
// Make newform 570.2.u.g in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_570_u();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_570_u_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_570_2_u_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_570_2_u_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 := [1, 0, 0, -1, 0, 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_570_u();"
function MakeCharacter_570_u()
    N := 570;
    order := 9;
    char_gens := [191, 457, 211];
    v := [9, 9, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_570_u_Hecke();"
function MakeCharacter_570_u_Hecke(Kf)
    N := 570;
    order := 9;
    char_gens := [191, 457, 211];
    char_values := [[1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [0, 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, 0], [0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0], [1, 0, -1, -1, 1, 1], [0, -1, 1, 3, 1, -1], [-1, -1, -4, 1, 0, 4], [1, 2, 3, -3, -2, -1], [2, -2, 0, -1, 4, 2], [0, 1, 0, -1, -1, 1], [6, -1, 1, -1, 6, 0], [0, -5, 1, 0, 4, 4], [3, 2, 2, 0, -5, 3], [-1, -9, -1, 0, 0, 0], [-2, 0, 2, -1, -7, -3], [3, 0, 1, 0, 3, 0], [-1, -5, 0, 6, 6, -4], [2, 7, 7, -7, -7, -2], [3, -4, 0, 1, 1, 2], [-5, 2, -1, 2, -5, 0], [-9, 0, 9, 4, 1, -5], [-5, 2, -10, -5, 0, 5], [1, 5, 2, 1, 0, -1], [-3, 5, -2, 3, -3, -3], [-7, -7, 1, 7, 0, -1], [-4, 1, -6, 6, -1, 4], [6, 6, 4, 3, -9, -4], [0, 4, -3, -5, -3, 4], [-1, -1, 6, 1, -5, -5], [0, 0, 0, -9, -1, -9], [2, 6, 6, 0, -2, -4], [3, 3, 10, 2, -5, -10], [10, -4, 4, -4, 4, -10], [12, -5, 0, -7, -7, -7], [11, 11, -1, 0, -11, 1], [9, -4, 4, -5, 0, 5], [-7, 10, 10, 0, -7, -3], [3, 0, -3, 2, -2, 5], [0, -6, 3, -2, 3, -6], [-6, -1, 0, 7, 7, -5], [-5, 10, -6, 6, -10, 5], [0, 8, 0, -8, 0, 8], [-11, 0, -2, 0, -11, 0], [-3, 2, 2, 0, 4, -6], [2, -14, -5, -7, 0, 7], [2, 3, -12, -2, 9, 9], [1, 10, -10, 10, 1, 0], [5, -5, 10, -10, 5, -5], [-4, 0, 4, -3, 14, -7], [0, -2, -2, 0, 5, -3], [1, -12, -12, 0, 9, 3], [2, 0, -2, -2, 10, 0], [0, -3, -10, 1, -10, -3], [0, 0, 2, -9, 9, -2], [1, -1, 0, 0, 0, 23], [-4, 7, 1, 7, -4, 0], [0, 5, 8, 8, 0, -8], [-3, -3, -10, -7, 0, 7], [-3, 0, 3, -7, 3, -10], [0, 2, 2, -13, 2, 2], [-5, -8, 0, 13, 13, -1], [0, 2, 1, -1, -2, 0], [0, -5, 7, 4, 7, -5], [14, -1, 0, -14, 0, 14], [-3, 9, -2, 3, -7, -7], [0, -9, 3, -9, 0, 0], [3, 3, -13, 5, -8, 13], [0, 2, 11, 6, 11, 2], [2, 0, -2, 2, 14, 4], [1, 0, -1, -12, 5, -11], [-5, 9, -13, 5, 4, 4], [0, 16, -2, 0, -2, 16], [-19, 11, -9, 9, -11, 19], [0, 0, 15, 11, -11, -15], [-5, -20, 10, 5, 10, 10], [3, 18, 18, 0, -5, -13], [-9, 3, -10, -1, 0, 1], [-6, 7, 4, 7, -6, 0], [10, 9, -1, 1, -9, -10], [3, -4, -7, 7, 4, -3], [1, 6, -29, 6, 1, 0], [17, -1, -1, 0, -11, 12], [-2, 5, -16, -14, 0, 14], [2, 