
// Make newform 570.2.u.a in Magma, downloaded from the LMFDB on 26 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_a();".
// 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_a();".
// 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], [-2, 1, -3, 2, 2, 2], [0, 1, 0, 0, 0, 1], [0, 0, -1, -4, 4, 1], [2, -4, 2, -2, 4, -2], [-2, -1, 2, -2, -1, -2], [-3, -2, 0, 5, 5, -2], [0, 2, -2, 2, 0, 0], [0, -2, -2, 0, 4, 4], [0, -5, -5, 0, 0, 5], [4, 6, 3, -1, 0, 1], [-6, 0, 6, 4, 2, -2], [-2, 4, 1, 4, -2, 0], [-3, 5, 0, -2, -2, -4], [-2, -5, -4, 4, 5, 2], [-4, 4, 0, 0, 0, -4], [-4, 0, 2, 0, -4, 0], [2, 0, -2, 0, -8, 2], [-2, 2, 2, 4, 0, -4], [4, -4, 8, 4, 0, -4], [0, 2, 4, 0, -6, -6], [-3, -3, 10, 2, 1, -10], [-2, -2, -2, 2, 2, 2], [-2, -2, -8, 4, -2, 8], [0, -3, 5, 10, 5, -3], [-4, 8, 0, 4, -8, -8], [2, 0, -2, -4, -4, -2], [8, 2, 2, 0, 0, -2], [11, 11, 4, -8, -3, -4], [5, 2, 3, -3, -2, -5], [2, -6, 0, 4, 4, -4], [-6, -6, -1, -8, 14, 1], [-4, 2, 8, 12, 0, -12], [-4, 2, 2, 0, 0, -2], [4, 0, -4, -3, 4, 1], [0, 4, 4, -8, 4, 4], [0, 11, 0, -11, -11, 4], [1, -14, 0, 0, 14, -1], [0, 10, -13, -2, -13, 10], [12, -4, -2, -4, 12, 0], [0, 6, 6, 0, 2, -8], [-4, 4, 4, 8, 0, -8], [-10, -12, 9, 10, 3, 3], [-14, 10, -4, 10, -14, 0], [11, -2, 4, -4, 2, -11], [3, 0, -3, 13, 2, 16], [12, -6, -6, 0, -2, 8], [-14, -2, -2, 0, -8, 10], [4, 0, -4, 0, -8, 4], [0, -6, 8, 10, 8, -6], [8, 8, -7, -4, -4, 7], [-8, 2, 0, 6, 6, 17], [0, 8, -2, 8, 0, 0], [-4, 8, -3, 1, 0, -1], [8, 18, 0, -8, 0, 8], [-2, 0, 2, 0, 24, -2], [0, -4, -4, -7, -4, -4], [7, 7, 0, -14, -14, 8], [-2, -10, -6, 6, 10, 2], [0, -19, 5, -8, 5, -19], [-16, 2, 0, 16, 0, -16], [12, -2, 14, -12, -12, -12], [10, -20, -6, -20, 10, 0], [-17, -17, 2, 9, 8, -2], [0, -7, 5, -6, 5, -7], [6, 0, -6, -12, 20, -6], [-4, 0, 4, 10, -4, 6], [4, -16, -2, -4, 18, 18], [0, -4, -4, 12, -4, -4], [-8, -16, -8, 8, 16, 8], [2, 2, 15, 4, -6, -15], [16, 9, -9, -16, 0, 0], [-9, 6, 6, 0, 4, -10], [-4, -21, -6, -2, 0, 2], [-10, 0, 10, 0, -10, 0], [9, -12, 0, 0, 12, -9], [-8, 26, -8, 8, -26, 8], [5, -5, -8, -5, 5, 0], [2, 8, 8, 0, 1, -9], [10, 14, 14, 4, 0, -4], [-2, -2, 14, 8, -6, -14], [6, -6, 0, 0, 0, 0], [4, 8, 0, 0, -8, -4], [2, 2, -14, -12, 10, 14], [6, -8, 1, -6, 7, 7], [-2, -2, -2, 0, -18, 20], [-18, 0, 18, 12, 18, -6], [6, 0, 3, -6, -3, -3], [0, -2, 4, -20, 4, -2], [0, 0, 0, 0, 0, 18], [-10, -11, 17, 