
// Make newform 560.4.g.c in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_560_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_560_g_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_560_4_g_c();".
// 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_560_4_g_c();".
// 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];
        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_560_g();"
function MakeCharacter_560_g()
    N := 560;
    order := 2;
    char_gens := [351, 421, 337, 241];
    v := [2, 2, 1, 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_560_g_Hecke();"
function MakeCharacter_560_g_Hecke(Kf)
    N := 560;
    order := 2;
    char_gens := [351, 421, 337, 241];
    char_values := [[1, 0], [1, 0], [-1, 0], [1, 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 := 4;
    raw_aps := [[0, 0], [0, 7], [10, -5], [0, 7], [37, 0], [0, 51], [0, -41], [-108, 0], [0, 70], [249, 0], [134, 0], [0, 334], [206, 0], [0, 376], [0, -287], [0, -6], [-2, 0], [-940, 0], [0, 106], [-456, 0], [0, 650], [-1239, 0], [0, -428], [220, 0], [0, 1055], [1960, 0], [0, -1825], [0, -144], [-1681, 0], [0, 798], [0, 434], [1290, 0], [0, -192], [1402, 0], [302, 0], [3167, 0], [0, 470], [0, 2390], [0, -2631], [0, -2243], [52, 0], [2462, 0], [-3159, 0], [0, 2060], [0, -1738], [-894, 0], [4083, 0], [0, -377], [0, -2551], [-74, 0], [0, 1888], [4997, 0], [-3830, 0], [3390, 0], [0, -7170], [0, 7672], [54, 0], [-2932, 0], [0, -3254], [3327, 0], [0, -4627], [0, -4083], [0, -4089], [4008, 0], [0, 7355], [0, 1684], [1460, 0], [0, -7514], [0, -2862], [6368, 0], [0, -3635], [7116, 0], [0, -319], [0, -11652], [7748, 0], [0, -8680], [1711, 0], [0, -1589], [-5147, 0], [9100, 0], [2618, 0], [-3695, 0], [-15779, 0], [0, 7238], [-2646, 0], [0, -5688], [3285, 0], [0, -14834], [-9972, 0], [0, 9096], [0, 15867], [242, 0], [0, -3558], [-1473, 0], [603, 0], [0, -18387], [-9018, 0], [4624, 0], [0, 5876], [-8537, 0], [0, -13060], [0, -21372], [0, 12704], [-8762, 0], [24764, 0], [0, -1811], [0, 10548], [0, 17439], [2451, 0], [-7792, 0], [0, -1937], [0, 5036], [0, 27286], [28538, 0], [-25007, 0], [-12130, 0], [0, -14385], [0, -2208], [0, 22448], [8791, 0], [-13180, 0], [0, -7164], [0, -12335], [0, 15436], [19184, 0], [32975, 0], [31497, 0], [-18610, 0], [0, -17508], [0, -4685], [-25925, 0], [0, -25578], [4291, 0], [0, 31528], [-23154, 0], [-13992, 0], [0, -21681], [0, -16903], [0, -18905], [5571, 0], [-10894, 0], [-30731, 0], [0, -1038], [0, 7958], [30666, 0], [-5354, 0], [0, -42890], [0, -22950], [-2824, 0], [0, 4866], [0, 10676], [-29856, 0], [0, 1944], [0, -14628], [0, -12858], [18324, 0], [-14751, 0], [47922, 0], [0, 44987], [-20356, 0], [0, -27786], [0, -48674], [0, -11168], [-20094, 0], [0, -49104], [0, -27751], [-37600, 0], [0, 10911]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_560_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_560_4_g_c();".
// 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_560_4_g_c(:prec:=2)
    chi := MakeCharacter_560_g();
    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_560_4_g_c();".
// 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_560_4_g_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_560_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![49, 0, 1]>,<11,R![-37, 1]>],Snew);
    return Vf;
end function;