
// Make newform 7056.2.b.x in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7056_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_7056_b_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_7056_2_b_x();".
// 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_7056_2_b_x();".
// 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 := [4, 0, -8, 0, 14, 0, -4, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [20, 0, 0, 0, 0, 0, 1, 0], [0, 2, 0, -28, 0, 7, 0, -2], [0, 22, 0, -21, 0, 7, 0, -1], [0, 6, 0, -70, 0, 21, 0, -6], [0, 52, 0, -49, 0, 14, 0, -3], [4, 0, -12, 0, 4, 0, -1, 0], [-18, 0, 56, 0, -14, 0, 4, 0]];
    Rf_basisdens := [1, 14, 14, 7, 14, 7, 2, 7];
    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_7056_b();"
function MakeCharacter_7056_b()
    N := 7056;
    order := 2;
    char_gens := [6175, 1765, 785, 4609];
    v := [1, 2, 2, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_7056_b_Hecke();"
function MakeCharacter_7056_b_Hecke(Kf)
    N := 7056;
    order := 2;
    char_gens := [6175, 1765, 785, 4609];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [-1, 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 := [[0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 1, 0, -1, 0], [0, 0, 2, 0, 1, 0, 0, 0], [0, 0, 2, 0, -1, 0, 1, 0], [0, -2, 0, 1, 0, 0, 0, 0], [0, 0, -1, 0, 1, 0, 1, -1], [-2, 1, 0, 1, 0, 0, 0, 0], [4, -2, 0, 0, 0, -1, 0, 0], [0, 1, 0, 2, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, -1, 1], [0, 0, -2, 0, 6, 0, 0, 1], [-2, 2, 0, 0, 0, -1, 0, 0], [-2, -2, 0, 0, 0, -2, 0, 0], [6, 2, 0, 2, 0, -1, 0, 0], [0, 0, 0, 0, -1, 0, -2, -2], [0, 0, 2, 0, -2, 0, 0, 2], [0, 0, 3, 0, 3, 0, 1, 2], [0, 0, 2, 0, -1, 0, -2, 2], [0, 0, -4, 0, -4, 0, 0, -1], [0, -2, 0, -3, 0, -1, 0, 0], [0, 0, 2, 0, 7, 0, 3, 2], [0, 0, -6, 0, 1, 0, 2, -2], [0, 0, -3, 0, 0, 0, 3, -1], [0, -6, 0, -2, 0, -1, 0, 0], [0, 0, -5, 0, 5, 0, 1, 1], [2, 3, 0, -2, 0, 2, 0, 0], [0, 0, 0, 3, 0, -1, 0, 0], [0, 0, -2, 0, -2, 0, 0, 0], [-8, -2, 0, 1, 0, -3, 0, 0], [-6, -1, 0, 1, 0, -2, 0, 0], [0, 4, 0, -1, 0, 1, 0, 0], [2, 0, 0, -2, 0, -2, 0, 0], [0, 0, -6, 0, 2, 0, -4, 1], [0, 0, 3, 0, 2, 0, -2, 0], [0, 0, 6, 0, 2, 0, 4, 0], [-10, 4, 0, -3, 0, -2, 0, 0], [0, 0, 10, 0, -1, 0, 1, 0], [0, 0, -7, 0, 5, 0, -1, 2], [0, 0, 1, 0, 6, 0, 0, 0], [0, 0, 7, 0, 3, 0, 5, 0], [-2, -8, 0, 2, 0, 2, 0, 0], [-4, 10, 0, -1, 0, 1, 0, 0], [0, 4, 0, 3, 0, 1, 0, 0], [0, 0, -4, 0, 0, 0, 4, 2], [-4, -8, 0, -2, 0, 2, 0, 