
// Make newform 847.2.f.s in Magma, downloaded from the LMFDB on 01 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_847_f();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_847_f_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_847_2_f_s();".
// 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_f_s();".
// 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 := [25, 25, 10, -5, 21, -11, 6, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [375, 43855, -84434, 40069, -18164, 7489, -2159, 555], [35160, -68120, 27673, -14943, 9063, -6608, -1002, -970], [-13875, -13500, 38305, -81659, 28414, -12059, 4159, -1604], [-90925, -19395, 35785, -82042, 21007, -19912, 2252, -2052], [-12035, -4035, 72554, -82441, 50856, -17466, 6691, -2667], [66675, 54640, 22635, 59219, -26434, 21519, -1464, 4024]];
    Rf_basisdens := [1, 1, 94655, 94655, 94655, 94655, 94655, 94655];
    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_f();"
function MakeCharacter_847_f()
    N := 847;
    order := 5;
    char_gens := [122, 365];
    v := [5, 2];
    // 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_847_f_Hecke();"
function MakeCharacter_847_f_Hecke(Kf)
    N := 847;
    order := 5;
    char_gens := [122, 365];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 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 := 2;
    raw_aps := [[0, 1, 0, 0, 0, 0, 0, 0], [1, 0, 0, -1, 0, 0, 0, 0], [1, 0, -1, -1, 0, 0, -1, -1], [0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [2, -3, -2, 0, 2, 0, 0, 2], [-2, 0, -1, -1, 0, 1, 2, 1], [-1, 1, 0, 1, 2, -1, 2, 0], [-2, 2, -1, 0, 0, 0, -1, -2], [0, -3, 0, 0, 3, 0, 0, 0], [3, -2, -3, 0, 1, 0, 0, 1], [0, 2, 2, -1, -1, -1, -1, -1], [-3, 2, 0, 3, 0, -2, 2, 0], [1, 0, -6, 0, 0, 0, -6, 0], [2, 4, 0, -2, -1, -4, 1, 0], [-1, -1, 1, 4, -3, 0, 0, -3], [0, 1, 6, -2, 1, -2, -2, -2], [-7, 0, 7, 7, 0, -1, 7, -5], [-2, -1, -2, 0, 1, -1, -2, 1], [4, 0, 1, 1, 0, -3, -4, -2], [0, 3, -4, -3, -5, 2, -3, 2], [-2, 3, 2, 2, -2, 0, 0, -2], [3, 0, 4, 4, 0, -5, -3, -3], [-1, -5, -3, 0, 1, -1, -3, 5], [-2, -5, 2, 6, 5, 0, 0, 5], [-7, 7, 7, -1, -3, 0, 0, -3], [0, -11, -9, -2, 4, 7, -2, 7], [-2, -1, 0, 2, -5, 1, -4, 0], [-7, -3, 1, 0, -2, 2, 1, 3], [-5, -2, 0, 5, -4, 2, 14, 0], [3, 0, 4, 4, 0, 3, -3, -2], [3, 2, 3, 0, 0, 0, 3, -2], [4, 0, -12, -12, 0, 0, -4, -2], [0, 1, -2, 1, 4, -5, 1, -5], [-4, 0, -5, -5, 0, 0, 4, 3], [-4, 