
// Make newform 882.2.u.d in Magma, downloaded from the LMFDB on 20 June 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_882_u();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_882_u_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_882_2_u_d();".
// 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_882_2_u_d();".
// 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], [-32, 192, -128, -100, 134, -5, -58, 36, 7, -10, 5, -1], [-96, 0, 128, -12, -94, 49, 50, -36, -11, 18, -1, -3], [64, -80, 40, 44, -52, 17, 22, -18, 1, 4, -3, 1], [-352, 384, 96, -380, 186, 165, -166, -4, 57, -22, -5, 1], [-112, 56, 76, -108, -3, 62, -18, -19, 12, 1, -3, 0], [-416, 224, 368, -444, -90, 305, -70, -112, 77, 6, -21, 5], [160, 64, -304, 156, 142, -167, 2, 72, -35, -10, 11, -3], [112, -104, -92, 168, -31, -93, 60, 19, -31, 5, 6, -3], [608, -864, -80, 788, -450, -275, 370, -32, -135, 62, 15, -15], [1248, -1216, -512, 1388, -378, -625, 478, 76, -197, 54, 25, -13]];
    Rf_basisdens := [1, 128, 128, 128, 64, 128, 32, 128, 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_882_u();"
function MakeCharacter_882_u()
    N := 882;
    order := 7;
    char_gens := [785, 199];
    v := [7, 3];
    // 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_882_u_Hecke();"
function MakeCharacter_882_u_Hecke( : Kf := false)
    N := 882;
    order := 7;
    char_gens := [785, 199];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 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 := [[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, -1, 1, 0, -1, 1, 0], [-1, -2, 1, -1, 1, -1, 1, 0, 0, -1, 0, 0], [0, -1, 1, 0, 0, -1, -1, 0, -1, 0, -1, 0], [2, 1, -3, 0, -2, 1, 0, 1, 0, 3, 0, 0], [2, 0, -3, 3, -1, -1, -1, 1, 1, 1, 1, 1], [2, -1, 1, -1, -2, 2, 1, 1, -1, 0, 0, 2], [-3, 0, 0, -3, 3, -2, 0, 0, 0, 0, 2, -1], [3, 3, -2, 2, -5, 3, 1, 2, 2, 1, -1, 2], [3, 1, -1, 4, -1, 1, 0, 3, 0, 4, 0, 1], [-1, -2, -3, 3, 1, 4, -3, 1, 1, -2, 3, 1], [2, 0, -2, 1, -2, -3, 4, -3, -1, 5, -3, 1], [3, 3, -5, 2, -2, 0, -1, 0, 0, 0, 3, -3], [-1, -6, 1, 0, 1, 1, -1, 2, 2, -3, 2, 0], [1, -2, 0, 1, -1, 0, -3, 3, 0, -2, 3, -2], [4, 4, -3, 1, -5, 1, 0, 0, -1, 1, -2, 2], [4, 7, -4, 4, -6, -3, 2, -1, -1, 5, -2, -1], [8, 2, -2, 0, -2, 2, 0, 2, 0, 0, 0, 2], [-3, 4, 0, -3, 1, -2, 1, -1, 0, 4, 1, -2], [0, 1, -2, 1, -2, 0, 0, 1, 3, 1, -2, -2], [8, -2, 2, 3, -2, 2, -4, 6, 4, -1, 0, 2], [0, 