
// Make newform 201.2.e.c in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_201_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_201_e_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_201_2_e_c();".
// 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_201_2_e_c();".
// 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, -3, 13, 4, 26, -5, 21, 0, 8, -2, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-47489, -9814, 39957, -144911, 58183, -53276, 48871, -25937, 7711, -1127], [-12403, 7742, -31521, 89131, -45899, 42028, -17723, 20461, -6083, 1899], [-6827, -90294, -66953, -116057, -47811, -95096, -25035, -22135, 2993, -3473], [14781, 129718, 138743, 276579, -19987, 242740, 20461, 93141, -22907, 12403], [129813, -53410, 217455, -404293, 316645, -289940, 93141, -141155, 41965, -14781], [-12251, 52952, 1699, 104757, -22367, 89918, -1369, 31695, -7629, 3911], [3317, 49012, 31963, 58697, 14424, 49566, 9085, 14961, -2953, 2133], [12005, -56110, -130277, -146785, -38715, -165532, -41255, -61375, 14061, -9205]];
    Rf_basisdens := [1, 1, 44248, 44248, 22124, 44248, 44248, 11062, 5531, 22124];
    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_201_e();"
function MakeCharacter_201_e()
    N := 201;
    order := 3;
    char_gens := [68, 136];
    v := [3, 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_201_e_Hecke();"
function MakeCharacter_201_e_Hecke(Kf)
    N := 201;
    order := 3;
    char_gens := [68, 136];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 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+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, 0, 0, 0, 0, 0, 0, 0, 0, 1], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 1, 1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, -1, 0, 0, -1, 0], [0, 0, -1, 0, 2, 2, 0, 0, 1, -1], [0, 0, 0, 0, 0, 0, 1, -1, -1, 1], [1, 0, 0, 0, 0, -1, -2, 1, 2, 0], [-2, -3, 0, -3, 0, 2, 0, 1, 0, -1], [-3, -2, 0, -2, 0, 3, -2, 0, 2, 2], [0, 3, -1, 0, 2, -4, 0, 0, -2, -1], [0, 0, -1, 0, -2, -4, 0, 0, -3, -1], [-6, -2, 0, -2, 0, 6, -3, -3, 3, -1], [0, -4, 1, 0, 1, 4, 0, 0, -1, 1], [1, 0, 1, 0, 1, 0, 3, 1, 0, 0], [0, 0, -1, 0, -1, -4, 0, 0, 3, -1], [1, 0, 3, 2, 1, 0, 3, 1, 0, 0], [-1, 0, -1, 3, -1, 0, -2, -1, 0, 0], [3, -1, 0, -1, 0, -3, 3, 2, -3, 2], [1, -2, -3, -1, 0, -1, 1, -3, 1, -3], [0, 2, 0, 0, 0, 5, 0, 0, 0, 0], [3, 2, 0, 2, 0, -3, 0, -1, 0, 0], [0, 4, -3, 0, -5, -4, 0, 0, -3, -3], [1, 2, 0, 2, 0, -1, 2, 3, -2, 0], [-4, 0, -2, -7, -6, 0, 1, -6, 0, 0], [3, 1, 0, 1, 0, -3, 1, 0, -1, -2], [0, 4, 4, 0, -7, -3, 0, 0, -2, 4], [0, 3, 1, 0, -5, -4, 0, 0, 0, 1], [9, 0, -1, 9, 2, 0, 2, 2, 0, 0], [-2, 0, 6, 3, -1, 0, 1, -1, 0, 0], [0, 5, -2, 0, -2, 1, 0, 0, -5, -2], [0, -5, 3, 0, 5, 6, 0, 0, 2, 3], [-1, 0, -3, -1, 2, 0, -2, 2, 0, 