0, 0, 4, -3, 0, 8, 0], [12, 5, 0, -12, 2, -5, -8, 0], [-1, 1, 1, -6, 2, 0, 0, 2], [-1, -5, 1, -4, -1, 0, 0, -1], [4, 0, 0, -4, 4, 0, -11, 0], [-12, -6, 0, 12, -2, 6, 10, 0], [5, 0, 0, 0, 0, -2, -5, 4], [0, 5, 7, -1, 1, -6, -1, -6], [9, 0, -9, -9, 0, -3, -9, 4], [-4, 5, 2, 0, -1, 1, 2, -5], [17, 2, 0, 0, 1, -1, 0, -2], [-1, -4, 1, 6, 3, 0, 0, 3], [5, 5, 0, -5, 6, -5, 4, 0], [0, -7, -13, -9, 3, 4, -9, 4], [8, 1, -8, -7, 3, 0, 0, 3], [-3, -1, 3, -5, -1, 0, 0, -1], [-6, -7, 0, 6, 3, 7, 12, 0], [-12, 4, 8, 0, 4, -4, 8, -4], [-5, 9, 5, -9, -11, 0, 0, -11], [0, 5, 15, 3, -6, 1, 3, 1], [2, 0, -10, 0, 0, 0, -10, 0], [6, 0, -15, -15, 0, -4, -6, 1], [0, 8, 2, 2, -6, -2, 2, -2], [6, 1, -6, 6, -6, 0, 0, -6], [12, 0, 0, 0, 0, -2, -12, -1], [12, 4, 0, -12, -2, -4, -11, 0], [0, -1, 5, 2, 3, -2, 2, -2], [3, 4, -10, 0, 5, -5, -10, -4], [11, -3, 0, -11, 5, 3, -1, 0], [16, 0, -15, -15, 0, 6, -16, 8], [3, -10, -3, 17, 7, 0, 0, 7], [10, -8, -4, 0, 9, -9, -4, 8], [0, 0, -6, -6, -4, 4, -6, 4], [12, 0, -11, -11, 0, 0, -12, -5], [2, -1, 0, -2, -2, 1, 20, 0], [-19, 0, 6, 0, -1, 1, 6, 0], [0, 2, 2, -7, 0, -2, -7, -2], [0, 8, 3, -11, -15, 7, -11, 7], [6, 3, 11, 0, 2, -2, 11, -3], [-2, 0, 10, 10, 0, -5, 2, -4], [-14, 1, 14, 20, -2, 0, 0, -2], [0, -13, -15, -12, 10, 3, -12, 3], [1, -1, -11, 0, 3, -3, -11, 1], [5, 0, -7, -7, 0, -2, -5, -2], [14, -3, -14, -23, 1, 0, 0, 1], [0, 3, 2, 0, -12, 12, 2, -3], [-7, -9, 0, 7, -3, 9, 4, 0], [-4, -7, 4, 27, 4, 0, 0, 4], [0, 18, 2, -6, -8, -10, -6, -10], [16, -1, -19, 0, 6, -6, -19, 1], [4, -1, 0, -4, 5, 1, -13, 0], [-7, 11, 7, 26, -7, 0, 0, -7], [-11, 0, -7, -7, 0, 12, 11, -4], [17, -10, -5, 0, 2, -2, -5, 10], [1, -3, -6, 0, -10, 10, -6, 3], [15, 0, 1, 1, 0, 2, -15, 8], [-8, 0, 3, 3, 0, -6, 8, -2], [8, 13, 0, -8, 4, -13, -11, 0], [0, 13, -10, -15, -12, -1, -15, -1], [0, -11, -11, -9, 2, 9, -9, 9], [-1, -2, 0, 1, 5, 2, -12, 0], [-2, -15, 0, 2, -1, 15, 2, 0], [0, -4, 12, -7, -4, 8, -7, 8], [8, 0, -6, -6, 0, 9, -8, 9], [11, 1, -11, -1, 7, 0, 0, 7], [-18, -6, 0, 18, 1, 6, -5, 0], [0, 10, 11, -1, 4, -14, -1, -14], [0, -16, 0, 15, 8, 0, 0, 8], [13, -6, 0, -13, -1, 6, 3, 0], [-4, 9, -2, 0, 2, -2, -2, -9], [-5, 0, 0, 0, 0, 9, 5, -1], [0, 0, 6, 4, -1, 1, 4, 1], [7, 1, -4, 0, -10, 10, -4, -1], [-27, 0, 17, 17, 0, 12, 27, 4], [0, -11, 20, -1, 4, 7, -1, 7], [-15, 22, 15, 10, -9, 0, 0, -9], [-1, -10, 0, 1, 0, 10, 4, 0], [-14, 7, 3, 0, 10, -10, 3, -7], [-1, 