
// Make newform 567.2.g.k in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_567_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_567_g_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_567_2_g_k();".
// 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_567_2_g_k();".
// 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 := [7, -38, 64, -40, 38, -10, 1, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [23150, -10595, 17846, -6480, -193, -619, 688, -14], [-21668, 13756, -23147, 8788, -716, 957, -743, 74], [-2218, -9416, 15724, -5887, 688, -648, 465, -139], [14432, -52069, 36565, -43953, 11731, -262, 2176, -1269], [18369, -49196, 25443, -33224, 9308, -288, 1557, -962], [88979, -241837, 128118, -149399, 42410, -2449, 7290, -4270], [-98045, 265634, -139939, 167278, -47477, 2087, -8118, 4805]];
    Rf_basisdens := [1, 8102, 8102, 8102, 8102, 4051, 8102, 8102];
    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_567_g();"
function MakeCharacter_567_g()
    N := 567;
    order := 3;
    char_gens := [407, 325];
    v := [1, 2];
    // 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_567_g_Hecke();"
function MakeCharacter_567_g_Hecke(Kf)
    N := 567;
    order := 3;
    char_gens := [407, 325];
    char_values := [[-1, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, -1, 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, 1, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-1, 1, 1, 1, 0, 0, 0, 0], [0, 0, -1, -1, 1, 0, 0, 0], [-1, -1, 0, -2, 0, 0, 0, 0], [1, 1, 0, 0, 0, -1, 0, 1], [1, 2, 0, 0, -1, -1, 0, 2], [0, 0, 0, -1, 1, 2, -1, 1], [3, 1, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -3, 0, -2], [0, 0, 0, -1, 1, 5, 2, 0], [0, 0, 0, 2, -2, -1, -1, -3], [0, -2, -3, 0, 0, 0, -3, -2], [0, 0, 0, -1, 1, 2, -1, 2], [-5, 0, -1, 0, -1, 5, -1, 0], [-2, -1, -3, 0, -1, 2, -3, -1], [0, 0, 0, 1, -1, 2, -1, 3], [6, 0, -4, 0, 1, -6, -4, 0], [0, 0, 0, 1, -1, 0, -2, -3], [3, 0, -3, 0, 0, 0, 0, 0], [1, -1, 3, 0, -3, -1, 3, -1], [1, 3, 3, 0, 3, -1, 3, 3], [0, 0, 0, 4, -4, 2, 1, -2], [0, 0, 0, 3, -3, -6, -1, -1], [0, 0, 0, 4, -4, 3, 4, -1], [4, -1, -1, 2, 0, 0, 0, 0], [-4, 0, -3, -6, 0, 0, 0, 0], [0, 0, 0, 0, 0, 3, -4, 2], [2, 3, 1, 0, -1, -2, 1, 3], [8, -2, -1, 0, 1, -8, -1, -2], [-6, 5, 4, 2, 0, 0, 0, 0], [-7, 1, -5, -2, 0, 0, 0, 