4, 2, 4, -1, 0, 0, -2, -3, 3, -3, 1], [0, -4, 5, -4, 0, 0, 0, 4, 3, -4, -4, 1], [-8, -7, 7, -6, 2, -2, 1, -4, -1, -5, 0, -2], [2, 0, -2, -1, 0, 1, 0, 1, 4, -6, 1, 1], [-5, 0, 5, -1, 0, 0, 1, 0, -3, 3, 0, 1], [0, 3, -5, 3, 0, 0, 0, 1, 2, -1, 0, -1], [0, 7, -4, 0, 0, -3, 1, -4, -1, 5, -1, 0], [0, 2, -2, 2, -3, 0, 0, -1, -3, 4, 3, 2], [1, 8, -6, 6, -9, 2, 0, 1, 1, 0, 0, 1], [-5, -5, 3, -2, 5, -2, -1, 0, 2, -2, 2, -2], [5, 5, 2, 1, 5, -3, -3, 0, 3, -3, 3, -3], [3, 0, -3, 5, -5, 0, 0, 0, -1, 4, 0, 0], [-4, -8, -3, 0, 4, -2, 3, -5, 2, 1, 2, 0], [-2, -6, 0, -2, -4, 2, -4, 4, 0, -6, 2, 4], [-12, -12, 9, -9, 9, -3, 2, 0, 3, -3, 0, 0], [-6, -6, 3, -6, 5, 0, -1, 0, 0, 0, 2, -2], [7, 9, -10, 10, -6, 2, -4, -3, -3, 10, 4, -3], [4, 7, 0, 4, -2, 0, -1, 1, 0, 7, 1, -5], [3, 9, 0, 0, -5, -1, -3, -4, -4, 7, 3, -4], [0, 4, -5, 4, 3, 0, 0, -7, -2, 4, 1, -5], [-5, 0, 5, -1, 4, -4, 1, -4, 0, 0, -4, -3], [1, 0, -1, 0, 5, -2, -3, -2, -3, 6, -2, -5], [6, 2, -2, 1, 4, -4, 2, -1, -2, 3, 0, -4], [-9, 0, 9, -8, 9, -4, 3, -4, 3, -14, -4, -1], [0, 10, -7, 10, -3, 0, 0, -3, 0, 6, 7, -3], [-8, -2, 0, -8, 11, 0, 4, -4, 0, -2, -4, 3], [-7, -8, 8, -2, -2, 2, -3, 3, 3, -5, 0, 2], [6, 6, 1, 0, -5, 2, 5, 0, -2, 2, -6, 6], [1, -3, 10, -10, 0, 0, 0, 3, 3, -2, 0, 3], [-3, -3, -3, -9, 0, 1, 0, 0, -1, 1, 3, -3], [1, 0, 0, 1, 4, 7, -7, 7, 0, 0, 0, 5], [-6, -6, 3, -20, 0, 0, 9, 0, 0, 0, -3, 3], [11, 5, -3, 3, -3, -1, -1, -2, -2, 13, 1, -2], [-1, -1, 1, -1, -9, 9, 2, 7, -2, 1, 0, 9], [0, -1, 17, -1, 3, 0, 0, 2, 0, -5, -6, 2], [-11, -1, 0, -11, 19, 1, 0, 0, 0, -1, -1, -1], [-4, -4, 8, -8, 10, -8, 0, -6, -6, 2, 0, -6], [9, 4, -13, 0, -9, -5, 3, -8, 2, 11, 2, 0], [8, 1, -3, 0, -8, -5, 0, -5, 0, 3, 0, 0], [-9, 9, -9, 0, 3, -3, 4, -4, -4, 4, 0, -3], [-4, -1, -4, 0, 4, 1, 1, 0, -1, 5, -1, 0], [-2, -11, 12, -12, 8, 5, 0, 3, 3, -5, 0, 3], [6, 5, -5, 9, -3, 3, -1, 5, 1, 8, 0, 3], [-1, 0, 0, -1, 0, 6, 4, -4, 0, 0, -10, 4], [-9, 1, -4, 4, -4, 2, 0, 3, 3, -12, 0, 3], [12, 12, -10, 7, -18, 5, 5, 0, -5, 5, -8, 8], [4, 12, 0, 4, -11, 5, -8, 8, 0, 12, 3, 3], [6, 6, 7, -2, 3, -2, 7, 0, 2, -2, -4, 4], [1, 1, 8, -7, 5, -5, 5, 0, 5, -5, -3, 3], [14, 14, -12, 17, -10, -3, 1, 0, 3, -3, 2, -2], [0, 14, -17, 14, -5, 0, 0, 1, -2, 4, 5, 3], [9, -2, 2, 1, -2, 2, 3, -2, -3, 4, 0, 2], [-3, -7, 14, 0, 3, 4, 1, 3, -2, -12, -2, 0], [0, 10, -1, 10, -1, 0, 0, -5, -6, 6, 9, 1], [-2, 0, 2, 6, -1, 1, -6, 1, 9, -17, 