
// Make newform 260.2.bk.a in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_260_bk();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_260_bk_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_260_2_bk_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_260_2_bk_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];
        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_260_bk();"
function MakeCharacter_260_bk()
    N := 260;
    order := 12;
    char_gens := [131, 157, 41];
    v := [12, 9, 11];
    // 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_260_bk_Hecke();"
function MakeCharacter_260_bk_Hecke(Kf)
    N := 260;
    order := 12;
    char_gens := [131, 157, 41];
    char_values := [[1, 0, 0, 0], [0, 0, 0, 1], [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 := [[0, 0, 0, 0], [-1, 0, 1, -1], [-2, 0, 0, 1], [-2, 1, -2, 0], [-3, 1, 2, 2], [-2, 0, 0, -3], [2, -2, -1, 1], [-1, -1, -2, 3], [-3, 4, -1, -1], [-2, 0, 1, 0], [-3, -6, 6, 3], [-4, 5, 2, -5], [5, 5, 0, 0], [-4, -4, 3, 1], [4, 0, -8, -4], [1, 4, -4, -1], [-5, 5, 6, 1], [9, 0, -9, 0], [0, -1, 14, -1], [-3, 3, 2, -1], [-8, -8, 0, 4], [2, 0, -4, 4], [4, 0, -8, 4], [-5, -9, -4, 4], [6, 1, -6, -2], [-6, -12, 3, 12], [-3, -6, -6, -3], [-1, -4, 5, 5], [-9, 4, 4, -9], [2, -2, 1, 3], [-7, -12, -5, 5], [-8, 16, 0, -8], [-10, 3, -10, 0], [-3, 0, -3, 0], [-5, -5, 12, -7], [-3, 2, 2, -3], [1, 4, 4, 1], [2, -3, -2, 6], [-4, -15, 2, 15], [2, 2, -13, 11], [0, 4, 7, 4], [4, 0, -8, -16], [15, 0, -15, 0], [0, 9, 10, 9], [0, -1, 6, -1], [5, 8, -5, -16], [0, -4, 1, -4], [4, -5, -2, 5], [-2, 3, 2, -6], [3, -8, 8, -3], [-9, 16, 16, -9], [3, 10, 10, 3], [5, 5, 4, -9], [-1, -16, -1, 0], [5, 14, 9, -9], [-5, 12, 17, -17], [-1, 20, -1, 0], [5, 5, -10, -10], [-2, 2, -5, -7], [-11, -4, -4, -11], [13, 4, -17, -17], [2, -13, -2, 26], [0, 0, 0, 4], [-2, 0, 4, 0], [-7, 4, -4, 7], [-12, -8, 0, 4], [-19, 19, 14, -5], [-13, -8, 8, 13], [-8, -8, 9, -1], [-13, -1, 12, -12], [4, 7, -2, -7], [11, 6, -6, -11], [-5, -4, 9, 9], [6, -6, 9, 15], [3, 3, -6, -6], [2, -29, 2, 0], [-22, 8, 0, -4], [-6, -1, -6, 0], [9, -5, -4, -4], [11, 11, -4, -7], [-42, 0, 21, 0], [1, 16, -16, -1], [-11, -13, -2, 2], [22, 22, -17, -5], [0, -4, 7, -4], [-9, -18, 18, 9], [-9, 9, -8, -17], [0, -9, 2, -9], [21, -21, -4, 17], [-32, 12, 0, -6], [13, 10, -10, -13], [-9, 1, 10, -10], [14, -9, -14, 18], [-6, 12, 3, -12], [23, -2, -2, 23], [16, -16, -11, 5], [-1, -19, 20, 20], [6, -16, 0, 8], [11, 4, -7, 7], [-11, 24, 24, -11], [19, -14, -14, 19], [12, 5, -6, -5], [-12, -12, -5, 17], [0, 20, 3, 20], [6, 0, -12, 32], [0, 24, 0, -12], [0, -17, -18, -17], [-24, -8, 0, 4], [-14, 0, 28, 12], [0, 8, 1, 8], [-4, -4, 21, -17], [-4, -21, 2, 21], [2, 9, -2, -18], [11, 10, -10, -11], [25, 25, -10, -15], [13, -16, 13, 0], [-18, -5, -18, 0], [5, 0, -5, 5], [3, 34, 31, -31], [-19, -8, -19, 0], [5, -9, 4, 4], [9, 14, -23, -23], [21, -36, -36, 21], [6, 5, -6, -10], [-7, -25, -18, 18], [4, 0, -8, 28], [-17, 17, -4, -21], [-35, -4, 35, 8], [-27, -18, 18, 27], [12, 0, -24, -6], [-13, -23, -10, 10], [-20, -5, 10, 5], [42, -16, -21, 16], [-17, 10, 7, 7], [5, 5, -20, 15], [19, -11, -8, -8], [-10, 37, -10, 0], [-14, -19, -14, 0], [-14, 14, 15, 1], [14, -24, -7, 24], [-15, -22, 22, 15], [-3, -19, -16, 16], [12, 12, 3, -15], [8, 0, -16, -12], [0, 8, -5, 8], [-9, 9, 18, 9], [-40, -16, 0, 8], [-9, -28, 28, 9], [6, 0, -12, 8], [8, 0, -16, 16], [22, 9, -22, -18], [2, 4, -1, -4], [9, -6, -6, 9], [3, -4, 1, 1], [41, 16, -25, 25], [-16, 48, 0, -24], [1, -40, 1, 0], [15, 15, -12, -3], [-19, 16, 16, -19], [1, 16, -16, -1], [28, -19, -14, 19], [-14, -14, 27, -13], [48, 12, 0, -6], [15, 12, -15, -24], [0, -9, -2, -9], [12, -12, 0, 6], [0, -20, -7, -20], [2, 2, 9, -11]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_260_bk_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_260_2_bk_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_260_2_bk_a(:prec:=4)
    chi := MakeCharacter_260_bk();
    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_260_2_bk_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_260_2_bk_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_260_bk();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, 4, 5, 2, 1]>],Snew);
    return Vf;
end function;