
// Make newform 841.2.e.a in Magma, downloaded from the LMFDB on 26 May 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_841_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_841_e_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_841_2_e_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_841_2_e_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 := [64, -96, 0, 104, -72, -28, 57, -14, -18, 13, 0, -3, 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], [96, 32, -176, 68, 102, -83, -14, 40, -7, -10, 7, 1], [-288, 256, 96, -196, 38, 59, -58, 4, 7, -10, 5, -1], [-32, 192, -128, -100, 134, -5, -58, 36, 7, -10, 5, -1], [64, -80, 40, 44, -52, 17, 22, -18, 1, 4, -3, 1], [224, 0, -288, 220, 134, -165, 6, 68, -25, -10, 5, -1], [96, 0, -128, 12, 94, -49, -50, 36, 11, -18, 1, 3], [16, -40, 20, 16, -29, 6, 10, -9, 0, 3, -1, 0], [32, -64, 96, -12, -86, 53, 26, -36, 9, 10, -5, 1], [16, 8, -12, 0, 31, -5, -16, 9, 5, -5, 0, 1], [-352, 352, 176, -420, 82, 199, -122, -40, 59, -14, -11, 3], [480, -416, -368, 740, -146, -435, 290, 80, -151, 30, 31, -15]];
    Rf_basisdens := [1, 128, 128, 128, 64, 128, 128, 16, 64, 32, 128, 128];
    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_841_e();"
function MakeCharacter_841_e()
    N := 841;
    order := 14;
    char_gens := [2];
    v := [11];
    // 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_841_e_Hecke();"
function MakeCharacter_841_e_Hecke( : Kf := false)
    N := 841;
    order := 14;
    char_gens := [2];
    char_values := [[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 := ISA(Type(Kf), Fld), 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]];
    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, 2, 0, -1, -1, 1, -1, 0, 0, -1, 0, -1], [0, -1, 0, 0, 0, 1, 1, 0, 1, 0, -1, 0], [0, 0, 1, 1, 0, 1, -1, 1, 1, 1, 1, 0], [0, -1, 0, 2, 0, 1, -1, 2, -1, 0, 1, 2], [0, 1, -1, -1, -2, 1, 1, -1, 0, -1, 0, -1], [-1, -1, 0, 1, -1, -1, -2, 0, 0, 2, 2, -1], [-2, -3, 1, 0, 2, 0, 2, 0, 1, 1, 0, 0], [-1, -1, -2, 1, 1, -1, 1, 1, -1, 0, -1, 2], [0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, -3], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 1, 0, -2, -2, 1, -1, 2, 0, -1], [1, -3, 2, 0, -1, 1, -4, 1, 2, 0, 0, 0], [1, 2, -1, -2, -2, 0, 0, -3, -1, -1, 0, -1], [6, 6, 0, -4, 0, 0, -2, -3, -4, -4, -3, -3], [4, 7, 1, -9, -4, -1, -3, -5, 0, -5, 0, -7], [1, 2, 0, -3, -3, -2, -1, -2, -2, -2, 0, -3], [4, 0, -2, -2, -2, -4, 0, 0, -2, 2, 0, -2], [1, 0, 0, 3, 0, 2, -3, 0, 2, 2, 3, -1], [3, 