10, 0, 1, -13, -10, -9, 0], [0, 4, -7, 0, 7, -11, 0, -11], [1, -1, 0, -1, 4, 1, 17, 0], [-7, 0, 10, 10, 0, -4, 7, -9], [-14, 5, 14, -2, 2, 0, 0, 2], [0, -8, 2, 5, 8, 0, 5, 0], [2, -12, 0, 0, -5, 5, 0, 12], [-3, 6, 11, 0, 16, -16, 11, -6], [-10, 14, 10, -1, -4, 0, 0, -4], [-4, 0, 20, 20, 0, -1, 4, 7], [-14, 10, 17, 0, 11, -11, 17, -10], [-22, 1, 22, 11, 0, 0, 0, 0], [35, 2, 0, -35, -2, -2, -20, 0], [-30, 0, 28, 28, 0, -1, 30, -11], [0, 11, -16, 1, -7, -4, 1, -4], [15, 4, -3, 0, 0, 0, -3, -4], [0, 1, -13, -13, 0, -1, -13, -1], [10, 1, -10, -13, -4, 0, 0, -4], [24, 0, -13, -13, 0, 3, -24, 8], [-5, 9, 0, 5, -10, -9, -12, 0], [16, -18, -16, -21, 7, 0, 0, 7], [-15, 11, 15, -13, -5, 0, 0, -5], [-16, 18, 22, 0, 11, -11, 22, -18], [-4, 3, 0, 4, 5, -3, -2, 0], [29, 0, -27, -27, 0, -1, -29, -1], [7, 0, -22, -22, 0, -3, -7, 12], [-16, 0, -16, -16, 0, 1, 16, -2], [5, 7, 0, -5, -9, -7, -32, 0], [0, -22, 13, 20, 8, 14, 20, 14], [-21, -1, 21, 8, -1, 0, 0, -1], [-6, 7, 6, -12, -10, 0, 0, -10], [0, -8, -26, 10, 0, 8, 10, 8], [-7, -11, 0, 7, 11, 11, 11, 0], [0, 0, 3, 3, 0, -9, 0, -5], [13, -9, 0, 0, 7, -7, 0, 9], [16, 2, -32, 0, 0, 0, -32, -2], [19, 0, -17, -17, 0, -3, -19, -7], [20, 13, 0, -20, -1, -13, 7, 0], [-10, -10, -6, 0, 1, -1, -6, 10], [-1, -6, 0, 1, 19, 6, -3, 0], [0, -3, -3, -7, -2, 5, -7, 5], [-6, 0, 14, 14, 0, -17, 6, 4], [-22, 12, 22, 1, -6, 0, 0, -6], [1, 0, -5, -5, 0, -9, -1, 1], [18, 0, -7, -7, 0, -15, -18, -9], [-28, 13, 28, 20, -10, 0, 0, -10], [12, 0, 1, 1, 0, -15, -12, -18], [-18, -4, 11, 0, -16, 16, 11, 4], [0, 22, 5, -13, -4, -18, -13, -18], [12, 9, -1, 0, -4, 4, -1, -9], [-2, 5, 0, 2, 0, -5, -18, 0], [11, 6, -11, -7, 2, 0, 0, 2], [0, -9, 9, 19, 20, -11, 19, -11], [8, -6, 15, 0, 18, -18, 15, 6], [0, 15, 31, 21, -14, -1, 21, -1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_847_f_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_847_2_f_s();".
// 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_f_s(:prec:=8)
    chi := MakeCharacter_847_f();
    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_f_s();".
// 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_f_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_847_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![25, 25, 10, -5, 21, -11, 6, -1, 1]>,<3,R![1, -2, 4, -3, 1]>,<13,R![841, -1015, 2456, -735, 611, 105, -4, -5, 1]>],Snew);
    return Vf;
end function;