
// Make newform 1287.2.a.g in Magma, downloaded from the LMFDB on 05 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1287_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_1287_2_a_g();".
// 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_1287_2_a_g();".
// 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 := [-2, 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_1287_a();"
function MakeCharacter_1287_a()
    N := 1287;
    order := 1;
    char_gens := [1145, 937, 496];
    v := [1, 1, 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;

function MakeCharacter_1287_a_Hecke(Kf)
    return MakeCharacter_1287_a();
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, 1], [0, 0], [2, 1], [-2, 2], [-1, 0], [-1, 0], [4, -1], [-6, 0], [-2, -2], [0, 3], [0, -3], [2, -4], [12, 0], [-2, -5], [0, -6], [-2, 8], [8, -2], [-2, 0], [4, 5], [0, -4], [4, -6], [-2, 5], [0, 8], [14, -1], [10, 0], [0, -7], [0, 6], [0, 6], [-4, 2], [6, -4], [10, -3], [0, -12], [-6, -9], [-10, 1], [-12, -6], [6, 0], [4, 0], [0, 9], [12, -4], [-16, 5], [10, 10], [-8, -12], [8, 8], [-16, -2], [12, -2], [-8, 2], [-6, 9], [-4, -1], [-6, -12], [18, 6], [0, -3], [10, 4], [-10, 8], [-4, 8], [18, 2], [-12, -14], [-2, 10], [-10, 4], [14, -4], [-16, -4], [-14, -9], [-12, 2], [-2, -20], [2, -2], [20, 0], [14, -1], [-16, -3], [6, 18], [0, 6], [2, -12], [18, 11], [14, 4], [12, 12], [6, -6], [16, -5], [0, -2], [-6, 6], [2, -16], [6, -5], [20, 6], [18, 10], [14, 16], [-12, -8], [26, -8], [-14, -5], [20, 4], [-6, 1], [-14, 0], [8, 8], [20, -3], [-6, -14], [6, 24], [-8, -9], [8, -2], [4, 13], [-12, -6], [-6, -7], [-18, -12], [-2, -3], [18, 12], [10, 3], [-20, 2], [12, -4], [12, 9], [22, -3], [2, -4], [28, -10], [8, -26], [24, -4], [22, -12], [-22, 1], [16, 14], [6, 29], [0, 15], [16, -23], [-22, 4], [-8, -9], [-16, -16], [30, 2], [0, -20], [14, 18], [10, -6], [-24, 13], [20, -16], [-32, -7], [36, 5], [-6, -6], [0, -32], [4, -2], [-24, -6], [22, 14], [16, 20], [28, -12], [-28, -16], [-12, -18], [-2, -24], [18, -17], [-26, -14], [2, -26], [-12, 5], [-10, -16], [12, 12], [28, 6], [-6, 8], [-26, -8], [-4, 16], [30, -12], [12, 3], [8, -6], [16, 26], [-30, 12], [-18, -20], [20, 6], [8, 28], [8, 4], [0, -4], [-34, 9], [-6, 7], [-14, 2], [-16, 16], [20, -22], [36, 7], [22, 10], [6, 2], [-30, 21], [12, 12], [-24, 18], [2, 26], [-4, -6], [30, -18], [12, -12], [-42, 12], [-36, 12], [-10, 8], [-40, 6], [22, -26], [-6, -18], [-30, -21], [22, 9], [-10, 14], [-8, -7], [-16, 20], [20, -24], [12, -28], [12, 14], [6, -12], [-26, 10], [-8, -9], [6, 0], [-36, -4], [0, -12], [0, 8], [-4, -22], [-10, 1], [0, -4], [30, -10], [-34, -4], [-30, 0], [-12, 0], [-12, -4], [-4, -20], [2, -3], [-30, -12], [14, 36], [-36, 22], [-6, -22], [-4, -3], [-4, -28], [-24, -8], [4, 22], [14, -6], [-30, -8], [-20, 16], [0, 8], [14, 4], [26, -26], [-54, 