0], [2, 8, 0, 1, 0, -2, 0, 0], [0, 0, 1, 0, 10, 0, -4, 0], [-2, 3, 0, 1, 0, 4, 0, 0], [0, 0, -7, 0, 3, 0, 7, -1], [0, 0, -1, 0, -4, 0, -2, 0], [10, -12, 0, -1, 0, -2, 0, 0], [0, 0, 1, 0, 0, 0, 1, -1], [0, 0, 1, 0, -7, 0, -5, 0], [0, 0, 5, 0, -2, 0, -1, -1], [4, 2, 0, 3, 0, -4, 0, 0], [-4, 2, 0, 0, 0, -4, 0, 0], [-14, 5, 0, -1, 0, 0, 0, 0], [-12, 6, 0, 0, 0, -5, 0, 0], [0, 0, -1, 0, 6, 0, -1, -3], [-16, 2, 0, -2, 0, -1, 0, 0], [-2, 0, 0, 3, 0, -4, 0, 0], [0, 0, 0, 0, 3, 0, 4, 4], [0, 10, 0, 1, 0, -1, 0, 0], [0, 0, 12, 0, 8, 0, -4, -2], [-4, 9, 0, 2, 0, -4, 0, 0], [0, 0, -3, 0, -5, 0, -1, 3], [0, 0, 1, 0, 6, 0, -6, 0], [0, 0, -3, 0, -10, 0, -3, -1], [0, 0, 9, 0, 15, 0, -1, -1], [4, 8, 0, -3, 0, 3, 0, 0], [-12, 2, 0, -2, 0, -2, 0, 0], [0, 0, 4, 0, 0, 0, 4, -2], [-8, -4, 0, 0, 0, 0, 0, 0], [-14, 5, 0, 1, 0, 4, 0, 0], [0, 0, 2, 0, -9, 0, 6, -2], [14, 11, 0, 1, 0, 0, 0, 0], [0, 0, 10, 0, 3, 0, 6, -2], [-6, 2, 0, -4, 0, 5, 0, 0], [0, 16, 0, 0, 0, 0, 0, 0], [0, 0, -13, 0, -1, 0, -3, 0], [0, 0, -7, 0, 2, 0, 0, 0], [-12, -4, 0, 3, 0, -1, 0, 0], [0, 0, -7, 0, -11, 0, 3, 0], [0, -10, 0, -5, 0, 1, 0, 0], [-8, -10, 0, -2, 0, 2, 0, 0], [0, 0, 10, 0, 7, 0, 1, -4], [0, 0, 2, 0, -6, 0, -2, 6], [10, -10, 0, 0, 0, -7, 0, 0], [-10, -2, 0, 6, 0, -1, 0, 0], [0, 0, 2, 0, 14, 0, 0, 3], [0, 0, -13, 0, -11, 0, 1, 1], [0, 0, -2, 0, 10, 0, 6, 3], [16, -2, 0, -5, 0, -1, 0, 0], [0, 0, 9, 0, 8, 0, -3, -5], [0, 0, 7, 0, -10, 0, -1, 1], [0, 8, 0, -1, 0, -1, 0, 0], [6, -6, 0, -4, 0, -4, 0, 0], [0, 0, -8, 0, 4, 0, 0, -5], [-4, 20, 0, -1, 0, 3, 0, 0], [14, 12, 0, -3, 0, 0, 0, 0], [14, 5, 0, 3, 0, -2, 0, 0], [0, 0, 12, 0, 12, 0, -2, -1], [0, 0, 7, 0, 0, 0, 0, 4], [10, -6, 0, 0, 0, -5, 0, 0], [0, 0, 0, 0, -9, 0, -5, 2], [0, 0, 15, 0, -13, 0, 1, 0], [0, 0, -2, 0, -1, 0, 0, 8], [-4, -8, 0, -1, 0, -1, 0, 0], [-6, -9, 0, 4, 0, -2, 0, 0], [-18, -5, 0, 1, 0, 2, 0, 0], [12, 8, 0, 3, 0, 3, 0, 0], [0, 0, 6, 0, 10, 0, -2, 2], [6, -7, 0, -3, 0, 2, 0, 0], [-8, 2, 0, 7, 0, -2, 0, 0], [2, -18, 0, -2, 0, -1, 0, 0], [-10, 11, 0, -3, 0, 2, 0, 0], [0, 0, -1, 0, 11, 0, -3, -4], [0, 0, -1, 0, 6, 0, 2, 4], [6, -13, 0, 2, 0, -2, 0, 0], [0, 0, 12, 0, -17, 0, 3, -2], [0, 0, -9, 0, 7, 0, -7, 2], [8, -4, 0, 8, 0, 6, 0, 0], [-18, 7, 0, 1, 0, 4, 0, 0], [18, -17, 0, 2, 0, 6, 0, 0], [4, 4, 0, 0, 0, -2, 0, 0], [16, 6, 0, -6, 0, 1, 0, 0], [0, 0, -7, 0, 6, 0, 4, 0], [0, 0, 10, 0, 2, 0, -6, 4], [0, 0, -19, 0, -7, 0, 3, -2], [0, 0, -12, 0, 8, 0, 6, 0], [2, 7, 0, 2, 0, 2, 0, 0], [0, 