0, -3, 1, 3, -3, -3, 2, -1, 1, -1, -1], [0, -7, 1, -1, 0, 1, 5, -1, 2, -1, -2, 3], [-2, 2, 2, 3, 2, 2, -1, 2, 0, -2, 2, 0], [7, 6, 1, -9, -7, 3, -1, -4, 1, -7, -7, -7], [1, 1, -4, -3, -3, -3, -9, -1, -3, 0, 3, 0], [4, -2, -3, 3, 0, -1, 5, 1, -1, 1, 3, 0], [0, -8, 0, -1, 5, -4, 4, -4, 0, 0, -4, 4], [3, 2, 3, -2, -4, 3, -1, 2, 8, 5, 2, -3], [8, 3, -3, -1, 0, 1, -3, 3, -6, -5, -2, -1], [1, 0, 3, -3, -1, 1, 1, -4, 5, 3, -3, -5], [0, -3, 0, -2, -2, -2, -2, 0, 2, 2, 0, 1], [0, 0, 0, -5, 4, -4, 0, 0, -4, -5, -4, -1], [10, 3, -1, -7, 4, -1, -5, -1, 0, -1, 0, -1], [-12, -8, -6, 6, 10, -6, 6, -1, 0, 6, -1, 9], [-1, 1, 0, -1, -2, -4, -1, -2, 0, 2, 2, -1], [0, -4, 0, 0, 4, 0, 8, -1, 4, -4, -7, -3], [0, 0, -4, 5, 2, -2, 7, -2, 1, 6, 2, 3], [2, 8, -2, -6, -8, 2, -8, -3, -2, 0, 7, -9], [9, 6, -5, -1, -3, 0, -8, 6, -10, -5, 5, 1], [-10, -7, 3, 13, 4, -1, 1, 3, 0, 3, 0, 9], [0, 1, 0, 0, -8, -3, -11, 6, 3, 6, 3, -4], [4, 5, 1, -4, -6, 2, 0, 0, -3, -1, -4, -6], [4, 3, 0, -7, 4, 0, -4, -3, -3, 0, 0, -7], [2, 8, 2, -9, -6, 0, 0, -12, 2, -5, -6, -9], [-4, 0, 0, 14, 6, 0, 8, 6, -2, 0, 2, 8], [5, -9, -7, 2, 7, 2, 2, 9, -3, 0, 2, 9], [-5, -4, 5, 0, 4, -5, -9, 0, -2, 7, 4, 3], [-5, -5, 2, 7, 9, -3, 1, 5, -5, -4, 3, -2], [-3, -2, -3, 9, 11, 5, 9, 6, -3, 1, 1, 9], [1, 0, -1, -7, -5, 5, 5, -4, 9, -1, 1, -5], [8, -3, -6, -9, -2, -4, -2, -3, -3, -4, -6, -3], [-9, 0, 0, 3, -4, 2, 1, 0, 2, 12, 3, 9], [3, 8, 1, -9, 1, -2, 0, -2, 4, -1, 3, -1], [0, -9, -4, 0, 10, -1, 11, -2, -3, -2, -1, 10], [13, 14, 4, -6, 1, -1, -15, 2, -8, -1, 1, -11], [1, 13, 8, 3, -15, 7, -7, -1, 5, 0, 7, -12], [2, 2, 6, -7, -7, 8, -4, -2, 8, 0, -8, 0], [0, 1, 6, -11, -4, 6, 12, -13, 13, -6, -6, -3], [11, 1, -10, 1, -6, 0, -5, 4, -10, 4, 4, 9], [1, -11, 0, -3, 13, -3, 5, -3, -3, 0, -3, 10], [-11, -13, 3, 4, 4, 3, 7, 4, 6, 4, 8, 8], [1, -3, -2, -2, 5, -5, -2, 3, -4, 2, -2, 2], [4, -1, -8, -14, 2, -7, 9, -14, -1, -4, -7, -2], [2, -1, 0, 6, -2, 6, 6, 0, 6, 4, -4, 7], [2, 2, -2, -8, -2, 0, 0, -8, 0, -2, -2, -6], [7, 10, 1, -3, -13, 5, -3, 6, 1, 1, 1, -5], [6, 8, -6, -7, -8, 6, 3, -6, 2, -8, 2, -4], [13, 8, -4, 9, -3, 2, -1, 6, -6, -2, 4, -11], [-6, -8, 0, -6, 0, 0, 12, -10, 6, 0, -6, 2], [17, 9, -4, -13, -1, 5, -2, -4, 0, -4, 0, -7], [-8, -10, -10, 2, 12, -16, 0, 2, -10, 10, 0, 14], [-4, 13, 3, -14, -2, -2, 0, -4, 3, -3, 0, -6], [18, -4, -2, -7, -4, 1, -16, 2, 2, 1, -2, -5], [-6, -1, 5, -3, 2, -11, -1, -1, 0, -1, 0, -7], [-9, 3, 2, -5, 11, -3, 3, -5, -3, 0, -3, 6], [13, 13, 3, -4, -1, 2, -6, 1, -1, -6, -2, -3], [-8, -5, -4, 3, 