
// Make newform 273.2.by.b in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_273_by();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_273_by_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_273_2_by_b();".
// 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_273_2_by_b();".
// 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];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_273_by();"
function MakeCharacter_273_by()
    N := 273;
    order := 12;
    char_gens := [92, 157, 106];
    v := [12, 6, 7];
    // 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_273_by_Hecke();"
function MakeCharacter_273_by_Hecke(Kf)
    N := 273;
    order := 12;
    char_gens := [92, 157, 106];
    char_values := [[1, 0, 0, 0], [-1, 0, 0, 0], [0, 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 := 2;
    raw_aps := [[1, 0, -1, 1], [0, -1, 0, 1], [0, 1, 1, 0], [0, 1, 0, 2], [-2, 0, 2, 2], [0, 2, 3, -2], [0, -3, 0, 6], [-2, -2, 2, 0], [6, 1, -3, -1], [0, 0, 7, 0], [-6, 2, 2, -6], [5, 0, -5, -5], [7, -3, -4, -4], [0, -8, 0, 0], [-4, 4, -4, 4], [6, 2, 0, -1], [3, -3, -1, 2], [2, 3, 2, 0], [-2, -7, -5, 5], [-4, 4, 4, 0], [5, 9, -9, -5], [7, 10, 0, -5], [-4, 8, 8, -4], [-6, -5, 1, -1], [-3, -3, 1, 2], [0, 4, -3, 4], [0, -8, 0, 4], [0, 2, 4, 2], [7, 2, -2, -7], [-5, 2, 5, -4], [-2, 13, 1, -13], [-3, 0, 6, 9], [-7, -7, 4, 3], [-3, -3, -3, 0], [-11, 11, 4, -7], [0, 0, 0, 0], [-5, 0, 10, -10], [1, 1, -7, 6], [4, -1, -5, 5], [6, 10, -6, -20], [14, -1, -7, 1], [1, 8, 0, -4], [16, -2, -16, 4], [5, -13, 8, 8], [-6, -5, 1, -1], [-22, 2, 22, -4], [0, 6, -2, 6], [16, -15, -1, -1], [-1, -1, -15, 16], [7, 14, -14, -7], [0, 0, 0, 2], [-4, 10, -10, 4], [11, -11, -16, -5], [15, 7, -15, -14], [0, -6, 1, -6], [0, -10, -6, -10], [4, -8, 4, 0], [0, -3, -3, 3], [9, 2, 9, 0], [-4, 11, 11, -4], [0, 2, -16, 2], [-13, 13, 4, -9], [-1, -8, 8, 1], [-5, -10, 0, 5], [6, 0, -12, 20], [-18, -13, 13, 18], [13, -13, -7, 6], [2, 0, -4, -21], [3, -5, -3, 10], [10, -7, -17, 17], [-1, -6, 7, 7], [-17, 6, 6, -17], [6, -17, -3, 17], [-26, 1, 26, -2], [16, 14, -2, 2], [-13, -13, 15, -2], [1, 0, -2, 28], [-17, 17, 15, -2], [-1, -5, 6, 6], [2, 2, -7, 5], [24, -10, -12, 10], [23, 13, -13, -23], [-8, 1, 9, -9], [2, -7, -2, 14], [0, 14, 16, 14], [0, 12, 0, -6], [5, 5, -7, 2], [-25, -11, 14, -14], [-4, -4, 19, -15], [0, -6, -6, 0], [-10, -24, 0, 12], [14, 6, -8, 8], [-19, 19, 7, -12], [-18, 3, 9, -3], [2, -12, 12, -2], [-20, 4, -20, 0], [-11, -9, 20, 20], [6, 0, -12, -17], [14, 13, -7, -13], [5, -7, -7, 5], [11, -2, 0, 1], [-13, -8, 21, 21], [11, 19, -11, -38], [-24, 0, 12, 0], [-9, 0, 18, -21], [-7, 5, 5, -7], [22, -8, -14, -14], [-16, -3, 3, 16], [-31, 6, 0, -3], [-6, -20, 3, 20], [-13, -19, -13, 0], [7, 2, -5, 5], [25, -25, -24, 1], [-9, -16, 16, 9], [6, 6, 18, -24], [3, -28, 3, 0], [8, 8, 14, -22], [0, 17, -13, 17], [0, -2, -30, -2], [12, 12, -12, -24], [-7, 9, 16, -16], [0, -37, 0, 37], [-6, 0, 12, 12], [13, 13, -13, 0], [-6, 15, -9, -9], [-4, 0, 8, 28], [6, 6, -11, 5], [3, -9, -3, 18], [29, 14, 0, -7], [-23, 1, 1, -23], [2, 8, -10, -10], [-6, -22, -16, 16], [8, -14, -4, 14], [0, 0, -16, 0], [19, -19, -19, 0], [-20, 7, 13, 13], [1, 1, -15, 14], [4, -4, -8, -4], [20, -10, -20, 20], [0, 2, -3, 2], [2, 8, 8, 2], [6, -7, -13, 13], [-14, 24, -14, 0], [-27, -8, -8, -27], [0, 15, -12, 15], [7, -7, 11, 18], [5, -7, 7, -5], [39, -20, 0, 10], [-13, 0, 26, -19], [10, 16, -16, -10], [31, -31, -36, -5], [0, -14, -15, -14], [4, 0, -8, 34], [4, 44, -2, -44], [10, -33, -5, 33], [-12, 8, 0, -4], [-15, 17, 15, -34], [-37, -37, 26, 11], [22, 0, -44, -9], [-11, -20, -20, -11], [20, -12, -8, -8], [-6, 38, -6, 0], [-3, -16, 16, 3], [-1, 27, -1, 0], [45, 20, -25, 25], [31, 14, -14, -31], [0, 1, 15, 1], [18, 1, 18, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_273_by_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_273_2_by_b();".
// 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_273_2_by_b(:prec:=4)
    chi := MakeCharacter_273_by();
    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_273_2_by_b();".
// 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_273_2_by_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_273_by();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, -4, 5, -2, 1]>,<5,R![1, 2, 2, -2, 1]>],Snew);
    return Vf;
end function;