
// Make newform 570.2.u.d in Magma, downloaded from the LMFDB on 25 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_d();".
// 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_d();".
// 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, -3, 1, -1, 1, -3], [1, 1, 0, 1, -2, 0], [1, 2, -1, 1, -2, -1], [2, 0, 0, 3, 0, 0], [4, -1, 0, -3, -3, 1], [-4, 3, -3, 3, -4, 0], [2, 3, -3, -2, 0, 0], [3, 2, 2, 0, -3, 1], [1, -1, 3, 2, 0, -2], [6, 0, -6, -5, -1, 1], [1, -4, 5, -4, 1, 0], [3, 1, 0, -4, -4, 8], [-4, -5, 3, -3, 5, 4], [-5, -2, 0, 7, 7, -8], [-1, 4, 5, 4, -1, 0], [5, 0, -5, 0, -1, 5], [-7, 0, -2, 5, 0, -5], [3, 13, 0, -3, 0, 3], [-1, 3, 0, 1, -3, -3], [-3, -3, 3, -1, 4, -3], [2, -7, 0, 0, 7, -2], [-8, -8, 2, -1, 9, -2], [0, 0, -5, -7, -5, 0], [5, 5, -8, -5, 3, 3], [8, 0, -8, 1, -1, 9], [-6, 2, 2, 0, -6, 4], [-3, -3, -2, -2, 5, 2], [-2, -8, -4, 4, 8, 2], [-4, -1, 0, 5, 5, -3], [-5, -5, -7, -6, 11, 7], [3, 2, 6, 3, 0, -3], [-3, -12, -12, 0, 9, 3], [3, 0, -3, 0, 12, 3], [0, 0, -7, -8, -7, 0], [-2, 9, 0, -7, -7, -1], [-9, 8, -10, 10, -8, 9], [0, -8, 8, 8, 8, -8], [5, 0, 4, 0, 5, 0], [-5, 10, 10, 0, 0, -10], [4, 0, 9, 5, 0, -5], [16, -5, -2, -16, 7, 7], [7, -12, 0, -12, 7, 0], [-1, -1, -8, 8, 1, 1], [8, 0, -8, -5, -6, 3], [2, 0, 0, 0, 3, -3], [11, 10, 10, 0, -15, 5], [-2, 0, 2, 14, -2, 12], [0, -13, 10, -5, 10, -13], [0, 0, 10, 3, -3, -10], [-11, -7, 0, 18, 18, -5], [0, 3, 9, 3, 0, 0], [6, -1, 6, 0, 0, 0], [3, 13, 6, 3, 0, -3], [-17, 0, 17, 7, -1, -10], [0, 8, -2, -15, -2, 8], [13, 2, 0, -15, -15, 7], [-6, 2, -5, 5, -2, 6], [0, 1, -5, 8, -5, 1], [4, -13, 8, 4, 0, -4], [11, -1, 10, -11, -9, -9], [-2, -3, 11, -3, -2, 0], [7, 7, 19, -1, -6, -19], [0, 6, -15, -4, -15, 6], [-10, 0, 10, 10, 2, 0], [13, 0, -13, 4, 15, 17], [1, -9, -1, -1, 10, 10], [0, -12, 22, 8, 22, -12], [3, -11, 1, -1, 11, -3], [4, 4, 1, -11, 7, -1], [-15, 4, 4, 15, -8, -8], [9, -14, -14, 0, 11, 3], [-3, 11, 8, 11, 0, -11], [-10, 3, -22, 3, -10, 0], [10, 1, -7, 7, -1, -10], [-11, -8, -3, 3, 8, 11], [1, 10, 11, 10, 1, 0], [-17, -3, -3, 0, -3, 6], [-16, -5, -12, 4, 0, -4], [2, 2, 0, -16, 14, 0], [5, -1, 0, -4, -4, 12], [6, -11, 3, -3, 11, -6], [6, 6, -5, -8, 2, 5], [12, 14, -11, -12, -3, -3], [-14, -2, -2, 0, -5, 7], [3, 0, -3, -1, -6, 2], [-10, 18, 9, 10, -27, -27], [0, 13, -10, 19, -10, 13], [-19, 