10, -6, -6], [3, 26, 5, 2, 0, -2], [-21, 0, 21, -6, -10, -27], [-17, 16, 4, 16, -17, 0], [4, 10, 0, -14, -14, 16], [0, 8, -12, 10, -12, 8], [4, 2, 18, 2, 4, 0], [-20, 20, 10, 20, -20, 0], [-6, 6, 0, 0, 0, 28], [6, 6, 12, 7, -13, -12], [20, -16, -2, -20, 18, 18], [0, 6, 6, 0, 9, -15], [-9, 10, 10, 0, 8, -18], [-10, -16, 8, 10, 8, 8], [12, -4, -2, 2, 4, -12], [-22, 12, 0, 10, 10, 4], [-12, 4, -14, 4, -12, 0], [28, -4, 9, -28, -5, -5], [2, -26, -26, 0, 13, 13], [7, 0, -7, 1, -16, 8], [2, 2, 18, 2, 2, 0], [0, 25, -15, -6, -15, 25], [-12, -2, 0, 14, 14, 24], [-14, 0, 14, 20, -3, 6], [-16, -22, -14, 2, 0, -2], [-22, -13, -13, 0, 10, 3], [28, -7, -3, -28, 10, 10], [5, 5, 4, -5, 0, -4], [-4, -10, 0, 14, 14, 14], [0, 18, -10, 2, -10, 18], [10, 3, -2, -10, -1, -1], [-24, 8, 8, 0, -16, 8], [-8, 13, 12, 8, -25, -25], [10, 10, -10, 10, -10, -10], [-10, -10, 18, -8, 18, -18], [-6, -8, 2, 8, 0, -8], [4, 0, -4, 16, 8, 20], [0, 13, -5, -28, -5, 13], [7, -20, 11, -11, 20, -7], [-8, 6, -25, 25, -6, 8], [22, -22, 6, -22, 22, 0], [4, 20, -9, -13, 0, 13], [-18, 3, 3, 0, 1, -4], [0, -4, -17, -4, 0, 0], [19, 19, 10, -22, 3, -10], [0, -18, 2, -12, 2, -18], [-6, 1, 1, 0, 11, -12], [0, 2, 2, 3, 2, 2], [6, 6, -2, -25, 19, 2], [-8, 0, 0, 8, 8, -26], [-15, -15, -20, -4, 19, 20], [4, 8, 14, 8, 4, 0], [4, -6, 6, -4, 0, 0], [6, 34, 10, 4, 0, -4], [-4, 35, -12, 12, -35, 4], [12, 14, 2, -2, -14, -12], [-17, -9, 0, 26, 26, -6], [0, 3, -8, -12, -8, 3], [9, -28, 4, -5, 0, 5], [40, 5, -3, -40, -2, -2], [-6, -6, 12, -6, -6, 0], [6, 6, 19, 10, -16, -19], [-38, 0, 38, 20, 6, -18], [-2, -6, -6, 0, 16, -10], [10, 22, -6, -10, -16, -16], [-6, -24, 15, -15, 24, 6], [8, 8, 28, -10, 2, -28], [2, 6, -6, 6, 2, 0], [-12, 14, -30, -18, 0, 18], [16, 0, 26, 10, 0, -10], [20, -16, 12, -12, 16, -20], [-18, -1, -4, 4, 1, 18], [0, -14, 24, -2, 24, -14], [2, 0, -2, -21, 2, -19], [-20, -20, -20, 0, 0, 0], [-12, -3, 20, -3, -12, 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_a();".
// 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_a(: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_a();".
// 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_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_570_u();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![5041, 639, 507, 88, 45, 6, 1]>,<11,R![1, -3, 9, -2, 3, 0, 1]>],Snew);
    return Vf;
end function;