
// Make newform 570.2.c.c in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_570_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_570_c_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_c_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_570_2_c_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 := [5, -26, 54, -66, 59, -34, 16, -4, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [-10, 46, -62, 82, -50, 34, -8, 3], [25, -51, 67, -82, 50, -34, 8, -3], [-30, 109, -118, 118, -60, 31, -7, 2], [50, -138, 191, -186, 110, -57, 14, -4], [45, -157, 229, -249, 155, -83, 21, -6], [85, -249, 298, -313, 170, -96, 22, -7]];
    Rf_basisdens := [1, 1, 5, 5, 5, 5, 5, 5];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    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_c();"
function MakeCharacter_570_c()
    N := 570;
    order := 2;
    char_gens := [191, 457, 211];
    v := [1, 1, 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_c_Hecke();"
function MakeCharacter_570_c_Hecke(Kf)
    N := 570;
    order := 2;
    char_gens := [191, 457, 211];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 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 := 2;
    raw_aps := [[0, 0, 0, 0, 0, 0, 1, 0], [-1, 0, 0, -1, 1, 0, 1, 1], [-1, 0, 1, -1, 1, 0, 2, 1], [1, -3, 1, 0, -1, 0, 2, 1], [1, 0, 0, 2, -2, 0, -1, -2], [0, 0, 0, -2, 0, 0, 1, 0], [0, -1, 3, 0, -1, 2, 1, -1], [-1, -1, 1, 2, -1, 0, 0, 1], [-1, -1, 5, 0, -1, 2, 2, -1], [-3, 1, 3, 0, 1, -2, 2, 1], [-1, 4, -2, 0, 2, 0, -3, -2], [-2, 0, 0, 0, 0, 0, 0, 0], [3, 0, -6, 0, 0, 0, -3, 0], [1, -2, 0, 0, 0, 0, 1, 0], [-4, 2, 2, 0, 2, -4, 2, 2], [0, 0, 0, 0, 0, 0, 1, 0], [4, -3, 1, 0, -3, 6, -1, -3], [1, 0, 0, 2, 0, 0, -1, 0], [-7, 0, 0, 0, 0, 0, 0, 0], [-1, -2, 8, 0, -2, 4, 3, -2], [0, -3, 3, 0, -3, 0, 3, 3], [4, -8, 0, 0, 0, 0, 4, 0], [0, 2, -6, 0, 2, -4, -2, 2], [1, -2, 4, 0, -2, 4, 1, -2], [-10, 0, 0, -4, 0, 0, 2, 0], [-1, 0, 0, -2, 2, 0, -9, 2], [-1, 0, 0, 10, 0, 0, -5, 0], [-3, 0, 0, -6, 6, 0, 16, 6], [-1, -3, 5, 0, -5, 0, 4, 5], [-1, 0, 0, -2, 2, 0, 7, 2], [10, 0, 0, -4, 0, 0, 2, 0], [0, 0, 0, 0, 0, 0, -10, 0], [0, -1, 3, 0, -1, 2, 1, -1], [-10, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 4, 0], [4, -6, -2, 0, 2, 0, 2, -2], [-2, 4, 0, 0, 0, 0, -2, 0], [-1, 6, -4, 0, 4, 0, -5, -4], [-3, 0, 0, -6, 6, 0, -9, 6], [4, 0, 0, 8, -8, 0, -8, -8], [6, -2, -6, 0, -2, 4, -4, -2], [-4, 6, 2, 0, -2, 0, -2, 2], [0, 0, 0, 0, 0, 0, -5, 0], [-6, 0, 0, 0, 0, 0, 0, 0], [3, -2, 0, 0, -2, 4, -1, -2], [-2, 0, 0, 14, 0, 0, -7, 0], [0, -5, 5, 0, -5, 0, 5, 5], [10, 0, 0, 8, 0, 0, -4, 0], [-3, 0, 0, -6, 6, 0, -4, 6], [0, 0, 0, 0, 0, 0, 0, 0], [-12, 6, 6, 0, 6, -12, 6, 6], [-2, 0, 0, -4, 4, 0, -9, 4], [-2, 8, -4, 0, 4, 0, -6, -4], [5, 0, 0, 10, -10, 0, 5, -10], [-3, 0, 0, -6, 6, 0, 11, 6], [-2, -4, 16, 0, -4, 8, 6, -4], [4, 2, -14, 0, 2, -4, -6, 2], [6, 0, 0, 2, 0, 0, -1, 0], [4, -10, 2, 0, -2, 0, 6, 2], [5, -4, 2, 0, -4, 8, -1, -4], [-6, 6, 6, 0, -6, 0, 0, 6], [4, 0, 0, 8, -8, 0, -3, -8], [-2, 0, 0, 0, 0, 0, 0, 0], [-6, 0, 0, -12, 12, 0, 11, 12], [4, 1, -9, 0, 9, 0, -5, -9], [0, 0, 0, 0, 0, 0, -17, 0], [4, -11, 3, 0, -3, 0, 7, 3], [5, 0, 0, -14, 0, 0, 7, 0], [-12, 8, 0, 0, 8, -16, 4, 8], [2, 0, 0, -4, 0, 0, 2, 0], [12, -7, -3, 0, -7, 14, -5, -7], [6, 0, 0, 12, -12, 0, -27, -12], [-2, 4, 0, 0, 0, 0, -2, 0], [20, 0, 0, -2, 0, 0, 1, 0], [-10, 23, -3, 0, 3, 0, -13, -3], [5, 0, 0, 10, -10, 