
// Make newform 23.9.b.a in Magma, downloaded from the LMFDB on 18 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_9_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_9_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], [-4, 2, 1], [8, 3, -2]];
    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 := 9;
    raw_aps := [[0, -3, -2], [0, 13, -17], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 6765, -1841], [0, 0, 0], [0, 0, 0], [279841, 0, 0], [0, 71757, 148783], [0, 220077, -34577], [0, 0, 0], [0, -621267, 369583], [0, 0, 0], [0, -558195, -982961], [0, 0, 0], [15279074, 0, 0], [0, 0, 0], [0, 0, 0], [0, 1159245, -6416561], [0, -6401043, 3304303], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [-156941086, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, -61420083, 775183], [0, 24913773, 64936783], [0, 0, 0], [0, -78516915, 56138479], [0, 0, 0], [0, -17350995, 131401519], [0, 0, 0], [0, -165139923, -7814897], [-1547830846, 0, 0], [1733781794, 0, 0], [0, 189204717, 133142863], [0, 0, 0], [0, 0, 0], [0, 24185613, -349721777], [0, 316609293, 127882543], [0, 0, 0], [1780491554, 0, 0], [-2786857726, 0, 0], [0, 0, 0], [0, 0, 0], [0, -463285875, -481404401], [0, -632338707, 573458863], [0, 0, 0], [0, 0, 0], [0, 353648493, 970315183], [0, 0, 0], [0, 1079612205, 487629199], [-6550973566, 0, 0], [0, -1041930387, -822835697], [0, 0, 0], [0, 0, 0], [0, 0, 0], [2598112802, 0, 0], [0, -339080403, 2364507823], [0, 0, 0], [-3238308958, 0, 0], [0, -867844467, -2723858897], [0, 0, 0], [-14213499166, 0, 0], [0, -1541507955, -3096872081], [0, 1423098573, -3805632497], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 2998473357, 4885996783], [0, 0, 0], [0, -6668259507, 1086369103], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 2299589133, 8607161743], [0, -8225277555, -2986342481], [-75850403326, 0, 0], [0, 0, 0], [0, -3410774835, 11242264879], [83765173634, 0, 0], [0, 0, 0], [0, 0, 0], [0, -12880376115, -1880526641], [0, 11244812205, -10239736241], [0, -4335429747, -14140654097], [0, 0, 0], [0, -15158854227, -2987610737], [0, 0, 0], [0, 0, 0], [0, 20259930573, -543846737], [0, -4465759155, 22583451439], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 22692251373, 10636511023], [0, -14637352275, 27688174159], [138950697602, 0, 0], [-1810573246, 0, 0], [0, -8363938323, -30107251217], [74119430402, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, -35513396883, -17496923537], [0, -5600724243, -44199703217], [0, 0, 0], [0, 0, 0], [0, -42181483347, -19332556817], [0, 0, 0], [0, -19210275123, 53534087023], [-36012592606, 0, 0], [0, 0, 0], [0, 0, 0], [-449887340158, 0, 0], [0, 0, 0], [0, 0, 0], [0, -57990600723, 52169338063], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 50756559117, -72696991217], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [801341462594, 0, 0], [0, -11060639955, -105796713521], [-833040839326, 0, 0], [0, -107316944787, -5123834897], [0, 0, 0], [908357618594, 0, 0], [0, 0, 0], [204586750562, 0, 0], [0, 121295236557, -63597065777], [0, -91283329203, 110994967663], [0, -41902145427, 138065388463], [-392609358046, 0, 0], [0, 0, 0], [-754063416286, 0, 0], [0, -112064173107, 118140090703], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 74922941037, -180514474577], [0, 0, 0], [0, 0, 0], [0, 107246349837, -183397810577], [0, 0, 0], [0, 171925531245, 96641605999], [0, 0, 0], [0, 0, 0], [0, 0, 0], [-1528978238206, 0, 0], [-1913159984926, 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_9_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_9_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, 9));
    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_9_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_9_b_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_23_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,9,sign)));
    Vf := Kernel([<2,R![1951, -768, 0, 1]>],Snew);
    return Vf;
end function;