
// Make newform 500.3.b.a in Magma, downloaded from the LMFDB on 04 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_500_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_500_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_500_3_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_500_3_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 := [1, 0, -1, 0, 1, 0, -1, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 2, 0, 0], [1, 8, 0, -6, 2, 4, -2, -2], [-2, 0, 4, -2, -1, 0, 3, -2], [-3, 0, 6, 2, -4, 0, 2, 2], [2, 0, 0, 0, 5, 0, -5, 0], [1, 4, 0, 2, 1, 2, -1, -6], [1, 0, -2, -4, -2, 2, -4, -4]];
    Rf_basisdens := [1, 1, 1, 1, 1, 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_500_b();"
function MakeCharacter_500_b()
    N := 500;
    order := 2;
    char_gens := [251, 377];
    v := [1, 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_500_b_Hecke();"
function MakeCharacter_500_b_Hecke(Kf)
    N := 500;
    order := 2;
    char_gens := [251, 377];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 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]];
    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 := 3;
    raw_aps := [[0, 1, 0, 0, 0, 0, 0, 0], [0, -1, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 1, -2, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 5, -4, 0, 0, 1], [-8, 0, 0, 0, 0, 1, 6, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-16, 0, 0, 0, 0, 5, 6, 0], [0, 17, 0, 13, -2, 0, 0, 2], [0, -5, 0, -13, -4, 0, 0, 4], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [15, 0, 12, 0, 0, 1, 0, 0], [0, -58, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 17, 0, -1, -16, 0, 0, -17], [-45, 0, 12, 0, 0, -19, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-33, 0, -18, 0, 0, -5, 0, 0], [0, 22, 0, 0, 0, 0, 0, 0], [0, -17, 0, 11, 14, 0, 0, -14], [22, 0, 0, 0, 0, 1, -24, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 43, 0, 11, -16, 0, 0, 16], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [81, 0, 12, 0, 0, 23, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 17, 0, 29, -46, 0, 0, -17], [0, -59, 0, -55, 2, 0, 0, -2], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [68, 0, 0, 0, 0, -37, 6, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 83, 0, 31, -26, 0, 0, 26], [0, 43, 0, -23, -20, 0, 0, -43], [-87, 0, 42, 0, 0, -43, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-99, 0, 42, 0, 0, -47, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -37, 0, 31, 34, 0, 0, -34], [-38, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [117, 0, 42, 0, 0, 25, 0, 0], [0, -158, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 43, 0, -53, 10, 0, 0, -43], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 67, 0, 19, -86, 0, 0, -67], [9, 0, -78, 0, 0, 29, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -47, 0, 67, -20, 0, 0, 47], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -43, 0, -107, -32, 0, 0, 32], [10, 0, 0, 0, 0, 37, -84, 0], [0, 0, 0, 0, 0, 0, 0, 0], [129, 0, 72, 0, 0, 19, 0, 0], [-237, 0, -18, 0, 0, -73, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [134, 0, 0, 0, 0, -85, 36, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 5, 0, -97, 92, 0, 0, -5], [-398, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-261, 0, 42, 0, 0, -101, 0, 0], [0, -85, 0, 53, 32, 0, 0, 85], [0, -89, 0, -29, 118, 0, 0, 89], [0, 0, 0, 0, 0, 0, 0, 0], [0, -59, 0, 65, 62, 0, 0, -62], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 83, 0, -73, -10, 0, 0, -83], [982, 0, 0, 0, 0, 0, 0, 0], [-94, 0, 0, 0, 0, 89, -84, 0], [0, -79, 0, -155, -38, 0, 0, 38], [-4, 0, 0, 0, 0, 59, -114, 0], [0, -41, 0, 115, -74, 0, 0, 41], [0, 0, 0, 0, 0, 0, 0, 0], [0, 562, 0, 0, 0, 0, 0, 0], [-44, 0, 0, 0, 0, -41, 126, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -538, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [8, 0, 0, 0, 0, -67, 126, 0], [0, 482, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [188, 0, 0, 0, 0, -127, 66, 0], [0, 89, 0, 95, -184, 0, 0, -89], [0, -478, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [146, 0, 0, 0, 0, -1, -144, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 167, 0, 223, 28, 0, 0, -28], [0, 0, 0, 0, 0, 0, 0, 0], [902, 0, 0, 0, 0, 0, 0, 0], [237, 0, -138, 0, 0, 125, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 277, 0, 113, -82, 0, 0, 82], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 382, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-39, 0, -168, 0, 0, 43, 0, 0], [-218, 0, 0, 0, 0, 151, -84, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -47, 0, -199, -76, 0, 0, 76], [0, 0, 0, 0, 0, 0, 0, 0], [411, 0, -108, 0, 0, 173, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-165, 0, -168, 0, 0, 1, 0, 0], [0, -698, 0, 0, 0, 0, 0, 0], [0, -298, 0, 0, 0, 0, 0, 0], [346, 0, 0, 0, 0, -131, -84, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 131, 0, -139, 8, 0, 0, -131], [0, 0, 0, 0, 0, 0, 0, 0], [501, 0, -78, 0, 0, 193, 0, 0], [0, 173, 0, -73, -100, 0, 0, -173], [0, 85, 0, 163, -248, 0, 0, -85], [0, 67, 0, -191, 124, 0, 0, -67], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-129, 0, -198, 0, 0, 23, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-118, 0, 0, 0, 0, 0, 0, 0], [0, -335, 0, -103, 116, 0, 0, -116], [0, 0, 0, 0, 0, 0, 0, 0], [0, 37, 0, 233, 98, 0, 0, -98], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 142, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_500_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_500_3_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_500_3_b_a(:prec:=8)
    chi := MakeCharacter_500_b();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    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_500_3_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_500_3_b_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_500_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![3481, 0, 10034, 0, 1651, 0, 74, 0, 1]>,<13,R![0, 1]>],Snew);
    return Vf;
end function;