], [-290878, 0, 0], [0, 0, 0], [0, 0, 0], [0, 111814, 321007], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, -381866, 72847], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [-1287646, 0, 0], [0, -324698, 223087], [274994, 0, 0], [0, 57046, -378881], [0, 0, 0], [1361234, 0, 0], [0, 0, 0], [1124018, 0, 0], [0, -524714, -24977], [0, -194906, -428993], [0, 305206, 404959], [889106, 0, 0], [0, 0, 0], [-679534, 0, 0], [0, -185834, -460241], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 532534, -170273], [0, 0, 0], [0, 0, 0], [0, -578234, 137023], [0, 0, 0], [0, 640918, 277183], [0, 0, 0], [0, 0, 0], [0, 0, 0], [-632446, 0, 0], [-250894, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_23_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_23_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_23_5_b_a(:prec:=3)
    chi := MakeCharacter_23_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_23_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_23_5_b_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_23_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,5,sign)));
    Vf := Kernel([<2,R![79, -48, 0, 1]>],Snew);
    return Vf;
end function;