
// Make newform 512.3.d.e in Magma, downloaded from the LMFDB on 03 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_512_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_512_d_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_3_d_e();".
// 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_3_d_e();".
// 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 := [9, 0, -4, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 7, 0, -1], [0, 2, 0, -2], [-4, 0, 2, 0]];
    Rf_basisdens := [1, 3, 3, 1];
    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_512_d();"
function MakeCharacter_512_d()
    N := 512;
    order := 2;
    char_gens := [511, 5];
    v := [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_512_d_Hecke();"
function MakeCharacter_512_d_Hecke(Kf)
    N := 512;
    order := 2;
    char_gens := [511, 5];
    char_values := [[-1, 0, 0, 0], [-1, 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 := 3;
    raw_aps := [[0, 0, 0, 0], [2, -1, 0, 0], [0, 0, 0, 1], [0, 0, -1, -2], [6, 1, 0, 0], [0, 0, 4, 1], [8, -4, 0, 0], [14, 5, 0, 0], [0, 0, -5, -2], [0, 0, 8, -5], [0, 0, -10, 4], [0, 0, -4, 11], [-14, 16, 0, 0], [50, -1, 0, 0], [0, 0, 16, -8], [0, 0, -20, -1], [10, 19, 0, 0], [0, 0, 12, 9], [22, 25, 0, 0], [0, 0, 13, -14], [-28, -4, 0, 0], [0, 0, 42, 4], [-38, 11, 0, 0], [36, 12, 0, 0], [0, -4, 0, 0], [0, 0, -12, -13], [0, 0, 1, -6], [-58, -7, 0, 0], [0, 0, -48, -9], [-42, -48, 0, 0], [0, 0, -10, 36], [-138, -15, 0, 0], [-82, 32, 0, 0], [-46, 47, 0, 0], [0, 0, 32, -3], [0, 0, -13, -18], [0, 0, -60, 9], [18, -33, 0, 0], [0, 0, 19, -26], [0, 0, -20, -19], [-54, -21, 0, 0], [0, 0, 40, -47], [0, 0, 30, -12], [-32, 28, 0, 0], [0, 0, 40, 53], [0, 0, -25, -2], [106, -53, 0, 0], [0, 0, 10, -36], [-110, 39, 0, 0], [0, 0, 120, 9], [196, -20, 0, 0], [0, 0, -62, 52], [-344, -20, 0, 0], [-166, 75, 0, 0], [70, 16, 0, 0], [0, 0, 45, 66], [0, 0, 40, 11], [0, 0, -40, 40], [0, 0, -4, -13], [300, 12, 0, 0], [222, 21, 0, 0], [0, 0, -104, 5], [302, 53, 0, 0], [0, 0, -123, -54], [-190, -112, 0, 0], [0, 0, 120, -41], [486, -39, 0, 0], [382, -32, 0, 0], [-38, 51, 0, 0], [0, 0, -48, 63], [250, -80, 0, 0], [0, 0, 93, 18], [0, 0, -60, -16], [0, 0, 20, 83], [554, 11, 0, 0], [0, 0, -4, -136], [0, 0, -100, -53], [0, 0, -80, 43], [136, 124, 0, 0], [274, 48, 0, 0], [86, -23, 0, 0], [0, 0, 80, 77], [0, 0, -162, -36], [88, -180, 0, 0], [0, 0, 137, 82], [78, -187, 0, 0], [-64, -148, 0, 0], [-478, 48, 0, 0], [0, 0, -40, -153], [0, 0, -10, -36], [30, -283, 0, 0], [0, 0, 30, 36], [0, 0, 35, 118], [-670, 47, 0, 0], [218, 155, 0, 0], [0, 0, -105, -42], [0, 0, -220, 61], [254, -64, 0, 0], [-522, -63, 0, 0], [0, 0, -20, 49], [-122, -47, 0, 0], [0, 0, 116, 133], [-770, -51, 0, 0], [-514, -64, 0, 0], [110, 5, 0, 0], [142, -128, 0, 0], [-298, -7, 0, 0], [242, -64, 0, 0], [0, 0, -223, 50], [196, -260, 0, 0], [0, 0, -200, -64], [0, 0, 80, 173], [220, 124, 0, 0], [470, -103, 0, 0], [0, 0, -45, 110], [-80, 28, 0, 0], [-170, -263, 0, 0], [0, 0, 345, -6], [0, 0, 136, 35], [-6, 267, 0, 0], [0, 0, -88, -127], [-400, -116, 0, 0], [0, 0, -20, -133], [18, -137, 0, 0], [-486, 99, 0, 0], [0, 0, 108, 133], [0, 0, 60, -177], [0, 0, -62, -188], [0, 0, -39, -62], [0, 0, 24, -9], [770, -49, 0, 0], [0, 0, 135, 30], [0, 0, 42, -140], [0, 0, 0, -51], [114, 144, 0, 0], [336, 204, 0, 0], [0, 0, -356, 63], [186, -45, 0, 0], [0, 0, -140, -99], [-670, 80, 0, 0], [70, -143, 0, 0], [0, 0, -220, 163], [0, 0, 385, -30], [-242, 253, 0, 0], [0, 0, -108, -171], [0, 0, 195, -90], [0, 0, -100, 263], [78, -192, 0, 0], [794, -61, 0, 0], [0, 0, 26, 140], [0, 0, 360, -57], [270, -96, 0, 0], [-562, -67, 0, 0], [0, 0, -295, 98], [130, 175, 0, 0], [0, 0, -180, 192], [0, 0, 55, 102], [-400, 188, 0, 0], [-668, 140, 0, 0], [0, 0, 112, -1], [910, 277, 0, 0], [-578, -224, 0, 0], [0, 0, -245, -106], [-686, 23, 0, 0], [-1352, 76, 0, 0], [0, 0, -119, -110], [0, 0, 208, 32], [0, 0, -180, 159]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_512_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_512_3_d_e();".
// 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_3_d_e(:prec:=4)
    chi := MakeCharacter_512_d();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    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_3_d_e();".
// 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_3_d_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_512_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![-6, -4, 1]>],Snew);
    return Vf;
end function;