
// Make newform 847.2.f.u in Magma, downloaded from the LMFDB on 20 April 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_u();".
// 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_u();".
// 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 := [16, 48, 136, 376, 1036, 324, 266, 118, 59, -15, 7, -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, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [24224, 0, 0, 0, 0, 5251, 0, 0, 0, 0, 15, 0], [0, 367616, 0, 0, 0, 0, -16203, 0, 0, 0, 0, -59], [-104932, 0, 0, 0, 0, 10727, 0, 0, 0, 0, 37, 0], [0, -577480, 0, 0, 0, 0, 37657, 0, 0, 0, 0, 133], [-944, -2832, -375640, -22184, -61124, -19116, -15694, 9241, -3481, 885, -413, 59], [-768, -2304, -259493, -18048, -49728, -15552, -12768, 7801, -2832, 720, -336, 48], [52464, 148648, 410968, 1132348, 332228, 290738, 128974, 64487, -16395, 7563, -1093, 1093], [5200, 15600, 1633540, 122200, 336700, 105300, 86450, -53167, 19175, -4875, 2275, -325], [72672, 205904, 569264, 1568504, 485285, 402724, 178652, 89326, -22710, 10583, -1514, 1514], [-229008, -648856, -1793896, -4942756, -1567258, -1269086, -562978, -281489, 71565, -33471, 4771, -4771]];
    Rf_basisdens := [1, 1, 66764, 133528, 33382, 66764, 133528, 33382, 133528, 66764, 66764, 66764];
    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, 0, 0], [0, 0, 0, 1, 0, 0, 0, 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 := [[1, 1, 1, 1, 0, 0, -1, -1, -1, 0, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, -1, -1, 2, -1, -1, 1, -2, 1, 1, -2, 1], [0, 0, 0, 0, 0, 0, -1, 2, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, -2], [2, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0], [4, -1, -1, 4, 0, 0, 1, -4, 1, 0, -4, 0], [0, -2, 0, -6, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 6, 1], [-2, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -2, -1], [4, 0, 0, 4, 2, 2, 0, -4, 0, -2, -4, -2], [0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, -6, 0, -1, 0, 0], [-2, 0, -4, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 4, -2, 0, -5, 0, 0], [0, -1, 0, -6, 0, -1, 0, 0, 0, 0, 0, 0], [10, 0, 0, 10, 2, 2, 0, -10, 0, -2, -10, -2], [0, 0, 0, 0, 0, 0, 2, 4, 0, 2, 0, 0], [-8, 0, -2, 0, -1, 0, 0, 0, 0, 0, 0, 0], [4, -2, -2, 4, 3, 3, 2, -4, 2, -3, -4, -3], [-6, 3, 3, -6, -5, -5, -3, 6, -3, 5, 6, 5], [0, -5, 0, -2, 0, 3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 4], [4, 0, -2, 0, 6, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -8, -1], [0, 0, 0, 0, 0, 0, -4, -8, 0, 4, 0, 0], [8, 0, 2, 0, -2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 2, 0, -3, 0, 0], [0, -6, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -4, 0, 0, 4, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -4, 0, 2, -2], [0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, -7], [8, -2, -2, 8, 2, 2, 2, -8, 2, -2, -8, -2], [12, 0, 0, 12, 4, 4, 0, -12, 0, -4, -12, -4], [0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 6, -3], [0, 0, 0, 0, 0, 0, 0, 0, -6, 0, -6, 1], [0, 0, 0, 0, 0, 0, -4, -4, 0, 3, 0, 0], [0, -2, 0, 14, 0, 3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -4, 4, 0, 