
// Make newform 570.2.u.h in Magma, downloaded from the LMFDB on 24 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_h();".
// 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_h();".
// 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], [3, 0, -1, -3, 1, 1], [0, 1, -1, -3, -1, 1], [1, 1, 0, 1, -2, 0], [-1, -2, 1, -1, 2, 1], [-2, 0, -2, 1, -2, 4], [2, 1, 0, -3, -3, 1], [0, -3, -1, -3, 0, 0], [6, 5, -3, -6, -2, -2], [-1, -2, -2, 0, 5, -3], [-1, 5, -1, 0, 0, 0], [2, 0, -2, 3, -3, 5], [-1, 2, -3, 2, -1, 0], [-1, -5, 0, 6, 6, -2], [2, 1, 1, -1, -1, -2], [3, 2, 0, -5, -5, 2], [7, -2, 7, -2, 7, 0], [-5, 0, 5, -2, -3, -7], [-1, 2, 8, 9, 0, -9], [-1, 5, -2, -1, 0, 1], [-1, 5, -6, 1, 1, 1], [3, 3, 7, -5, 2, -7], [0, 7, 0, 0, -7, 0], [0, 0, 6, -3, 3, -6], [0, -2, -1, 7, -1, -2], [-3, 7, -6, 3, -1, -1], [-2, 0, 2, 3, 7, 1], [-10, -2, -2, 0, -2, 4], [9, 9, 2, 0, -9, -2], [14, -8, 8, -8, 8, -14], [-8, 1, 0, 7, 7, -1], [-11, -11, 7, 8, 3, -7], [5, 6, -2, -7, 0, 7], [-3, 6, 6, 0, 3, -9], [-13, 0, 13, 12, 4, -1], [0, -2, 3, 6, 3, -2], [4, 3, 0, -7, -7, -1], [3, -4, -8, 8, 4, -3], [0, 4, -12, -4, -12, 4], [-11, -2, 0, -2, -11, 0], [-5, -4, -4, 0, -4, 8], [6, 4, -1, -7, 0, 7], [-2, -3, 10, 2, -7, -7], [-3, 4, -14, 4, -3, 0], [-3, 11, 4, -4, -11, 3], [14, 0, -14, -5, -6, 9], [4, -16, -16, 0, 9, 7], [-5, -2, -2, 0, 7, -5], [14, 0, -14, 2, 2, 16], [0, 7, -6, 13, -6, 7], [0, 0, 14, -5, 5, -14], [-1, 5, 0, -4, -4, -7], [12, -3, -9, -3, 12, 0], [8, 13, -2, -10, 0, 10], [5, -1, 10, 5, 0, -5], [7, 0, -7, 5, -3, 12], [0, 16, -10, -3, -10, 16], [-9, 0, 0, 9, 9, 9], [-12, -6, 1, -1, 6, 12], [0, -1, -9, 10, -9, -1], [-10, -13, -8, 2, 0, -2], [-1, -11, 8, 1, 3, 3], [-8, -7, -1, -7, -8, 0], [-15, -15, 3, 15, 0, -3], [0, -8, -13, 12, -13, -8], [-2, 0, 2, -2, 14, -4], [-1, 0, 1, -2, 25, -3], [5, 17, -3, -5, -14, -14], [0, 4, 6, 4, 6, 4], [7, 1, 11, -11, -1, -7], [2, 2, 1, 17, -19, -1], [-3, 16, 0, 3, -16, -16], [-11, -12, -12, 0, 15, -3], [-9, 13, -2, 7, 0, -7], [14, -11, 6, -11, 14, 0], [18, -3, 13, -13, 3, -18], [7, -14, -1, 1, 14, -7], [17, -18, -1, -18, 17, 0], [11, 9, 9, 0, -1, -8], [4, 1, -2, -6, 0, 6], [-2, -2, 4, 0, 2, -4], [11, 7, 0, -18, -18, 2], [-8, -7, -21, 21, 7, 8], [-12, -12, -9, 0, 12, 9], [8, -8, 7, -8, 1, 1], [-4, -4, -4, 0, 11, -7], [9, 0, -9, -3, 18, 6], [0, -20, 23, 0, -3, -3], [0, 9, 10, 5, 10, 9], [-9, 7, 0, 2, 2, 8], [-3, 