0], [-9, -1, 1, 0, 0, 0, 0, 0], [9, -2, -1, 0, 1, -9, -1, -2], [-3, -2, -1, 6, 0, 0, 0, 0], [-1, 0, 3, 3, 0, 0, 0, 0], [0, 0, 0, -1, 1, -4, 5, 4], [0, 0, 0, 1, -1, 0, -2, 6], [-1, 4, 2, 0, 1, 1, 2, 4], [-3, 6, 4, 0, 3, 3, 4, 6], [-6, 5, -1, 0, 3, 6, -1, 5], [3, 0, -8, -7, 0, 0, 0, 0], [-1, 0, 3, 0, -2, 1, 3, 0], [0, 0, 0, 0, 0, -5, -6, -3], [-7, -3, 2, 1, 0, 0, 0, 0], [8, -3, 4, 0, 2, -8, 4, -3], [1, 1, 3, 0, -6, -1, 3, 1], [0, 0, 0, -2, 2, 0, -5, -3], [3, 2, -5, 0, 0, 0, 0, 0], [-10, 0, 3, 6, 0, 0, 0, 0], [0, 0, 0, 4, -4, -1, 0, -1], [10, -5, -2, 0, -4, -10, -2, -5], [-4, 0, -3, -6, 0, 0, 0, 0], [2, 0, -7, -5, 0, 0, 0, 0], [-8, 1, 6, 2, 0, 0, 0, 0], [-7, 5, -3, -2, 0, 0, 0, 0], [-14, -4, 3, 0, -1, 14, 3, -4], [0, 0, 0, 7, -7, 9, -5, -3], [-18, -3, -2, 2, 0, 0, 0, 0], [0, 0, 0, -7, 7, -5, 5, -2], [0, 0, 0, 1, -1, 3, 10, 0], [9, -5, 6, 0, -3, -9, 6, -5], [-10, 8, 9, 9, 0, 0, 0, 0], [0, 0, 0, -2, 2, -1, 5, 0], [6, 2, 5, 0, -2, -6, 5, 2], [1, -6, -1, 0, -4, -1, -1, -6], [-12, 2, 2, 0, 7, 12, 2, 2], [3, -6, -1, 0, -2, -3, -1, -6], [0, 0, 0, -1, 1, 4, 6, -3], [0, 0, 0, -3, 3, 10, 0, 8], [-4, -7, 0, -11, 0, 0, 0, 0], [0, 0, 0, 2, -2, 13, -2, -7], [-13, 1, 0, 0, 0, 0, 0, 0], [3, -2, 7, 2, 0, 0, 0, 0], [18, -4, -2, -4, 0, 0, 0, 0], [-3, 1, -4, -9, 0, 0, 0, 0], [-6, 5, 10, 9, 0, 0, 0, 0], [0, 0, 0, -4, 4, 5, 11, 3], [-12, 7, 11, 3, 0, 0, 0, 0], [0, 0, 0, -3, 3, 13, -3, 3], [-11, -2, 4, 0, -4, 11, 4, -2], [0, 0, 0, 3, -3, -2, 6, 3], [-5, -1, 9, 0, 5, 5, 9, -1], [-10, -7, 0, -3, 0, 0, 0, 0], [10, 8, 3, 0, -3, -10, 3, 8], [-23, 0, -4, 0, -1, 23, -4, 0], [3, 3, -3, -3, 0, 0, 0, 0], [9, -3, -10, 0, -8, -9, -10, -3], [0, 0, 0, -4, 4, 10, -7, 7], [-9, -3, 5, 0, 1, 9, 5, -3], [0, 0, 0, -8, 8, 5, -3, -1], [-2, -4, -13, 5, 0, 0, 0, 0], [5, 2, -11, 0, -4, -5, -11, 2], [0, 4, 0, 0, -6, 0, 0, 4], [3, 6, 4, 8, 0, 0, 0, 0], [12, -12, -3, -6, 0, 0, 0, 0], [10, 1, -12, 8, 0, 0, 0, 0], [-11, 8, 9, 0, 8, 11, 9, 8], [0, 0, 0, 4, -4, -6, -2, -3], [0, 0, 0, 4, -4, 0, -2, -5], [0, 0, 0, 6, -6, -11, -3, 2], [2, -4, -5, 0, 7, -2, -5, -4], [0, 0, 0, -3, 3, -21, 1, 0], [-4, -3, 6, 0, 7, 4, 6, -3], [0, 0, 0, -8, 8, 15, 7, 1], [-6, -9, 2, 0, -5, 6, 2, -9], [0, 0, 0, 6, -6, 15, -6, 1], [0, 0, 0, -2, 2, -19, 9, 2], [0, 0, 0, -2, 2, 2, 5, 9], [0, 0, 0, -2, 2, 9, 1, -1], [-3, 10, 7, 8, 0, 0, 0, 0], [0, -6, 