0], [-8, 0, 4, 0, 1, 0, -2, 1, 0, 0], [-3, 0, -1, 1, 2, 0, -2, 2, 0, 0], [2, 0, 0, -5, 0, 0, -3, 0, 0, 0], [-3, 7, 0, 7, 0, 3, -5, -1, 5, -2], [2, 0, 0, 0, 0, -2, -1, 0, 1, 5], [0, -5, 3, 0, 2, 0, 0, 0, 4, 3], [0, 0, -2, 0, -1, 1, 0, 0, 6, -2], [3, 1, 0, 1, 0, -3, 3, 0, -3, 4], [-8, 0, -4, 3, 4, 0, 1, 4, 0, 0], [0, -10, 3, 0, 6, 10, 0, 0, -3, 3], [0, 5, 0, 5, 0, 0, 4, 3, -4, 3], [0, 0, -6, 1, 6, 0, 5, 6, 0, 0], [0, -5, 5, 0, 5, 0, 0, 0, 2, 5], [12, 1, 0, 1, 0, -12, 6, 6, -6, -1], [-8, -4, 0, -4, 0, 8, -5, 2, 5, 3], [8, 0, -4, -1, -8, 0, -3, -8, 0, 0], [0, -3, -4, 0, 6, -3, 0, 0, 3, -4], [-6, 4, 0, 4, 0, 6, -1, -1, 1, 3], [0, 4, 4, 0, -3, -7, 0, 0, -2, 4], [0, 1, 1, 0, -1, 6, 0, 0, -2, 1], [-3, 0, 3, -5, -1, 0, -2, -1, 0, 0], [-2, 3, 0, 3, 0, 2, 0, 9, 0, 1], [0, -5, 7, 0, 7, -2, 0, 0, 0, 7], [4, 0, -6, 6, 9, 0, 2, 9, 0, 0], [-11, 0, -3, -8, -6, 0, -3, -6, 0, 0], [9, 0, -1, 10, 2, 0, 5, 2, 0, 0], [6, 0, 2, 0, 1, 0, 8, 1, 0, 0], [1, 0, 0, 0, 0, -1, 6, -6, -6, -2], [13, 0, 5, 9, -3, 0, 6, -3, 0, 0], [5, 0, 3, 5, -7, 0, 0, -7, 0, 0], [6, -1, 0, -1, 0, -6, -2, -1, 2, -1], [-13, 0, -3, -9, -4, 0, -6, -4, 0, 0], [11, 0, -5, 0, 12, 0, 3, 12, 0, 0], [-11, -11, 0, -11, 0, 11, -3, 3, 3, -4], [0, 9, -4, 0, 5, -3, 0, 0, -3, -4], [1, 13, 0, 13, 0, -1, 3, -1, -3, -10], [0, -4, 0, 0, 12, 5, 0, 0, 10, 0], [4, 0, -2, 2, -14, 0, 4, -14, 0, 0], [-1, -13, 0, -13, 0, 1, 1, -3, -1, 6], [-13, 0, 11, -1, -5, 0, -6, -5, 0, 0], [0, 0, 5, 0, -5, 0, 0, 0, -7, 5], [0, 8, -7, 0, -4, 4, 0, 0, 1, -7], [-10, -10, 0, -10, 0, 10, -5, -3, 5, -9], [-4, 2, 0, 2, 0, 4, -1, 3, 1, 3], [-8, 0, 0, 0, 0, 8, -5, -2, 5, -7], [-19, 0, -1, -8, 3, 0, -7, 3, 0, 0], [30, 0, 0, 15, 6, 0, 5, 6, 0, 0], [0, -7, -3, 0, 8, 0, 0, 0, 2, -3], [10, -3, 0, -3, 0, -10, 6, 4, -6, -5], [7, 3, 0, 3, 0, -7, -5, 0, 5, 2], [0, -9, 11, 0, 6, 10, 0, 0, 4, 11], [0, -6, 0, 0, 4, 5, 0, 0, 6, 0], [-13, 9, 0, 9, 0, 13, 3, -3, -3, -8], [0, 5, 0, 0, 10, 1, 0, 0, 3, 0], [0, 6, 1, 0, 6, -6, 0, 0, 3, 1], [0, -2, -3, 0, -6, -16, 0, 0, -1, -3], [-8, 0, 2, -12, 6, 0, -6, 6, 0, 0], [2, 5, 0, 5, 0, -2, 0, -7, 0, 3], [-1, 2, 0, 2, 0, 1, -4, -4, 4, -12], [4, 5, 0, 5, 0, -4, -4, 1, 4, -1], [2, -10, 0, -10, 0, -2, -1, -10, 1, 1], [4, 0, -2, -12, 1, 0, 0, 1, 0, 0], [12, 12, 0, 12, 0, -12, 3, -2, -3, -9], [0, 5, 5, 0, -4, 16, 0, 0, -4, 5], [-2, 0, 6, -3, 3, 0, -5, 3, 0, 0], [-12, 0, -4, -14, -17, 0, -8, -17, 0, 0], [1, 12, 0, 12, 0, -1, -2, 0, 2, -12], [-3, 0, 3, 2, 15, 0, 7, 15, 0, 0], [0, -13, 5, 0, 1, 8, 0, 0, -4, 5], [0, 3, -4, 0, 2, 11, 0, 0, -1, -4], [18, 0, 4, 8, 9, 0, -6, 9, 0, 0], [0, 1, 0, 0, -2, 15, 0, 0, 5, 0], [0, 7, -3, 0, 1, -8, 0, 0, 4, -3], [0, 15, -9, 0, 1, -8, 0, 0, -4, -9], [2, -10, 0, -10, 0, -2, 1, -5, -1, 5], [-10, 4, 0, 4, 0, 10, -3, 0, 3, 11], [0, -15, -5, 0, 2, 2, 0, 0, -2, -5], [-16, -7, 