1, -5], [-9, 0, 9, 3, -4, -2, 3, -2, -2, 4, -2, 1], [0, 4, 3, 4, -4, 0, 0, 0, -1, 4, 2, 1], [8, 5, -8, 8, -4, 5, 6, -1, -1, 9, -6, -1], [11, 9, 0, 11, -3, 2, -4, 4, 0, 9, 2, 7], [10, 5, 9, -9, -11, 4, 1, 6, 6, 4, -1, 6], [2, 2, 7, 2, -3, 4, 13, 0, -4, 4, -10, 10], [1, 0, -1, 12, -11, -6, 5, -6, 4, 4, -6, -1], [7, -4, 0, 0, -7, 3, -3, 6, -6, 6, -6, 0], [-8, -21, 15, 0, 8, 3, -5, 8, -1, -14, -1, 0], [-2, 0, 2, -15, 10, 3, 2, 3, 7, -19, 3, 5], [-5, -5, -2, 1, -4, -2, 15, 0, 2, -2, -2, 2], [10, 7, 3, -3, -14, -1, 9, 7, 7, 3, -9, 7], [2, -8, 0, 2, -7, 5, -4, 4, 0, -8, -1, 13], [-2, 10, 0, -2, -13, 2, -2, 2, 0, 10, 0, 5], [5, 10, -7, 0, -5, 5, -1, 6, 2, 5, 2, 0], [-2, 0, 2, -8, 8, 9, -9, 9, 5, -11, 9, 0], [-7, -4, 4, 0, -2, 2, 0, -5, 0, 0, 0, 2], [0, -12, 12, -12, -3, 0, 0, 15, 5, -12, 3, 10], [-8, -1, 21, 0, 8, 5, -4, 9, 0, -21, 0, 0], [-1, 7, -7, -7, 5, -5, 2, -7, -2, -5, 0, -5], [5, -2, 2, -7, -2, 2, -3, 2, 3, -10, 0, 2], [2, 2, 14, -2, 10, -10, 0, 0, 10, -10, -4, 4], [-11, 12, -2, 0, 11, -11, -6, -5, -5, 7, -5, 0], [-19, 0, 19, -10, 7, 7, -4, 7, -2, 1, 7, 3], [-20, -1, 1, -4, 10, -10, 6, -16, -6, 2, 0, -10], [0, 8, -14, 8, -13, 0, 0, 6, 4, 7, 8, 2], [-11, -4, 4, -1, 2, -2, -1, 2, 1, -2, 0, -2], [-11, -5, 1, -1, 3, 5, -17, 2, 2, -13, 17, 2], [2, -6, 4, 0, -2, 0, -2, 2, 5, -9, 5, 0], [9, -4, 4, -3, -1, 1, -3, 7, 3, -6, 0, 1], [0, 0, 0, 13, -21, 7, 1, 7, 6, -5, 7, 8], [2, -22, 10, 0, -2, 0, 0, 0, 0, -10, 0, 0], [-2, -15, -1, 0, 2, -7, 6, -13, -9, 10, -9, 0], [-4, 0, 0, 2, -1, 1, 0, -11, 0, 2, 0, 1], [0, 11, 2, 11, 3, 0, 0, 1, -3, -4, 6, 4], [-6, -11, -4, 4, 10, 5, -4, 1, 1, -7, 4, 1], [1, -6, 6, -3, 4, -4, -11, 6, 11, -14, 0, -4], [-1, -1, -12, -2, 3, -2, -13, 0, 2, -2, 15, -15], [-18, 0, 0, -18, 10, 0, 2, -2, 0, 0, -2, -2], [4, 4, 12, -18, 16, 0, 8, 0, 0, 0, 4, -4], [9, 9, -4, 2, 1, 6, -8, 0, -6, 6, 5, -5], [16, 16, -20, 6, -21, 3, -2, 0, -3, 3, -1, 1], [-2, -4, 0, -2, 1, 8, -10, 10, 0, -4, 2, 11], [0, 30, 2, 30, -13, 0, 0, -1, -1, 14, -3, 0], [-22, -4, 11, 0, 22, -2, -2, 0, 4, -15, 4, 0], [0, 14, -8, 14, -8, 0, 0, -4, -10, 12, -7, 6], [6, 0, -6, -1, -9, 2, 8, 2, -2, -6, 2, 10], [-6, 0, 6, 16, -10, 0, -6, 0, -4, -4, 0, -6], [0, 10, 10, 10, -2, 0, 0, -3, 5, 5, 5, -8], [-8, -1, 0, -8, 5, -3, 12, -12, 0, -1, -9, -7], [-5, -5, 9, 14, 2, -4, 9, 0, 4, -4, -7, 7], [8, 0, -8, -4, 0, 3, 1, 3, 6, 13, 3, 4], [25, 23, -27, 0, -25, -4, 4, -8, -5, 32, -5, 0], [-5, -6, 