
// Make newform 287.2.r.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_287_r();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_287_r_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_287_2_r_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_287_2_r_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_287_r();"
function MakeCharacter_287_r()
    N := 287;
    order := 12;
    char_gens := [206, 211];
    v := [4, 9];
    // 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_287_r_Hecke();"
function MakeCharacter_287_r_Hecke(Kf)
    N := 287;
    order := 12;
    char_gens := [206, 211];
    char_values := [[-1, 0, 1, 0], [0, 0, 0, 1]];
    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, 1, 0, 0], [1, 1, 0, 0], [1, 0, 1, 0], [0, -1, 0, -2], [3, 2, -1, 1], [-1, -2, 2, 1], [-2, -2, 4, 4], [-2, -2, -3, 5], [-4, -2, 4, 4], [6, 2, -2, -6], [0, -1, 1, -1], [-4, 2, 4, -4], [4, 0, 0, 5], [2, 0, -4, -2], [3, 3, 1, -4], [0, -1, -1, 1], [0, -1, -9, -1], [0, 7, 0, 0], [-2, 1, 1, 1], [7, -1, 1, -7], [-4, 8, 2, -8], [-2, 2, -5, -7], [-1, 2, 0, -1], [-5, 5, 5, 0], [0, -4, -4, 0], [-6, 2, 4, 4], [1, 7, 1, 0], [-1, -11, 1, 22], [-8, 4, 4, 4], [-4, -18, 0, 9], [5, 14, 0, -7], [0, 0, 0, 0], [8, -5, -3, -3], [-1, 26, 0, -13], [3, 3, 3, -6], [0, 1, 1, -1], [-14, -8, 6, -6], [19, -3, -19, 6], [-1, 8, -8, 1], [-4, 7, -4, 0], [3, 2, -5, -5], [5, 2, 2, 5], [1, 1, 11, -12], [4, 5, 1, -1], [-4, 0, 8, -7], [0, 9, -9, -9], [-12, 5, 5, -12], [2, 8, 0, -4], [9, -1, -8, -8], [0, 0, -6, 6], [-15, 15, 1, -14], [-2, 13, -13, 2], [4, -22, -2, 22], [-1, 0, 2, 3], [-12, 12, -4, -16], [-8, -5, 13, 13], [0, 0, -11, 0], [10, -2, -10, 4], [0, -9, 10, -9], [8, -6, -6, 8], [0, -3, 9, -3], [-12, 14, 14, -12], [0, 0, 0, 22], [0, 1, -1, -1], [-18, -18, 22, -4], [16, -16, 4, 20], [11, -11, -15, -4], [3, 0, -6, 16], [13, -7, -6, -6], [-2, 0, 4, 0], [0, 5, 12, 5], [-3, 9, 3, -18], [14, -21, -7, 21], [-1, -8, 1, 16], [1, -18, 0, 9], [9, 9, -10, 1], [-20, -3, 10, 3], [9, -9, -9, 0], [5, -22, 5, 0], [0, 15, 8, 15], [4, 0, -8, 6], [1, 20, -20, -1], [-32, 4, 16, -4], [9, -28, 0, 14], [20, -20, -3, 17], [8, 8, 8, 0], [-8, 0, 16, 27], [2, 2, 14, -16], [26, -18, 0, 9], [-1, -9, 9, 1], [6, 0, -6, 0], [-13, -16, -3, 3], [-22, 15, 11, -15], [7, -14, 0, 7], [-15, 15, -3, -18], [12, -7, 7, -12], [0, 0, -6, 6], [10, 21, 11, -11], [-32, -2, 32, 4], [0, -14, 0, 0], [-19, -4, -4, -19], [14, -7, -7, -7], [-3, -13, -10, 10], [2, -12, 2, 0], [9, 9, 0, 0], [-20, -11, 9, -9], [-8, -7, 7, 8], [-4, -4, -8, 12], [0, 16, 16, 16], [5, 2, 2, 5], [-17, 11, -17, 0], [10, -2, -5, 2], [-10, 0, 20, 27], [0, -17, 15, -17], [-19, 22, 0, -11], [18, 1, -19, -19], [-5, -1, 1, 5], [2, 29, -1, -29], [14, -14, -12, 2], [33, 4, -4, -33], [2, -2, -1, 2], [-13, -10, -10, -13], [7, 24, 7, 0], [-10, -7, 17, 17], [-21, -21, 24, -3], [6, 22, 0, -11], [18, -18, 6, 24], [-8, -8, 23, -15], [-3, 0, 0, 3], [6, 22, 6, 0], [0, -11, -27, -11], [8, 0, -16, -2], [9, -9, -18, -9], [12, 6, 6, 12], [0, 14, 0, -28], [-25, -28, 0, 14], [-8, 9, -1, -1], [-20, -2, 10, 2], [0, -24, 0, 12], [8, 0, -8, 8], [-3, 0, 6, 37], [38, -1, -38, 2], [1, 28, 27, -27], [-5, -22, 22, 5], [4, -36, -2, 36], [22, -41, -41, 22], [14, 0, -28, 29], [0, 29, -2, 29], [3, -19, 3, 0], [-7, 21, -7, 0], [9, -8, -9, 16], [4, 0, -8, -29], [-9, 4, 4, -9], [-8, -8, 9, -1], [2, -17, -1, 17], [-8, 0, 16, -6], [8, 8, 11, -19], [7, 7, -25, 18], [18, 4, 4, 18], [-4, -15, 2, 15], [-20, -10, 20, 20], [-30, 32, 0, -16], [-11, 25, 25, -11], [-17, 17, 9, -8], [-26, -3, 23, -23], [0, -6, 8, -6], [27, 18, -9, 9], [-18, 9, 27, -27]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_287_r_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_287_2_r_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_287_2_r_b(:prec:=4)
    chi := MakeCharacter_287_r();
    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_287_2_r_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_287_2_r_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_287_r();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 0, -1, 0, 1]>,<3,R![1, -2, 5, -4, 1]>],Snew);
    return Vf;
end function;