17, 0, 2, 2, -6], [19, 7, 0, -19, -7, -7], [-12, 0, -25, -13, 0, 13], [-7, 0, 7, 29, -12, 22], [-4, 15, 15, 15, -4, 0], [0, 6, 0, -6, -6, 6], [0, 10, 0, -17, 0, 10], [12, -21, -10, -21, 12, 0], [5, -10, 5, -10, 5, 0], [-16, 16, 0, 0, 0, 9], [21, 21, 3, -19, -2, -3], [9, 3, -20, -9, 17, 17], [-22, 3, 3, 0, 4, -7], [-4, 12, 12, 0, -17, 5], [11, -7, -1, -11, 8, 8], [3, -14, -2, 2, 14, -3], [-6, 5, 0, 1, 1, -1], [-7, 2, 27, 2, -7, 0], [16, 2, 5, -16, -7, -7], [7, 4, 4, 0, -11, 7], [-5, 0, 5, -5, 20, -10], [-12, -11, 0, -11, -12, 0], [0, 2, -2, -15, -2, 2], [7, 17, 0, -24, -24, 8], [31, 0, -31, -21, -11, 10], [-1, 7, 6, 7, 0, -7], [-5, -23, -23, 0, 26, -3], [-8, -10, 12, 8, -2, -2], [-19, -19, 15, 17, 2, -15], [-9, 10, 0, -1, -1, 9], [0, -28, 11, -5, 11, -28], [-17, -28, 19, 17, 9, 9], [24, 9, 9, 0, -9, 0], [23, 7, -5, -23, -2, -2], [1, -26, -3, 3, 26, -1], [-17, -17, 1, -11, 28, -1], [-7, 32, 2, 9, 0, -9], [-17, 0, 17, -6, -18, -23], [0, 1, -3, 22, -3, 1], [-13, 5, 1, -1, -5, 13], [-4, -4, -7, 7, 4, 4], [8, -2, 17, -2, 8, 0], [17, -18, -7, -24, 0, 24], [-23, 7, 7, 0, -1, -6], [3, 0, -14, 0, 3, 0], [-5, -5, 35, 6, -1, -35], [0, 6, -23, -15, -23, 6], [1, -25, -25, 0, 4, 21], [0, 21, -29, 2, -29, 21], [25, 25, 4, 10, -35, -4], [-2, -1, 0, 3, 3, 25], [15, 15, 27, -10, -5, -27], [-15, 18, -15, 18, -15, 0], [12, 14, 2, -12, -16, -16], [21, -6, 6, -15, 0, 15], [-2, -22, 11, -11, 22, 2], [-15, -11, -14, 14, 11, 15], [-15, -7, 0, 22, 22, -1], [0, -14, -2, 18, -2, -14], [-8, -9, 6, 14, 0, -14], [-3, -8, 24, 3, -16, -16], [6, -13, -27, -13, 6, 0], [-29, -29, 16, 12, 17, -16], [18, 0, -18, 11, 21, 29], [1, 24, 24, 0, -27, 3], [-13, -10, 9, 13, 1, 1], [-23, 7, -12, 12, -7, 23], [-3, -3, 44, 1, 2, -44], [24, -26, 7, -26, 24, 0], [0, -19, 4, 4, 0, -4], [7, 14, -6, -13, 0, 13], [35, -23, 20, -20, 23, -35], [5, -11, 11, -11, 11, -5], [0, -19, 9, -31, 9, -19], [-8, 0, 8, -23, 0, -31], [-11, -18, 1, 12, 0, -12], [-8, -6, 0, -6, -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_d();".
// 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_d(: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_d();".
// 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_d( : 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![3249, 1026, 495, 60, 27, 3, 1]>],Snew);
    return Vf;
end function;