0, -5, -13, -2, 7, -7, 0, -6, -5, -10], [-1, 0, 1, 7, 11, -4, -14, -4, -9, 20, -4, -18], [4, 4, 9, 18, 13, -6, -10, 0, 6, -6, 4, -4], [-11, -14, 0, -11, 23, -10, 13, -13, 0, -14, -3, -2], [-14, -14, 0, -14, -4, -2, -4, 4, 0, -14, 6, 8], [0, -13, 11, -13, -11, 0, 0, 10, 7, 1, 11, 3], [17, 23, -20, 0, -17, 4, -4, 8, 7, 13, 7, 0], [5, 0, -5, -12, 17, -15, 10, -15, -16, -9, -15, -5], [-10, 7, -12, 0, 10, 5, 4, 1, 2, 10, 2, 0], [0, 9, -30, 9, -10, 0, 0, 4, 0, 6, 7, 4], [-6, -7, 0, -6, 17, -9, -3, 3, 0, -7, 12, -4], [-4, 10, -9, 9, -12, 0, -3, 2, 2, -6, 3, 2], [-6, -6, 1, -13, 2, 1, 16, 0, -1, 1, 1, -1], [-7, -7, -3, -8, -5, -9, -5, 0, 9, -9, -2, 2], [-8, -16, 0, -8, -6, -2, 1, -1, 0, -16, 1, -5], [-14, -14, 0, -19, 6, -3, -15, 0, 3, -3, 6, -6], [-1, 18, -10, 10, -18, -5, -2, 0, 0, -1, 2, 0], [0, -8, 35, -8, 14, 0, 0, 0, -3, -14, -10, 3], [10, -7, 0, 10, 13, -1, -4, 4, 0, -7, 5, -3], [8, 15, -15, -4, 11, -11, 3, -7, -3, -1, 0, -11], [-6, 0, 6, -6, 5, -10, 11, -10, -13, 24, -10, 1], [-5, 7, -7, 25, -7, 7, -2, -1, 2, 23, 0, 7], [-10, -14, 14, -30, 0, 0, 0, 2, 0, -30, 0, 0], [9, 0, -9, 0, -1, -1, 2, -1, 8, -15, -1, 1], [0, -1, 32, -1, 15, 0, 0, 1, 7, -16, -10, -6], [9, 2, -24, 24, 6, 1, 6, -8, -8, 17, -6, -8], [-1, -10, -5, 5, 1, 7, -5, 9, 9, -10, 5, 9], [-5, -23, 5, 0, 5, 4, -14, 18, 17, -22, 17, 0], [12, 12, 11, -18, -6, 2, 1, 0, -2, 2, -17, 17], [0, 0, -12, 11, -12, -3, -4, 0, 3, -3, 0, 0], [-7, -7, -5, 8, -3, -16, -7, 0, 16, -16, 2, -2], [-1, -6, 0, -1, 1, 12, -7, 7, 0, -6, -5, 10], [-5, -13, 4, 0, 5, -2, 3, -5, -1, -3, -1, 0], [0, 11, -3, 11, 9, 0, 0, -14, -3, 5, -7, -11], [18, 0, -18, 2, -6, -2, 6, -2, -4, -4, -2, 4], [3, 0, -3, 10, -8, -1, -1, -1, 3, 3, -1, -2], [12, 17, -27, 0, -12, -1, -1, 0, 9, 18, 9, 0], [18, 22, -20, 20, -12, -8, 2, -10, -10, 28, -2, -10]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_882_u_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_882_2_u_d();".
// 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_882_2_u_d(:prec:=12)
    chi := MakeCharacter_882_u();
    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_882_2_u_d();".
// 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_882_2_u_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_882_u();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![1, 25, 376, 1096, 878, -480, 203, -114, 62, -3, 13, 1, 1]>],Snew);
    return Vf;
end function;