0, -10, 0, -1, 0, 3, -2], [0, 0, 1, 0, 0, 0, -12, -4], [0, 0, -14, 0, 9, 0, 7, -4], [12, -20, 0, 1, 0, -5, 0, 0], [0, 0, 15, 0, 2, 0, -1, -7], [10, -6, 0, 6, 0, -4, 0, 0], [8, 8, 0, 0, 0, 4, 0, 0], [-16, -14, 0, 1, 0, 3, 0, 0], [0, 0, 0, 0, 4, 0, 6, -7], [0, 0, 3, 0, -7, 0, 1, -3], [0, 0, -4, 0, 11, 0, 0, 0], [-10, 0, 0, 5, 0, 4, 0, 0], [0, 0, 7, 0, 8, 0, 10, 4], [0, 0, 6, 0, -5, 0, 3, -6], [-4, -14, 0, -5, 0, -2, 0, 0], [0, 0, -3, 0, 11, 0, -9, -3], [-34, 9, 0, 0, 0, 2, 0, 0], [0, 0, -4, 0, -15, 0, -3, 2], [0, 0, -14, 0, 6, 0, -2, 5], [-6, 2, 0, 2, 0, 7, 0, 0], [0, 0, -2, 0, 6, 0, -4, 5], [0, 0, 13, 0, 9, 0, -5, 4], [0, 0, 6, 0, -10, 0, -12, -3], [0, 0, -22, 0, 3, 0, 7, -2], [0, 0, -6, 0, -5, 0, 4, 0], [0, 0, -7, 0, 24, 0, 1, -5], [0, 0, 7, 0, 1, 0, -7, 1], [-2, -18, 0, 6, 0, 4, 0, 0], [0, 0, 2, 0, 6, 0, -6, -2], [16, 6, 0, -5, 0, 5, 0, 0], [-2, -3, 0, -5, 0, -2, 0, 0], [20, -4, 0, -2, 0, 6, 0, 0], [0, 0, -6, 0, -14, 0, -6, 3], [0, 0, -1, 0, -10, 0, -2, 0], [12, 14, 0, -2, 0, -2, 0, 0], [0, 0, 1, 0, -6, 0, -3, 11], [0, 0, 13, 0, -13, 0, 3, -7], [0, 0, 10, 0, -7, 0, 6, -10], [0, 0, -9, 0, -5, 0, 5, 6], [18, 11, 0, 4, 0, -2, 0, 0], [8, -8, 0, -1, 0, -7, 0, 0], [0, 0, 1, 0, -20, 0, 3, 7], [0, 0, 4, 0, 8, 0, 2, -7], [-22, -3, 0, 5, 0, -2, 0, 0], [4, 26, 0, 0, 0, -1, 0, 0], [0, 0, 7, 0, -16, 0, -6, 4], [0, 0, 12, 0, -8, 0, -12, 1], [-8, 2, 0, 3, 0, 7, 0, 0], [-12, -13, 0, -8, 0, 0, 0, 0], [0, 0, -5, 0, 20, 0, 1, 1], [0, 0, -11, 0, 15, 0, -5, -3], [0, 0, -8, 0, 1, 0, 1, 10], [-4, 10, 0, 2, 0, 6, 0, 0], [-4, -12, 0, -6, 0, 8, 0, 0], [-6, 13, 0, 4, 0, 10, 0, 0], [8, 28, 0, -4, 0, 2, 0, 0], [0, 0, -1, 0, 2, 0, -2, -8], [0, 0, -7, 0, -11, 0, -13, -2], [0, 0, 8, 0, 0, 0, -12, 8], [0, 0, -7, 0, -2, 0, 9, 7], [0, 0, -3, 0, 15, 0, 7, -7], [0, 0, 8, 0, -13, 0, 11, -2], [18, 5, 0, -2, 0, 2, 0, 0], [0, 13, 0, -4, 0, -4, 0, 0], [0, 0, -20, 0, -11, 0, -5, 4], [22, 4, 0, -3, 0, 6, 0, 0], [6, 10, 0, 10, 0, -8, 0, 0], [-32, -4, 0, -3, 0, 1, 0, 0], [0, 0, 16, 0, 11, 0, -8, 4], [0, 0, -27, 0, -14, 0, -6, 0], [-28, 12, 0, 2, 0, 6, 0, 0], [0, 0, -10, 0, 23, 0, 5, 0], [36, -16, 0, 0, 0, 4, 0, 0], [0, 0, 17, 0, 5, 0, -9, 2], [-4, -12, 0, -13, 0, -5, 0, 0], [-16, 10, 0, 4, 0, -7, 0, 0], [28, -17, 0, 0, 0, 0, 0, 0], [0, 0, -13, 0, -10, 0, 1, -3], [0, 0, -2, 0, 18, 0, -2, -9], [8, 18, 0, 1, 0, 1, 0, 0], [-10, -12, 0, 3, 0, 6, 0, 0], [0, 0, -1, 0, 6, 0, -16, 4], [0, 0, 6, 0, -2, 0, 8, -3], [0, 0, 5, 0, 0, 0, -5, -9], [0, 0, -15, 0, -5, 