4, -6, -10, 7, -7, 6, 4, 9], [-4, 6, 4, 2, -6, -4, 2, -1, 2, 2, -3, 9], [8, -1, -7, 8, 4, 0, -2, 18, -9, -7, 2, 12], [-1, 18, 10, -4, -7, 7, -11, -8, 6, 1, -1, -7], [0, 12, 8, 1, 0, -1, -8, -4, -4, -1, 8, -7], [2, -2, 0, 3, 18, -6, -14, 0, -6, 1, -4, 3], [10, -3, -1, 3, -12, -1, -13, 3, -2, 3, 6, -9], [8, 0, -6, 4, -2, -6, -12, 1, -2, 4, 1, -1], [-13, -13, 6, 23, 17, -1, -5, 7, 5, 12, 1, 6], [-12, -1, 0, 5, 2, -8, -14, 3, -3, 8, 0, -1], [-5, -5, -12, -3, -5, -5, 7, -7, -3, 4, 5, 2], [-13, 8, 11, -1, 7, 1, 15, -4, 11, -5, -5, -11], [20, 14, 8, -11, -14, 8, -9, -2, 6, -2, -2, -14], [2, -4, -14, 15, 2, 4, -4, 12, -18, -5, 6, 15], [-10, -3, 0, -6, 8, -1, -11, 0, -1, 4, 3, -2], [-6, -10, 0, 4, 6, -12, 0, -6, -8, 0, -8, 0], [6, 3, 0, 2, -22, 7, 21, 0, 7, -4, 1, -2], [8, 2, 10, -2, -16, 10, -6, 7, 6, -4, 7, -9], [2, -12, -2, 10, 12, 2, -4, 11, -8, 6, 5, 17], [10, 1, 2, -1, 2, -2, 6, 3, -3, 2, -2, 3], [-2, 2, 2, 12, -2, -2, 2, 9, -4, 6, -1, 3], [-13, 7, 10, 3, 5, 1, 3, -9, -1, 0, 1, 8], [7, 13, -6, -1, -3, -9, -22, -4, -12, 2, -2, -17], [5, 28, 15, -27, -1, 2, 0, -14, 8, -15, -7, -15], [4, -22, 0, 22, 10, 0, 0, 0, 0, 0, 0, 10], [-10, -2, 12, -3, -5, -10, 6, -22, 10, 0, -10, -8], [5, 5, 0, 3, 9, 7, 17, -7, 1, -12, -7, -6], [7, -11, -7, -3, 11, 7, 6, 2, 1, -8, -4, 0], [-2, 10, 2, 18, 6, 0, 0, 6, -6, 2, 8, 0], [-21, -17, 4, 19, 11, -13, 6, 6, 0, 6, 0, 9], [5, 8, -5, 9, 1, -1, -9, 12, -19, -11, 11, 7], [2, 1, 0, -10, -6, 16, 0, 0, 16, -12, 6, -17], [6, 19, 16, -12, -14, 5, -11, -6, 11, 0, -3, -22], [-17, -21, -1, -4, 18, -5, 23, -8, -2, 0, -8, 20], [-21, -5, -6, -9, 11, -9, 13, -3, 15, 0, -9, 2], [-11, -10, -3, -7, 7, -19, -3, -20, -3, -1, -1, -5], [4, 3, -8, 1, -6, 10, 2, -1, 1, -10, 8, -5], [-18, -18, -2, 25, 13, -12, -4, 10, 0, 24, 12, 12], [23, 5, 0, -5, -4, 10, 1, 2, 0, -8, -8, 11], [3, 3, -6, -13, -1, -17, -3, -11, -9, 0, -17, -14], [-2, 29, 24, 0, -6, 13, -3, -6, 11, -4, -3, -28], [-8, 11, 0, -18, 0, -3, -1, 0, -3, -10, 1, 0], [0, 7, 0, -13, -14, 0, 0, -7, -7, 0, 0, -13], [19, 16, 1, 5, -7, 7, -9, 4, -9, -11, 11, -15], [-9, -15, 5, 9, -11, 5, 0, -2, 9, 4, -2, 2], [-25, -12, -1, 5, 19, -7, 7, -8, -1, -1, -1, -7], [-3, 11, 3, -13, -11, -3, -10, -10, 3, 0, 4, 4], [-1, -10, 1, 4, 10, -1, 23, -10, 10, -9, -10, 1], [-7, -9, -13, 24, 11, -4, 8, 9, -23, 0, -4, 35], [8, 5, 3, 1, -2, 13, 15, 11, 6, -9, 2, 13], [6, 13, -2, -3, 2, -10, -4, -15, -15, -10, -2, -1], [5, 0, -2, 3, -11, 0, -3, -2, -2, 0, -2, 5], [16, 12, -8, -6, 6, 0, 7, 2, -16, -14, -12, 9], [4, 2, -4, 24, -2, 4, 28, 