
// Make newform 847.2.a.l in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_847_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_847_2_a_l();".
// 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_847_2_a_l();".
// 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 := [5, 5, -4, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-3, -1, 1, 0], [0, -3, -1, 1]];
    Rf_basisdens := [1, 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_847_a();"
function MakeCharacter_847_a()
    N := 847;
    order := 1;
    char_gens := [122, 365];
    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;

function MakeCharacter_847_a_Hecke(Kf)
    return MakeCharacter_847_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 := [[0, 1, 0, 0], [-1, 0, -1, 0], [-1, -1, 0, 0], [-1, 0, 0, 0], [0, 0, 0, 0], [0, -1, -2, 2], [1, -1, 3, -1], [1, -2, -1, -1], [-3, 0, -1, -2], [0, 0, 0, -3], [0, 1, 3, -1], [-2, 1, 1, -1], [-1, 0, -3, -2], [5, 0, 6, 0], [-3, -1, -2, 4], [-4, 4, -1, 3], [-6, 2, 2, 1], [-7, 5, 0, 1], [-4, -1, -2, 1], [1, -2, 5, -3], [-4, 2, -3, 5], [2, 1, 2, -2], [-4, 3, -7, 5], [-4, -1, -3, 5], [-6, 0, -2, -5], [-1, 4, 7, -3], [9, -7, 2, 4], [-6, 5, -2, 1], [6, -2, -1, -3], [-9, -4, 5, -2], [-4, 2, -7, -3], [-6, 0, -3, 2], [-12, -2, -8, 0], [-2, -5, 1, -4], [5, -3, 9, 0], [4, 3, -4, 0], [-4, 2, -12, 5], [6, -3, -1, -2], [-4, -6, 1, -1], [-7, -4, 4, 0], [2, -2, 12, -6], [0, 4, 5, -2], [-7, 6, 1, 1], [9, -4, 0, 3], [2, -1, -2, 5], [17, -1, 0, -2], [6, -1, 1, 3], [-9, 6, -5, 5], [-13, 4, -9, -3], [7, -4, 8, -3], [-5, -2, 3, -1], [6, -3, -6, 7], [4, 4, -8, 4], [9, 2, -5, 11], [-15, -1, -3, -6], [8, 0, 10, 0], [-15, 1, -9, -4], [2, -2, 2, 6], [6, -5, -6, -6], [0, 1, -12, 2], [1, 2, 12, -4], [5, -2, 2, -3], [7, 5, 10, 4], [-10, 5, -11, -3], [-15, 8, 1, 6], [-17, 3, 3, -7], [6, -9, -4, 8], [-6, 4, -6, 4], [11, 5, -1, 0], [22, 2, 2, 1], [-13, 1, 6, 0], [2, -2, -7, 0], [-3, -7, 11, -15], [-17, 2, -11, 3], [10, -4, 8, -5], [-20, 1, -14, 2], [15, -3, 12, 10], [-10, -3, -11, 1], [-7, -2, -2, -2], [-23, -2, -14, 1], [2, 12, 2, -3], [3, -3, 7, -9], [27, -3, 4, 4], [-2, 10, 6, -8], [3, 6, 19, -1], [9, 5, -4, -1], [-26, -4, -7, 7], [7, 4, 18, -12], [-12, 2, 5, -10], [-5, 10, -6, 3], [1, 8, 16, 2], [-3, 2, 5, 6], [3, 4, -8, 13], [-10, -1, -15, 12], [11, -9, 9, 2], [-13, -5, -1, 2], [0, -1, 2, -15], [-12, -8, 7, -4], [6, -9, -2, -9], [-1, 8, -11, 7], [-23, -1, -18, 6], [11, -14, -1, -4], [15, -8, 0, 8], [16, 1, 13, 6], [6, 2, 2, 9], [0, -1, -5, 9], [-6, -1, -4, -1], [-3, -10, 4, 1], [17, 4, -10, 12], [20, 7, -1, -4], [10, 13, 15, -9], [3, 0, -1, 10], [-11, -10, 3, -7], [10, -13, 1, 10], [7, 11, 0, 7], [-18, 4, -1, -1], [10, -9, 3, -4], [2, -7, -14, -2], [-2, 0, -5, 8], [-2, -5, 0, -12], [8, -16, 11, -6], [-1, 10, 10, -4], [-20, -7, -16, 1], [3, -11, 17, -10], [-11, -1, -22, 0], [15, 2, 35, -2], [28, -11, -2, -1], [16, 4, -1, -7], [12, 0, -3, -4], [-13, -1, -13, 0], [-13, -3, -10, -4], [13, -8, -11, -3], [17, -10, 5, 9], [21, 11, 16, -7], [-13, 6, 15, -5], [-6, 11, -22, 18], [6, 5, 4, 3], [27, 1, -2, 1], [-22, 12, -15, -3], [16, 2, 32, -1], [-27, 9, 5, -7], [13, 14, 20, -8], [-8, 2, -21, 1], [-12, -3, 6, -10], [26, -8, -10, 0], [-4, 11, 7, -11], [-3, 5, -3, 9], [-13, 7, 0, -9], [-16, 0, -32, -2], [-17, -7, 2, -3], [27, 1, 20, -13], [-16, -1, -6, 10], [4, 19, 1, -6], [-3, 5, -7, 2], [14, 4, 8, -17], [-1, -6, -22, 6], [5, -1, 4, 9], [-7, -9, 11, -15], [20, 3, 28, -10], [-1, 18, -13, 15], [-7, 16, 11, 4], [5, -18, -13, 4], [-11, -4, 1, 9], [20, 0, 2, 5], [7, -8, 11, -2], [-9, 11, -19, 20], [23, -18, 15, 6], [31, -1, 21, 14]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_847_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_847_2_a_l();".
// 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_847_2_a_l(:prec:=4)
    chi := MakeCharacter_847_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_847_2_a_l();".
// 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_847_2_a_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_847_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![5, 5, -4, -2, 1]>,<3,R![-1, 1, 1]>],Snew);
    return Vf;
end function;