
// Make newform 490.4.e.e in Magma, downloaded from the LMFDB on 29 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_490_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_490_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_490_4_e_e();".
// 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_490_4_e_e();".
// 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, -1, 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_490_e();"
function MakeCharacter_490_e()
    N := 490;
    order := 3;
    char_gens := [197, 101];
    v := [3, 2];
    // 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_490_e_Hecke();"
function MakeCharacter_490_e_Hecke(Kf)
    N := 490;
    order := 3;
    char_gens := [197, 101];
    char_values := [[1, 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 := 4;
    raw_aps := [[0, -2], [1, -1], [0, -5], [0, 0], [9, -9], [-51, 0], [81, -81], [0, 86], [0, -48], [211, 0], [254, -254], [0, 20], [-74, 0], [-318, 0], [0, -167], [170, -170], [854, -854], [0, -580], [58, -58], [152, 0], [702, -702], [0, 419], [-124, 0], [0, -768], [-1085, 0], [538, -538], [0, -1623], [0, -1306], [1395, -1395], [-996, 0], [1906, 0], [0, 288], [-916, 916], [640, 0], [0, -330], [-2485, 2485], [1114, -1114], [0, -1448], [4251, 0], [0, 2829], [-716, 716], [3614, 0], [0, 4253], [4664, -4664], [3546, 0], [-54, 54], [-4215, 0], [3179, 0], [-4951, 4951], [0, -266], [0, -296], [6173, 0], [3404, -3404], [-652, 0], [0, 5554], [7070, -7070], [-1856, 1856], [0, 1236], [-3844, 3844], [-7605, 0], [-4913, 4913], [-4633, 0], [-271, 0], [-6054, 6054], [0, -9529], [0, -8506], [0, 10388], [11872, 0], [5308, -5308], [-12470, 0], [-671, 671], [0, -1608], [5681, -5681], [0, 3848], [5344, 0], [0, 5920], [10743, -10743], [0, -8663], [0, 4459], [13430, -13430], [-11094, 0], [-12739, 0], [-7455, 7455], [-12210, 0], [0, -428], [0, 7398], [5807, 0], [0, -18002], [-18964, 0], [2648, 0], [0, 16395], [-15804, 15804], [-132, 132], [-15727, 0], [0, 17213], [-5583, 0], [0, 4216], [13378, -13378], [0, 3208], [0, -6351], [2436, 0], [-7508, 7508], [-18996, 18996], [0, 3982], [14956, -14956], [-1445, 1445], [-6764, 0], [0, 4895], [5677, -5677], [-3330, 0], [0, -1957], [-7734, 7734], [-19452, 0], [-20884, 20884], [-18721, 0], [11562, -11562], [23551, 0], [-22204, 22204], [0, -1442], [339, 0], [-21372, 21372], [6616, 0], [0, -8737], [-1884, 1884], [0, -33356], [19107, 0], [0, 19117], [0, 36584], [25040, 0], [0, -13345], [15037, -15037], [29934, 0], [0, -8725], [-13624, 0], [0, 24168], [12278, 0], [4051, -4051], [-30643, 30643], [-15549, 0], [12675, -12675], [38008, 0], [0, -5277], [22350, -22350], [-33192, 0], [-1694, 1694], [17564, 0], [10450, 0], [30102, -30102], [0, -22268], [0, 24842], [0, -24766], [9840, 0], [-11500, 0], [0, 32768], [-42548, 42548], [29160, 0], [0, 30731], [0, -318], [-19689, 0], [45740, -45740], [0, 51474], [-31554, 0], [-20394, 0], [0, -18562], [162, -162], [52755, -52755], [20836, -20836], [4221, -4221]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_490_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_490_4_e_e();".
// 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_490_4_e_e(:prec:=2)
    chi := MakeCharacter_490_e();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_490_4_e_e();".
// 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_490_4_e_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_490_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![1, -1, 1]>,<11,R![81, -9, 1]>],Snew);
    return Vf;
end function;