5, 0, -2, 0, 5, -6], [-11, -8, 4, 0, -7, 11, 4, -8], [-15, 6, 10, 2, 0, 0, 0, 0], [4, 5, 11, 16, 0, 0, 0, 0], [-15, 2, 10, -3, 0, 0, 0, 0], [-10, 1, -5, 0, 2, 10, -5, 1], [24, -4, -10, 0, -9, -24, -10, -4], [-9, 10, 11, 3, 0, 0, 0, 0], [0, 0, 0, -2, 2, 26, -5, -2], [0, 0, 0, 2, -2, -16, 2, -11], [0, 0, 0, -5, 5, -15, 1, 0], [13, -15, -13, 0, -4, -13, -13, -15], [-23, -1, 3, 0, -4, 23, 3, -1], [7, 3, 3, 0, 6, -7, 3, 3], [-21, 9, 18, 9, 0, 0, 0, 0], [-10, 0, -14, 0, -7, 10, -14, 0], [0, 0, 0, -8, 8, -1, 0, 14], [0, 0, 0, 11, -11, 14, 2, -4], [-11, 0, 11, 1, 0, 0, 0, 0], [14, 3, 4, 0, 14, -14, 4, 3], [-2, 4, -2, 0, -4, 2, -2, 4], [10, -5, -10, -11, 0, 0, 0, 0], [19, 5, -1, -5, 0, 0, 0, 0], [18, -11, 8, -3, 0, 0, 0, 0], [-2, 12, 9, 0, 6, 2, 9, 12], [-24, -1, 5, 0, 6, 24, 5, -1], [0, 0, 0, -5, 5, 15, 7, 15], [-4, 1, -13, 0, -2, 4, -13, 1], [10, -1, -12, 0, -16, -10, -12, -1], [-19, 9, -6, 3, 0, 0, 0, 0], [9, 3, -2, 0, -3, -9, -2, 3], [0, 0, 0, -4, 4, -1, 2, 6], [3, -6, -18, 3, 0, 0, 0, 0], [-21, 2, -7, 0, -2, 21, -7, 2], [0, 0, 0, 6, -6, 3, -6, -8], [0, 0, 0, -9, 9, -8, 6, 3], [-4, 2, -6, -17, 0, 0, 0, 0], [5, -15, -9, 0, 0, 0, 0, 0], [0, 0, 0, -5, 5, 17, 9, 8], [21, -3, 1, -4, 0, 0, 0, 0], [15, 12, 3, -9, 0, 0, 0, 0], [2, -12, -12, 0, 0, 0, 0, 0], [7, -8, -9, 2, 0, 0, 0, 0], [-7, -6, 12, -12, 0, 0, 0, 0], [0, 0, 0, -8, 8, 8, -2, 1], [0, 0, 0, -3, 3, 22, 18, 12], [24, 3, 7, 0, 0, -24, 7, 3], [12, 2, -2, 11, 0, 0, 0, 0], [0, 0, 0, -7, 7, 10, 0, -3], [3, 18, 4, 0, 3, -3, 4, 18], [-20, 0, -14, -4, 0, 0, 0, 0], [-14, 1, 6, 0, 12, 14, 6, 1], [0, 0, 0, 7, -7, 8, 15, 2], [0, 0, 0, -2, 2, -4, 8, 9], [-3, 11, 10, 9, 0, 0, 0, 0], [10, -6, -3, 0, -9, -10, -3, -6], [9, 12, 1, 14, 0, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_567_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_567_2_g_k();".
// 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_567_2_g_k(:prec:=8)
    chi := MakeCharacter_567_g();
    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_567_2_g_k();".
// 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_567_2_g_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_567_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![9, 18, 54, -30, 39, -6, 7, -1, 1]>,<13,R![49, -140, 400, -70, 107, -40, 25, -5, 1]>],Snew);
    return Vf;
end function;