0, 0, 0], [-12, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [6, 0, 7, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, 4, 4, 0, 4, 4, -4, 0, -4, -4, 0, -4], [0, 0, 0, 0, 0, 0, 0, 0, 3, 0, -4, 8], [0, -2, 0, -16, 0, 2, 0, 0, 0, 0, 0, 0], [-4, 0, 0, -4, 3, 3, 0, 4, 0, -3, 4, -3], [12, -6, -6, 12, 2, 2, 6, -12, 6, -2, -12, -2], [0, 0, 0, 0, 0, 0, 0, 0, -4, 0, -8, 4], [-6, 0, 7, 0, -1, 0, 0, 0, 0, 0, 0, 0], [-12, -1, -1, -12, 2, 2, 1, 12, 1, -2, 12, -2], [0, 4, 0, -10, 0, 0, 0, 0, 0, 0, 0, 0], [16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, -2, 0, 2, 0, 0], [0, 4, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0], [-8, 4, 4, -8, -4, -4, -4, 8, -4, 4, 8, 4], [0, 0, 0, 0, 0, 0, -6, 4, 0, -6, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -10, 0, 0, 2], [0, -3, 0, -10, 0, 5, 0, 0, 0, 0, 0, 0], [16, 0, 6, 0, 2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 10, -3], [0, 0, 0, 0, 0, 0, -2, 4, 0, -3, 0, 0], [8, 2, 2, 8, -1, -1, -2, -8, -2, 1, -8, 1], [6, 0, -6, 0, 5, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 8, 0, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 8, 6, 0, -2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 10, -5], [16, 0, -2, 0, 7, 0, 0, 0, 0, 0, 0, 0], [0, 4, 0, 10, 0, -2, 0, 0, 0, 0, 0, 0], [0, -5, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0], [-12, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -4, 2, 0, -7, 0, 0], [0, -7, -7, 0, 6, 6, 7, 0, 7, -6, 0, -6], [0, -6, 0, 16, 0, 1, 0, 0, 0, 0, 0, 0], [-14, 0, -10, 0, 2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 8, -2, 0, 4, 0, 0], [-2, -7, -7, -2, 9, 9, 7, 2, 7, -9, 2, -9], [-6, 0, 3, 0, -5, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 20, 6], [2, -4, -4, 2, -10, -10, 4, -2, 4, 10, -2, 10], [0, -2, 0, 12, 0, 5, 0, 0, 0, 0, 0, 0], [-12, 0, -8, 0, 4, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -10, 0, 2, -3], [-6, -14, -14, -6, 3, 3, 14, 6, 14, -3, 6, -3], [0, 0, 0, 0, 0, 0, 0, -4, 0, -4, 0, 0], [18, 0, 3, 0, 3, 0, 0, 0, 0, 0, 0, 0], [2, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -3, 8, 0, 8, 0, 0], [0, 0, 0, 0, 0, 0, -8, 12, 0, 4, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 8, 0, -10, -11], [0, 4, 0, -14, 0, -2, 0, 0, 0, 0, 0, 0], [0, 6, 0, -4, 0, 6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -4, 0, -8, 4], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -12, 3], [0, 14, 0, 12, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 10, -4, 0, -6, 0, 0], [-36, -4, -4, -36, 0, 0, 4, 36, 4, 0, 36, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -8, -8], [0, 0, 0, 12, 0, -4, 0, 0, 0, 0, 0, 0], [8, 2, 2, 8, -2, -2, -2, -8, -2, 2, -8, 2], [0, 0, 0, 0, 0, 0, 0, 0, 6, 0, -8, 2], [8, 0, 16, 0, -8, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, -28, 0, 3, 0, 0], [0, -3, 0, -18, 0, 5, 0, 0, 0, 0, 0, 0], [-10, 0, -15, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 6, -8, 0, 6, 0, 0], [0, -9, 0, -30, 0, -1, 0, 0, 0, 0, 0, 0], [4, -8, -8, 4, -4, -4, 8, -4, 8, 4, -4, 4], [0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 24, -6], [-16, 0, 4, 0, -2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -12, 