
// Make newform 464.4.k.a in Magma, downloaded from the LMFDB on 15 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_464_k();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_464_k_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_464_4_k_a();".
// 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_464_4_k_a();".
// 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, 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_464_k();"
function MakeCharacter_464_k()
    N := 464;
    order := 4;
    char_gens := [175, 117, 321];
    v := [2, 4, 3];
    // 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_464_k_Hecke();"
function MakeCharacter_464_k_Hecke(Kf)
    N := 464;
    order := 4;
    char_gens := [175, 117, 321];
    char_values := [[-1, 0], [1, 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 := 4;
    raw_aps := [[0, 0], [0, 0], [0, -4], [0, 0], [0, 0], [0, 18], [-99, -99], [0, 0], [0, 0], [-142, -65], [0, 0], [-305, 305], [-351, 351], [0, 0], [0, 0], [-572, 0], [0, 0], [-649, -649], [0, 0], [0, 0], [-845, 845], [0, 0], [0, 0], [-923, -923], [611, -611], [1269, 1269], [0, 0], [0, 0], [0, -1460], [-1665, 1665], [0, 0], [0, 0], [2191, 2191], [0, 0], [0, -3514], [0, 0], [-1819, 1819], [0, 0], [0, 0], [0, -4082], [0, 0], [3942, 0], [0, 0], [2645, 2645], [-1174, 0], [0, 0], [0, 0], [0, 0], [0, 0], [4537, -4537], [598, 0], [0, 0], [0, -5272], [0, 0], [8096, 0], [0, 0], [-2367, 2367], [0, 0], [-1316, 0], [-7430, 0], [0, 0], [-2983, -2983], [0, 0], [0, 0], [8712, 0], [7475, -7475], [0, 0], [-6391, 6391], [0, 0], [-8964, 0], [0, -3298], [0, 0], [0, 0], [-12922, 0], [0, 0], [0, 0], [-7483, 7483], [-9614, 0], [2398, 0], [3887, 3887], [0, 0], [12151, -12151], [0, 0], [-11107, 11107], [0, 0], [0, 0], [2997, 2997], [0, 16506], [-8671, 8671], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-17996, 0], [0, -23738], [0, 0], [15191, 15191], [0, 0], [0, -24836], [0, 0], [11263, -11263], [0, 0], [-12025, 12025], [0, 0], [0, 15502], [0, 0], [3509, 3509], [0, 0], [0, 19548], [5499, -5499], [0, 0], [0, 0], [-6989, -6989], [0, 0], [0, 0], [-17185, 17185], [0, 0], [22068, 0], [0, -34632], [15471, -15471], [0, 0], [0, 0], [0, 20030], [0, 36810], [0, 0], [0, 0], [-14993, 14993], [0, 0], [0, 0], [0, 0], [-6265, -6265], [27320, 0], [-15947, 15947], [-11345, -11345], [0, 0], [29115, 29115], [-8217, 8217], [0, 0], [0, 1850], [0, 0], [0, 0], [-9287, -9287], [0, 0], [-32923, 32923], [-20056, 0], [0, 0], [0, 0], [-29844, 0], [-29479, -29479], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-47480, 0], [0, -51946], [0, -48460], [0, 0], [-56758, 0], [0, 0], [0, 0], [-22936, 0], [0, 0], [0, 0], [43525, 43525]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_464_k_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_464_4_k_a();".
// 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_464_4_k_a(:prec:=2)
    chi := MakeCharacter_464_k();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_464_4_k_a();".
// 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_464_4_k_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_464_k();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![0, 1]>],Snew);
    return Vf;
end function;