
// Make newform 512.2.g.d in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_512_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_512_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_512_2_g_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_512_2_g_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, 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_512_g();"
function MakeCharacter_512_g()
    N := 512;
    order := 8;
    char_gens := [511, 5];
    v := [8, 7];
    // 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_512_g_Hecke();"
function MakeCharacter_512_g_Hecke(Kf)
    N := 512;
    order := 8;
    char_gens := [511, 5];
    char_values := [[1, 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, 0, 0], [1, 0, 0, 1], [2, 2, -1, 1], [-1, 0, -1, 0], [1, -1, -2, -2], [0, 1, -1, 0], [0, 2, 0, 2], [3, -2, 2, -3], [-3, 0, 3, 4], [2, 1, 1, 2], [-4, 0, 0, 0], [0, 0, 1, -1], [3, 0, -3, 4], [3, -3, 4, 4], [0, -4, 6, -4], [4, -4, 1, 1], [-1, -1, -4, 4], [0, -1, -1, 0], [3, 2, 2, 3], [3, 4, 3, 0], [-7, 0, 7, 0], [0, 0, -6, 0], [1, 4, -4, -1], [-3, -8, -3, 0], [-10, -6, 0, 6], [-4, -4, 5, -5], [-1, -12, -1, 0], [1, -1, -2, -2], [-6, -5, 5, 6], [0, -4, 12, -4], [-4, 12, 0, -12], [-1, 10, 10, -1], [3, 0, -3, -8], [-9, 9, 4, 4], [10, -10, -5, -5], [-7, 0, 7, -12], [0, -1, -1, 0], [9, -4, -4, 9], [-9, -8, -9, 0], [4, 1, -1, -4], [1, -8, 8, -1], [-6, 6, 7, 7], [-12, 0, 0, 0], [-10, 6, 0, -6], [2, 2, 11, -11], [-1, 24, -1, 0], [9, 4, -4, -9], [-4, -12, 0, 12], [-7, -8, -8, -7], [-6, -6, -17, 17], [-3, 0, 3, -8], [0, 8, -6, 8], [0, 6, 0, 6], [11, 11, -4, 4], [6, 0, 0, 0], [3, 4, 3, 0], [10, 7, -7, -10], [0, 0, -18, 0], [0, 0, 1, 1], [9, 4, 9, 0], [9, 9, -2, 2], [14, 14, -1, 1], [-9, -2, 2, 9], [-3, 0, 3, -8], [-1, 12, -1, 0], [2, -11, -11, 2], [-3, 3, -2, -2], [0, 12, 0, 12], [-1, -1, 8, -8], [-18, -7, -7, -18], [6, 0, 0, 0], [15, 4, 15, 0], [0, 0, 6, 0], [-12, 12, -11, -11], [-21, -21, 4, -4], [0, -12, 0, 12], [-4, -4, -19, 19], [-6, -17, 17, 6], [0, 2, 0, 2], [-13, -12, -13, 0], [-13, -2, -2, -13], [6, 6, 7, -7], [0, -4, 18, -4], [0, -6, 24, -6], [17, 0, -17, 0], [5, 5, -10, 10], [6, -18, 0, 18], [-1, 0, 1, -12], [-2, 7, -7, 2], [0, -12, -6, -12], [-17, 10, -10, 17], [12, 12, 0, -12], [11, 0, 11, 0], [-17, 17, 16, 16], [-3, 4, -4, 3], [-3, 0, 3, -20], [20, 7, 7, 20], [-9, 0, 9, -8], [-3, 3, 10, 10], [12, 11, 11, 12], [-9, 14, 14, -9], [16, 13, -13, -16], [7, 10, -10, -7], [-9, -8, -9, 0], [-3, -3, -2, 2], [2, 12, 0, -12], [13, -13, -2, -2], [0, 20, 0, 20], [21, 0, -21, -8], [5, 24, 5, 0], [-16, -12, 0, 12], [-6, -6, 19, -19], [-3, 0, 3, 28], [-3, 3, -14, -14], [-7, 0, 7, 36], [-18, 18, 0, -18], [-21, 2, 2, -21], [-9, 4, -9, 0], [10, -17, 17, -10], [-11, -8, 8, 11], [18, -18, -17, -17], [14, 6, 0, -6], [20, 20, 5, -5], [-5, 5, -8, -8], [21, -8, 8, -21], [8, -5, -5, 8], [12, 12, 1, -1], [0, -4, -30, -4], [-7, 0, 7, -24], [6, 17, 17, 6], [9, -4, -4, 9], [-9, -32, -9, 0], [0, 12, -6, 12], [6, -6, -17, -17], [27, 4, 27, 0], [14, -6, 0, 6], [20, 20, -7, 7], [-3, 4, -4, 3], [20, 7, 7, 20], [15, 0, -15, -20], [-3, 3, -26, -26], [4, -4, -11, -11], [-19, 0, 19, 24], [17, 17, 14, -14], [12, -25, -25, 12], [-21, 16, -21, 0], [-12, 12, 25, 25], [-21, 16, -21, 0], [-9, -9, 16, -16], [-12, 24, 0, -24], [6, -17, 17, -6], [0, -16, 0, -16], [15, 34, -34, -15], [9, 0, -9, 16], [3, -3, -8, -8], [0, -28, 6, -28], [17, 0, -17, 12], [-18, 6, 0, -6], [-19, 0, 19, 0], [-2, 7, -7, 2], [19, -14, 14, -19], [-9, -8, -9, 0], [23, -24, 23, 0], [7, -7, 16, 16], [0, -10, 0, -10], [-3, 0, 3, -32], [-16, 0, 0, 0], [-24, -24, -11, 11]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_512_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_512_2_g_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_512_2_g_d(:prec:=4)
    chi := MakeCharacter_512_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_512_2_g_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_512_2_g_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_512_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2, -4, 6, -4, 1]>,<5,R![98, -84, 34, -8, 1]>],Snew);
    return Vf;
end function;