0, -7, 0, 16, 0, -8, 0, -1], [0, 5, 16, 0, -8, 11, 0, 0, -1, 16], [5, 11, 0, 11, 0, -5, 1, 8, -1, 0], [-2, 0, 0, -4, -10, 0, -12, -10, 0, 0], [10, -1, 0, -1, 0, -10, -6, 6, 6, 5], [-15, -10, 0, -10, 0, 15, 0, -4, 0, 0], [0, 8, -10, 0, -15, 3, 0, 0, 4, -10], [-9, 0, -3, -15, -10, 0, 0, -10, 0, 0], [9, 0, 0, 0, 0, -9, 12, -2, -12, -2], [-7, 3, 0, 3, 0, 7, -5, 19, 5, 0], [0, 3, 0, 0, -1, -25, 0, 0, 3, 0], [-2, 0, -2, -11, 3, 0, -9, 3, 0, 0], [-19, 0, -7, 1, 0, 0, 4, 0, 0, 0], [0, -11, 4, 0, -5, -19, 0, 0, -5, 4], [-16, -2, 0, -2, 0, 16, -5, 2, 5, 9], [0, -3, -12, 0, 2, -13, 0, 0, 3, -12], [0, 14, -2, 0, -10, -19, 0, 0, -12, -2], [-8, 0, 0, 0, 0, 8, 1, 0, -1, -5], [-3, -8, 0, -8, 0, 3, -8, -1, 8, 12], [12, 17, 0, 17, 0, -12, 14, -5, -14, -7], [0, -13, 5, 0, -14, 4, 0, 0, 12, 5], [-13, -8, 0, -8, 0, 13, -12, 6, 12, 6], [-12, -4, 0, -4, 0, 12, -19, -11, 19, 3], [28, 0, -2, -7, 1, 0, -5, 1, 0, 0], [0, -1, 2, 0, -5, -7, 0, 0, -13, 2], [5, 0, -1, -2, -5, 0, -9, -5, 0, 0], [0, 10, 2, 0, -14, -21, 0, 0, -4, 2], [0, -6, -13, 0, 7, -2, 0, 0, 7, -13], [-9, 2, 0, 2, 0, 9, 6, 1, -6, -4], [0, 5, -9, 0, 10, 4, 0, 0, 12, -9], [-13, 0, 5, 7, 11, 0, 12, 11, 0, 0], [0, -3, 2, 0, 11, -17, 0, 0, 3, 2], [6, 3, 0, 3, 0, -6, -6, -12, 6, 11], [-12, -5, 0, -5, 0, 12, 8, -4, -8, -1], [0, -9, 0, -9, 0, 0, -6, -3, 6, 7], [12, 0, 4, 3, -1, 0, 15, -1, 0, 0], [0, -21, 14, 0, 6, 11, 0, 0, -1, 14], [-13, 2, 0, 2, 0, 13, -4, -6, 4, -10], [-27, 0, 5, 1, -4, 0, -4, -4, 0, 0], [0, 1, -4, 0, 0, -9, 0, 0, -5, -4], [-9, 0, 5, 6, -12, 0, 11, -12, 0, 0], [-15, 3, 0, 3, 0, 15, 5, 4, -5, 16], [17, 5, 0, 5, 0, -17, 5, 3, -5, 8], [0, 12, -10, 0, -3, 7, 0, 0, -14, -10], [12, -15, 0, -15, 0, -12, 8, -4, -8, 13], [0, 6, -2, 0, 1, -13, 0, 0, -12, -2], [-2, 0, -12, -6, -1, 0, 8, -1, 0, 0], [26, 22, 0, 22, 0, -26, 11, 15, -11, -9], [19, 0, 7, -1, 8, 0, -4, 8, 0, 0], [16, 0, -4, -12, -9, 0, -2, -9, 0, 0], [-10, 0, 0, 8, 4, 0, 8, 4, 0, 0], [-18, 0, 8, -7, -13, 0, -7, -13, 0, 0], [19, 0, 1, -13, -9, 0, 8, -9, 0, 0], [0, 14, -4, 0, 11, 17, 0, 0, -6, -4], [0, 9, 3, 0, -4, -10, 0, 0, -10, 3], [-8, 7, 0, 7, 0, 8, -4, -3, 4, 3], [2, 0, -4, 5, 15, 0, -3, 15, 0, 0], [10, 0, 18, -1, -8, 0, 3, -8, 0, 0], [1, 0, -11, 1, 8, 0, -8, 8, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_201_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_201_2_e_c();".
// 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_201_2_e_c(:prec:=10)
    chi := MakeCharacter_201_e();
    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_201_2_e_c();".
// 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_201_2_e_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_201_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 7, 49, 16, 56, 1, 57, 0, 8, 0, 1]>],Snew);
    return Vf;
end function;