
// Make newform 805.2.a.f in Magma, downloaded from the LMFDB on 12 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_805_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_805_2_a_f();".
// 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_805_2_a_f();".
// 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, -2, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-2, 0, 1]];
    Rf_basisdens := [1, 1, 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_805_a();"
function MakeCharacter_805_a()
    N := 805;
    order := 1;
    char_gens := [162, 346, 281];
    v := [1, 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;

function MakeCharacter_805_a_Hecke(Kf)
    return MakeCharacter_805_a();
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 := [[-1, 0, -1], [-1, 2, -1], [-1, 0, 0], [-1, 0, 0], [-2, 0, 1], [4, -3, 4], [2, -2, -1], [-2, 1, 1], [-1, 0, 0], [-6, 2, -4], [-2, 1, -1], [0, 0, -3], [1, -4, -2], [1, 0, 1], [6, -5, 1], [-3, 5, 5], [-8, 3, 0], [-1, -5, 1], [8, -5, 6], [-8, 1, -7], [-5, 5, -1], [-9, 3, -6], [-6, -7, 4], [0, 4, -3], [-5, 8, -8], [6, -5, -1], [5, -6, 0], [-11, 4, -8], [-3, 3, 1], [-13, -5, 3], [10, -1, 9], [12, 0, 2], [3, -10, 8], [5, -5, 9], [-12, 4, -1], [13, -6, 6], [-13, 3, 0], [-4, 12, -10], [-1, 16, -8], [-6, 8, -5], [-6, -2, 5], [3, 2, -4], [-14, 11, -4], [8, 5, -4], [4, 9, 3], [1, -18, 8], [2, 1, 8], [16, -6, -1], [-7, -5, -1], [-9, 14, -3], [5, 1, 6], [-6, 5, 4], [-10, 13, -4], [-16, 6, 7], [-16, 16, -14], [-7, 9, -17], [0, 19, -8], [-9, 8, -14], [-5, -6, 4], [3, 1, 8], [16, -15, 19], [6, 7, 8], [1, 6, -17], [-8, 4, 2], [-5, -1, -20], [11, -5, 15], [1, -2, 2], [2, -11, -1], [-25, -1, -1], [19, -9, 21], [7, -3, 12], [-1, -14, 9], [14, -4, -3], [3, 2, -13], [-6, 5, 0], [0, 6, 9], [-2, -2, 16], [3, 5, -14], [6, -17, 16], [22, -5, 14], [5, -15, 13], [4, -10, -3], [-10, -3, 7], [-23, -2, -5], [2, 11, -18], [-7, -7, 1], [-3, 16, -6], [2, 16, -3], [22, -5, 23], [-7, 8, -3], [-4, 1, -10], [-11, 9, -13], [8, -27, 15], [-2, 4, 11], [-1, -3, -5], [-1, 12, 11], [5, 10, -11], [19, -2, 6], [-33, 10, -2], [11, 7, -12], [-11, 18, 0], [7, -7, -22], [1, 1, 17], [-19, 6, -13], [-24, 1, -9], [13, -17, 16], [5, 0, -3], [31, -8, 14], [7, -17, 10], [7, -8, -11], [-15, 12, 4], [-1, -2, -3], [-10, -5, 14], [-1, -20, 0], [-30, 4, 11], [-7, -4, 17], [-2, 1, -19], [-28, -9, 4], [-31, 4, -9], [2, -6, -24], [-23, 23, -26], [27, -7, -13], [34, -12, 26], [13, -20, -23], [-28, 1, 1], [6, 14, 6], [3, -4, -19], [2, -16, 0], [5, -12, -2], [-9, 33, -1], [-3, -3, 17], [14, 5, -3], [1, -4, -24], [3, -7, -22], [13, -31, 8], [18, -4, 22], [26, 18, -4], [-36, 24, -32], [4, -2, 7], [15, -19, -8], [3, -10, -10], [-18, 10, -35], [-3, 0, -28], [6, -27, 8], [19, 21, -3], [-23, -15, 3], [18, -19, -12], [-11, 5, -15], [-13, -17, -7], [-4, 33, -4], [-23, 26, -19], [-14, -4, -13], [-13, 31, -19], [-6, -16, 20], [9, -14, 13], [-18, -11, 20], [-32, -1, 10], [16, 15, -4], [-13, 23, -19], [-21, 23, -23], [-34, 1, -18], [-3, -5, -13], [20, 10, -1], [-7, 13, 11], [24, -12, -22], [16, 6, 11], [2, 0, 9], [10, 8, 12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_805_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_805_2_a_f();".
// 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_805_2_a_f(:prec:=3)
    chi := MakeCharacter_805_a();
    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_805_2_a_f();".
// 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_805_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_805_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, -1, 2, 1]>,<3,R![7, -7, 0, 1]>,<11,R![7, 14, 7, 1]>],Snew);
    return Vf;
end function;