10], [0, 6, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -14, 0, 6, 11], [0, 0, 0, 0, 0, 0, -5, 4, 0, -6, 0, 0], [8, -5, -5, 8, 10, 10, 5, -8, 5, -10, -8, -10], [0, -10, 0, -4, 0, -9, 0, 0, 0, 0, 0, 0], [10, 0, 8, 0, 10, 0, 0, 0, 0, 0, 0, 0], [4, 0, -8, 0, -5, 0, 0, 0, 0, 0, 0, 0], [-12, 2, 2, -12, 2, 2, -2, 12, -2, -2, 12, -2], [0, 0, 0, 0, 0, 0, -1, 26, 0, 7, 0, 0], [12, 0, 8, 0, -4, 0, 0, 0, 0, 0, 0, 0], [-28, 7, 7, -28, -10, -10, -7, 28, -7, 10, 28, 10], [0, 0, 0, 0, 0, 0, 0, 0, -14, 0, 4, 2], [0, 0, 0, 0, 0, 0, 2, 12, 0, -7, 0, 0], [0, 3, 0, -24, 0, -6, 0, 0, 0, 0, 0, 0], [-28, 0, -7, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -11, 0, 14, 0, 13, 0, 0, 0, 0, 0, 0], [0, 8, 8, 0, 8, 8, -8, 0, -8, -8, 0, -8], [0, 0, 0, 0, 0, 0, 4, -8, 0, -12, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 7], [10, 4, 4, 10, 0, 0, -4, -10, -4, 0, -10, 0], [6, 5, 5, 6, 5, 5, -5, -6, -5, -5, -6, -5], [-18, 0, -7, 0, 9, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -14, 0, 2, -2], [0, 0, 0, 0, 0, 0, 10, 20, 0, -6, 0, 0], [0, 0, 0, 0, 0, 0, 0, 32, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 4, 28, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 10, 0, -20, -6], [0, -4, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0], [22, 14, 14, 22, -5, -5, -14, -22, -14, 5, -22, 5], [-14, -4, -4, -14, -2, -2, 4, 14, 4, 2, 14, 2], [0, 12, 0, -16, 0, -13, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -15, 0, -8, 2], [0, 0, 0, 0, 0, 0, -1, -6, 0, 7, 0, 0], [-30, 0, 3, 0, -5, 0, 0, 0, 0, 0, 0, 0], [-12, 0, -7, 0, 2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -16, -4, 0, 8, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -10, 0, -4, 6], [-4, 0, -2, 0, -7, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 4, -12], [0, 8, 0, 4, 0, 12, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 10, 4, 0, 10, 0, 0], [-28, -6, -6, -28, 6, 6, 6, 28, 6, -6, 28, -6], [0, 0, 0, 0, 0, 0, 0, -40, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 4, 30, 0, -8, 0, 0], [-14, 7, 7, -14, -9, -9, -7, 14, -7, 9, 14, 9], [0, 0, 0, 0, 0, 0, -9, 18, 0, 7, 0, 0], [-14, 0, -6, 0, 3, 0, 0, 0, 0, 0, 0, 0], [0, 4, 0, -32, 0, -12, 0, 0, 0, 0, 0, 0], [-14, 0, -4, 0, -10, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 21, 0, 4, -6], [-8, 2, 2, -8, -1, -1, -2, 8, -2, 1, 8, 1], [0, -17, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0], [-8, 0, -2, 0, 6, 0, 0, 0, 0, 0, 0, 0], [0, 7, 0, 18, 0, -1, 0, 0, 0, 0, 0, 0]];
    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_u();".
// 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_u(:prec:=12)
    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_u();".
// 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_u( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_847_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![4096, -2560, 2624, -1768, 1441, -214, 273, -44, 69, -20, 9, -2, 1]>,<3,R![16, 48, 136, 376, 1036, 324, 266, 118, 59, -15, 7, -1, 1]>,<13,R![16777216, 524288, 2113536, 393728, 284688, -115264, 38136, -8768, 2692, -416, 62, -8, 1]>],Snew);
    return Vf;
end function;