0, -19, -10], [-7, 0, 0, -14, 14, 0, 21, 14], [-8, 18, -2, 0, 2, 0, -10, -2], [-9, 4, 6, 0, 4, -8, 5, 4], [-1, 2, 0, 0, 0, 0, -1, 0], [3, 0, 0, 6, -6, 0, -19, -6], [-5, 15, -5, 0, 5, 0, -10, -5], [13, -10, 4, 0, -10, 20, -3, -10], [-5, 0, 0, -2, 0, 0, 1, 0], [-1, 2, 0, 0, 0, 0, -1, 0], [-5, 2, 4, 0, 2, -4, 3, 2], [-2, -6, 22, 0, -6, 12, 8, -6], [-6, 5, 7, 0, -7, 0, 1, 7], [5, 0, 0, 10, -10, 0, -5, -10], [-4, 8, 0, 0, 0, 0, -4, 0], [7, 0, -14, 0, 0, 0, -7, 0], [0, 0, 0, 0, 0, 0, 14, 0], [-10, 0, 0, 16, 0, 0, -8, 0], [-5, 0, 0, -10, 10, 0, 35, 10], [25, 0, 0, 10, 0, 0, -5, 0], [-9, 5, 3, 0, 5, -10, 4, 5], [-2, 4, -8, 0, 4, -8, -2, 4], [14, -12, 8, 0, -12, 24, -2, -12], [29, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, -18, 0, 0, 9, 0], [-12, 0, 0, 0, 0, 0, 0, 0], [12, -10, 6, 0, -10, 20, -2, -10], [-6, 0, 0, -12, 12, 0, 2, 12], [-2, 4, -8, 0, 4, -8, -2, 4], [-29, 0, 0, 2, 0, 0, -1, 0], [4, -5, -3, 0, 3, 0, 1, -3], [11, -8, 2, 0, -8, 16, -3, -8], [-2, -4, 16, 0, -4, 8, 6, -4], [17, -14, 8, 0, -14, 28, -3, -14], [7, -18, 4, 0, -4, 0, 11, 4], [30, 0, 0, -4, 0, 0, 2, 0], [-5, 12, -2, 0, 2, 0, -7, -2], [10, -6, -2, 0, -6, 12, -4, -6], [-15, 0, 0, -10, 0, 0, 5, 0], [-4, 0, 0, 12, 0, 0, -6, 0], [-5, 6, -8, 0, 6, -12, -1, 6], [2, -16, 12, 0, -12, 0, 14, 12], [-1, 1, -1, 0, 1, -2, 0, 1], [-2, 6, -14, 0, 6, -12, -4, 6], [-10, 5, 5, 0, 5, -10, 5, 5], [9, -21, 3, 0, -3, 0, 12, 3], [0, 0, 0, 8, 0, 0, -4, 0], [10, 0, 0, 20, -20, 0, -7, -20], [0, 0, 0, 0, 0, 0, -24, 0], [6, 0, 0, 12, 0, 0, -6, 0], [-4, 0, 0, -8, 8, 0, -16, 8], [22, 0, 0, -4, 0, 0, 2, 0], [6, 0, 0, 12, -12, 0, 3, -12], [-3, 3, 3, 0, -3, 0, 0, 3], [2, -6, 2, 0, -2, 0, 4, 2], [13, 0, 0, 14, 0, 0, -7, 0], [9, 0, 0, 18, -18, 0, -3, -18], [7, -18, 4, 0, -4, 0, 11, 4], [-4, 12, -4, 0, 4, 0, -8, -4], [1, 0, 0, 2, -2, 0, -16, -2], [-15, 0, 0, 20, 0, 0, -10, 0], [10, 0, 0, 20, -20, 0, -29, -20], [-15, 0, 0, -4, 0, 0, 2, 0], [2, 0, 0, 4, -4, 0, 21, -4], [3, 0, 0, 6, -6, 0, -14, -6], [-10, 15, 5, 0, -5, 0, -5, 5], [0, 0, 0, 0, 0, 0, -30, 0], [5, -13, 3, 0, -3, 0, 8, 3], [-5, 0, 0, -10, 10, 0, 18, 10], [9, -23, 5, 0, -5, 0, 14, 5], [-10, 10, -10, 0, 10, -20, 0, 10], [14, -34, 6, 0, -6, 0, 20, 6], [0, 0, 0, 0, 0, 0, -12, 0], [-7, 0, 0, -6, 0, 0, 3, 0], [-1, 0, 0, -2, 2, 0, 17, 2], [-10, 0, 0, 6, 0, 0, -3, 0], [6, 0, 0, 12, -12, 0, -16, -12], [2, -16, 12, 0, -12, 0, 14, 12], [-5, 0, 0, -10, 10, 0, -7, 10], [-27, 0, 0, -20, 0, 0, 10, 0], [-8, 2, 10, 0, 2, -4, 6, 2], [-8, 0, 0, 6, 0, 0, -3, 0], [3, 0, 0, 6, -6, 0, -34, -6], [-6, 25, -13, 0, 13, 0, -19, -13], [-1, 3, -7, 0, 3, -6, -2, 3], [10, -6, -2, 0, -6, 12, -4, -6], [5, 0, 0, 10, -10, 0, -29, -10], [10, -24, 4, 0, -4, 0, 14, 4], [-8, 12, -20, 0, 12, -24, -4, 12], [5, 0, 0, 10, -10, 0, 3, -10], [-10, 0, 0, -20, 20, 0, 26, 20], [7, -8, -6, 0, 6, 0, 1, -6], [9, -20, 2, 0, -2, 0, 11, 2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_570_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_570_2_c_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_570_2_c_c(:prec:=8)
    chi := MakeCharacter_570_c();
    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_c_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_570_2_c_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_570_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![245, 0, 34, 0, 1]>,<11,R![100, 0, 24, 0, 1]>,<29,R![125, 0, -70, 0, 1]>,<37,R![2, 1]>],Snew);
    return Vf;
end function;