3], [8, 31], [4, -4], [54, 9], [-14, -12], [-46, -2], [6, 23], [12, 3], [-28, 8], [-10, -10], [30, -30], [-16, 8], [-6, -3], [32, 8], [-18, 36], [10, -33], [22, 12], [-36, 1], [-36, 6], [0, -28], [-10, 34], [0, -22], [18, 22], [-18, 2], [36, 6], [-38, 3], [-36, 12], [-14, -38], [16, -10], [8, -32], [16, -13], [0, 34], [14, 6], [4, -4], [14, 0], [20, -4], [20, 32], [-20, -20], [12, -23], [14, -8], [-44, 4], [62, -4], [6, 14], [2, 32], [54, -9], [-20, -32], [30, -12], [14, -10], [-26, 31], [-44, 12], [54, -9], [-10, 29], [-2, -36], [-24, 7], [-14, -40], [-8, -1], [-14, 34], [30, -15], [42, 6], [40, 9], [-44, 8], [40, -6], [-24, -6], [32, -28], [0, 46], [-8, 3], [-12, -24], [-28, 2], [38, 6], [-40, -4], [-46, 16], [-12, 24], [-8, 9], [24, -35], [22, -10], [-18, 22], [4, 34], [14, 32], [-26, 28], [-28, 13], [4, -10], [-18, 40], [-24, 2], [-22, -12], [10, -40], [-12, -4], [-22, 39], [-30, -34], [8, 50], [-22, -4], [38, -22], [-30, 24], [-60, 0], [-10, 10], [-60, -18], [-6, 21], [0, -33], [52, -12], [12, -24], [6, 6], [56, 20], [38, 4], [66, 6], [36, -23], [-10, -45], [-6, 48], [12, 35], [0, -15], [40, 26], [-16, -32], [20, -3], [-28, 27], [24, 34], [0, -33], [8, 18], [-42, -4], [-22, -9], [-6, 0], [26, -30], [40, 22], [-74, 0], [12, -4], [70, -10], [-6, -21], [-46, 0], [6, 22], [-20, -14], [-40, 6], [-6, 45], [-28, 14], [6, 24], [24, -27], [14, 20], [-18, 16], [-14, 2], [18, -30], [-74, -11], [70, -1], [-58, 0], [4, 29], [-12, -10], [-18, 0], [60, 9], [-4, -14], [86, -4], [-24, 7], [40, 4], [-10, -40], [-76, -12], [44, 4], [28, 28], [-2, -52], [-30, -6], [-28, 8], [-68, 2], [54, -12], [-32, -25], [18, -9], [-10, 8], [20, 38], [40, -26], [-24, -30], [60, 13], [38, 36], [-78, -13], [10, -5], [-26, 14], [24, -48], [-38, -24], [48, 0], [-40, 15], [-54, 24], [38, -14], [-62, 14], [22, 12], [-38, -5], [-4, 32], [20, -8], [16, -10], [-34, -20], [18, -12], [22, -20], [44, -6], [-12, -34], [-10, -8], [-22, -10], [2, 38], [6, 7], [12, 12], [42, -21], [46, -18], [-48, 18], [-28, -9], [16, -20], [-6, 0], [-36, 1], [12, 20], [6, -33], [22, 30], [-10, -50], [-54, 4], [44, -22], [18, 1], [-60, -22], [10, -22], [-38, -4], [60, -4], [32, -34], [34, 12], [30, 41], [28, -40], [88, -14], [-8, 25], [0, -10]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1287_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1287_2_a_g();".
// 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_1287_2_a_g(:prec:=2)
    chi := MakeCharacter_1287_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 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_1287_2_a_g();".
// 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_1287_2_a_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1287_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, -2, 1]>,<5,R![2, -4, 1]>],Snew);
    return Vf;
end function;