7, -10, 3, 3, 3], [8, -14, 15, 7, 0, -7], [-9, 0, 9, 13, 10, 4], [16, 1, -1, 1, 16, 0], [0, -6, 0, 6, 6, 26], [0, -8, 0, 23, 0, -8], [26, -11, 8, -11, 26, 0], [-5, 0, -5, 0, -5, 0], [6, -6, 0, 0, 0, -15], [-5, -5, -9, 1, 4, 9], [33, 1, 2, -33, -3, -3], [-4, 9, 9, 0, -4, -5], [-10, -8, -8, 0, 3, 5], [7, -9, 1, -7, 8, 8], [-7, 24, 4, -4, -24, 7], [-6, 23, 0, -17, -17, -15], [13, -22, -7, -22, 13, 0], [0, 14, 9, 0, -23, -23], [-1, -14, -14, 0, 21, -7], [-9, 0, 9, -1, 12, -10], [16, -9, 28, -9, 16, 0], [0, -10, 4, -9, 4, -10], [-7, 1, 0, 6, 6, 32], [-5, 0, 5, 13, 9, 8], [11, -13, 20, 9, 0, -9], [-5, 19, 19, 0, -2, -17], [-20, 18, -8, 20, -10, -10], [-11, -11, -17, -1, 12, 17], [-17, 12, 0, 5, 5, 9], [0, -14, 11, -5, 11, -14], [31, -6, 19, -31, -13, -13], [10, -9, -9, 0, 19, -10], [15, -3, -17, -15, 20, 20], [-13, 22, -5, 5, -22, 13], [7, 7, -1, -29, 22, 1], [-13, -2, -2, 11, 0, -11], [19, 0, -19, -6, -8, 13], [0, 1, 9, -22, 9, 1], [-15, -25, -11, 11, 25, 15], [14, 12, 13, -13, -12, -14], [-10, 16, 17, 16, -10, 0], [9, -10, 1, -8, 0, 8], [25, -23, -23, 0, 7, 16], [23, -28, -6, -28, 23, 0], [15, 15, -7, 8, -23, 7], [0, -10, 9, -5, 9, -10], [11, -33, -33, 0, 18, 15], [0, 7, -1, -36, -1, 7], [1, 1, -4, -4, 3, 4], [-2, 13, 0, -11, -11, -17], [-23, -23, -3, 8, 15, 3], [1, -2, 47, -2, 1, 0], [24, 22, -18, -24, -4, -4], [7, -20, -18, -25, 0, 25], [-2, 10, -13, 13, -10, 2], [-17, -17, -30, 30, 17, 17], [23, -7, 0, -16, -16, 11], [0, -2, -18, 18, -18, -2], [-28, -9, -2, 26, 0, -26], [-35, 12, 2, 35, -14, -14], [6, -13, -1, -13, 6, 0], [25, 25, -14, 2, -27, 14], [-14, 0, 14, 7, 3, -7], [-17, 14, 14, 0, -9, -5], [5, 0, -3, -5, 3, 3], [19, 31, -6, 6, -31, -19], [19, 19, -4, 9, -28, 4], [-8, 2, 13, 2, -8, 0], [-2, -39, -2, 0, 0, 0], [-19, 6, -14, 5, 0, -5], [-15, -23, 8, -8, 23, 15], [-1, 17, 5, -5, -17, 1], [0, 11, -21, -9, -21, 11], [-10, 0, 10, 31, -2, 21], [23, 8, 5, -18, 0, 18], [-8, 10, 32, 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_h();".
// 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_h(: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_h();".
// 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_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_570_u();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![361, -456, 405, -178, 57, -9, 1]>,<11,R![289, 408, 423, 182, 57, 9, 1]>],Snew);
    return Vf;
end function;