2, -16, 0, -2, 16], [3, -15, 0, 12, 12, 6], [12, 5, 13, -13, -5, -12], [4, 4, -5, -10, 6, 5], [6, 4, 13, -6, -17, -17], [4, -6, -6, 0, 17, -11], [-21, 0, 21, -3, -6, -24], [-6, 6, -9, 6, 3, 3], [0, 13, -20, -5, -20, 13], [-5, 7, 0, -2, -2, 6], [-19, -11, 8, 19, 3, 3], [-4, -10, -15, -11, 0, 11], [11, 0, -11, 5, -16, 16], [2, 11, 1, 11, 2, 0], [-16, 2, 0, 14, 14, -2], [0, 14, -6, 19, -6, 14], [-2, -19, 0, -19, -2, 0], [3, 16, -11, 16, 3, 0], [-14, 22, 0, -8, -8, -11], [21, 21, 3, -5, -16, -3], [-1, -3, 14, 1, -11, -11], [-18, -7, -7, 0, 14, -7], [-18, -10, -10, 0, 9, 1], [-19, 1, 13, 19, -14, -14], [7, 26, 0, 0, -26, -7], [18, -11, 0, -7, -7, -5], [-13, -6, 13, -6, -13, 0], [-8, 6, 1, 8, -7, -7], [5, -12, -12, 0, 3, 9], [-9, 0, 9, 9, -16, 0], [4, -11, 24, -11, 4, 0], [0, 8, 10, 9, 10, 8], [3, -11, 0, 8, 8, -18], [-1, 0, 1, 3, -9, 2], [7, -1, 8, 1, 0, -1], [-7, -9, -9, 0, 20, -11], [-32, 2, 0, 32, -2, -2], [11, 11, -11, 1, -12, 11], [-3, 18, 0, -15, -15, -19], [0, 6, 17, -1, 17, 6], [-5, -18, 13, 5, 5, 5], [0, 13, 13, 0, -23, 10], [1, -1, -5, -1, 6, 6], [-15, 16, -3, 3, -16, 15], [-9, -9, -15, -1, 10, 15], [7, -20, 10, 3, 0, -3], [9, 0, -9, 6, 26, 15], [0, -3, -9, 8, -9, -3], [-21, 7, -9, 9, -7, 21], [28, -6, 17, -17, 6, -28], [32, -12, 3, -12, 32, 0], [-3, 32, 1, 4, 0, -4], [27, 3, 3, 0, 7, -10], [11, -20, -14, -20, 11, 0], [5, 5, -5, -18, 13, 5], [0, -2, -3, -9, -3, -2], [45, -7, -7, 0, 0, 7], [0, 3, -3, -24, -3, 3], [-5, -5, -2, 6, -1, 2], [20, -11, 0, -9, -9, -17], [-7, -7, 1, 22, -15, -1], [-15, 16, 17, 16, -15, 0], [-8, 22, -26, 8, 4, 4], [27, -2, 10, -17, 0, 17], [-8, -4, -23, 23, 4, 8], [-23, 17, -14, 14, -17, 23], [-3, -19, 0, 22, 22, -3], [0, 6, -14, -18, -14, 6], [-8, -31, -12, -4, 0, 4], [13, 14, 0, -13, -14, -14], [18, 3, -9, 3, 18, 0], [-17, -17, -20, 26, -9, 20], [-2, 0, 2, -1, 15, -3], [7, -8, -8, 0, 23, -15], [-17, -10, 7, 17, 3, 3], [1, 21, 20, -20, -21, -1], [-1, -1, -26, -19, 20, 26], [18, -26, -3, -26, 18, 0], [12, -31, 6, -6, 0, 6], [-13, 6, 10, 23, 0, -23], [25, -1, 4, -4, 1, -25], [5, 23, 3, -3, -23, -5], [0, -19, 1, -35, 1, -19], [-14, 0, 14, -7, 8, -21], [-25, -2, 9, 34, 0, -34], [-8, 10, 20, 10, -8, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_570_u_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_570_2_u_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_570_2_u_g(:prec:=6)
    chi := MakeCharacter_570_u();
    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_570_2_u_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_570_2_u_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_570_u();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![1, 0, 3, -2, 9, -3, 1]>,<11,R![289, -408, 423, -182, 57, -9, 1]>],Snew);
    return Vf;
end function;