3, 0, -4, -3, 1], [4, 15, -8, -9, -6, -2, -22, 11, -11, 2, 8, 5], [-3, -2, -5, 3, 5, 3, 3, 2, -5, -1, -1, 1], [-10, -32, -8, 6, -14, -8, 4, 3, -2, 6, 3, 9], [6, -5, -13, 3, 10, -1, 3, -7, 0, -7, 0, 13], [0, -11, 0, 12, -6, 1, 19, 0, 1, 12, -13, 14], [4, 14, 0, 1, 0, -2, -6, 0, -2, -3, 6, 5], [-10, -18, -22, 11, 22, -14, 18, 8, -8, 7, 4, 15], [-25, -34, -21, 6, 12, -21, 19, -4, -14, 7, -4, 23], [3, 25, 11, -6, -13, 6, -14, -5, -3, 0, 6, -19], [3, 3, -4, -15, -15, -9, -33, 5, -9, 0, 9, 0], [3, -5, -23, 7, 15, -23, -10, -4, -19, 4, -4, 10], [10, 9, 0, 0, 20, -10, -2, 0, -10, -10, -18, 17], [-4, -14, -4, 4, 2, -20, 0, -6, -4, 4, 0, -4], [-13, -18, -6, -16, 9, -9, 9, -2, 10, 11, -11, 11], [16, 33, 13, -43, -30, 3, -23, -7, 0, -7, 0, -33], [26, 30, 0, -16, 4, 0, -10, -11, -22, -22, -11, -13], [5, -14, -7, -4, 6, 0, 4, 13, 1, -7, -8, 11], [-20, -15, 4, 3, 2, -16, 10, -15, 15, 16, -4, -3], [5, -8, -11, -19, -11, 0, 12, -12, 8, -11, -19, 11], [10, -13, 7, -11, -2, 11, 21, -7, 14, -15, -22, 3], [-10, -9, -1, 15, 2, 3, 13, 5, 0, 5, 0, 3], [-6, -9, 8, -8, 12, -7, 15, -6, 15, 8, -3, -2], [14, 13, -4, -2, -22, -1, -33, 22, -3, 10, 11, -12], [-3, -19, -21, 0, 0, -3, 5, 14, 0, 14, 0, 16], [16, 17, -7, -1, 0, 1, -3, 7, -14, -9, -2, 3], [-22, -4, 16, 16, -4, 0, 12, -8, 22, 16, -6, 8], [-3, 18, 11, -1, -3, -5, 15, -2, 11, 3, 3, -3], [-8, 6, 4, 2, -10, 0, 8, -16, 2, 4, 2, 2], [-21, -2, -5, -9, 1, -1, 1, 2, 1, -1, 1, 23], [-9, -24, -9, 24, 22, 0, 0, 9, -7, 9, 2, 31], [-5, -7, 0, -1, -14, 14, 21, -6, 12, -8, 8, -1], [10, 7, -1, 1, 0, 9, -13, 11, -2, -9, 2, -11], [1, 5, -2, -7, 1, -3, -3, -1, -3, 0, -3, -6], [30, 30, -12, -19, -15, -10, -26, -2, -14, -8, 10, -4], [-5, -13, -8, -5, 19, -3, 9, 5, 1, 0, -3, 14], [-16, -10, 6, -6, 0, 6, 22, -15, -2, -8, -15, 5], [3, -11, -6, -3, 0, 0, 11, 0, 6, 0, 0, -3], [12, 15, 0, 10, -12, 9, 7, 0, 9, -2, 5, 8], [-12, 7, 6, -3, 12, -2, 6, -1, -1, -2, 6, -9], [-16, -2, -2, -2, -8, -10, -9, -6, 0, -6, 0, 9]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_841_e_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_841_2_e_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_841_2_e_a(:prec:=12)
    chi := MakeCharacter_841_e();
    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_841_2_e_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_841_2_e_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_841_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, -7, 18, 42, 114, 28, -111, -84, 4, 35, 23, 7, 1]>],Snew);
    return Vf;
end function;