0, -5, 5], [-10, 15, 0, 3, 0, -4, 0, 0], [-18, 1, 0, 2, 0, 2, 0, 0], [-4, 2, 0, 11, 0, 8, 0, 0], [10, -14, 0, 2, 0, -12, 0, 0], [0, 0, 16, 0, 24, 0, -2, 2], [12, 14, 0, 11, 0, -9, 0, 0], [36, 3, 0, 6, 0, 4, 0, 0], [0, 0, -22, 0, 11, 0, -9, 6], [0, 0, 3, 0, 23, 0, -7, -12], [12, 2, 0, 11, 0, 4, 0, 0], [0, 0, -11, 0, 5, 0, -1, -6], [18, 21, 0, 6, 0, 2, 0, 0], [16, 10, 0, 2, 0, -3, 0, 0], [0, 0, -26, 0, -6, 0, -4, 3], [40, 6, 0, -1, 0, 1, 0, 0], [-16, 10, 0, -2, 0, 9, 0, 0], [-4, 10, 0, -13, 0, -5, 0, 0], [0, 0, -9, 0, 10, 0, -14, 8], [10, -16, 0, -4, 0, 8, 0, 0], [0, 0, -15, 0, 19, 0, -9, 3], [0, -12, 0, 6, 0, 0, 0, 0], [0, 0, -5, 0, 17, 0, -3, 3], [-20, -4, 0, -5, 0, 9, 0, 0], [-36, 16, 0, -2, 0, -12, 0, 0], [-10, 2, 0, -8, 0, 8, 0, 0], [0, 0, -1, 0, -8, 0, -13, 5], [10, 8, 0, -3, 0, 10, 0, 0], [0, -6, 0, 9, 0, 2, 0, 0], [-14, -6, 0, -2, 0, 1, 0, 0], [0, 0, -24, 0, 20, 0, 8, -5], [0, 0, 7, 0, -9, 0, -3, -4], [-20, 6, 0, 2, 0, -6, 0, 0], [0, 0, -5, 0, 0, 0, -7, 5], [0, 0, -11, 0, 5, 0, 7, 2], [0, 0, -13, 0, -10, 0, -10, -8], [0, 0, -3, 0, 30, 0, 1, 3], [0, 0, 29, 0, -1, 0, 3, 1], [-4, 17, 0, 4, 0, -4, 0, 0], [4, 2, 0, -1, 0, 12, 0, 0], [0, 0, 23, 0, 16, 0, 1, -7], [0, 0, 1, 0, -14, 0, 12, 4], [0, 0, 20, 0, -16, 0, -12, 1], [0, 0, 19, 0, 25, 0, 9, 7], [0, 0, 16, 0, -13, 0, 16, -4], [0, 0, -1, 0, -20, 0, 6, 0], [0, 0, -2, 0, 3, 0, 5, 12], [16, 4, 0, -2, 0, 4, 0, 0], [30, -16, 0, -4, 0, -4, 0, 0], [0, 0, 28, 0, -7, 0, -7, -2], [0, 0, 4, 0, 12, 0, -12, -1], [2, 15, 0, -3, 0, 0, 0, 0], [0, 0, -15, 0, 6, 0, 18, -4], [0, 0, 4, 0, -4, 0, -10, -6], [0, 0, -3, 0, -16, 0, 6, 12], [0, 0, -8, 0, 0, 0, 16, 6], [0, 0, 11, 0, -22, 0, -2, 8], [0, 10, 0, 4, 0, -5, 0, 0], [0, 0, -5, 0, 13, 0, -15, 1], [-36, -8, 0, -10, 0, 2, 0, 0], [-10, 24, 0, -2, 0, -2, 0, 0], [18, 0, 0, -1, 0, 4, 0, 0], [36, 8, 0, 6, 0, -2, 0, 0], [0, 0, 30, 0, 18, 0, 12, 4], [-8, -14, 0, 1, 0, -11, 0, 0], [0, 0, -24, 0, -1, 0, 6, 2], [0, -24, 0, -7, 0, 3, 0, 0], [0, 0, 15, 0, -41, 0, -7, 0], [-22, -18, 0, 8, 0, 0, 0, 0], [48, 2, 0, -5, 0, 1, 0, 0], [28, 10, 0, -7, 0, 6, 0, 0], [0, 0, -24, 0, -7, 0, 11, 0], [-10, 14, 0, -4, 0, 14, 0, 0], [-28, 4, 0, -8, 0, 0, 0, 0], [-6, -1, 0, 17, 0, 10, 0, 0], [18, -24, 0, 3, 0, 4, 0, 0], [-6, 12, 0, 0, 0, -8, 0, 0], [0, 0, 13, 0, -8, 0, -13, -5], [12, -12, 0, 12, 0, 8, 0, 0], [0, 0, -4, 0, 11, 0, -11, -8], [34, -8, 0, 3, 0, 4, 0, 0], [24, -20, 0, -9, 0, -11, 0, 0], [0, 0, 0, 0, 29, 0, 12, 0], [-38, -9, 0, 5, 0, 0, 0, 0], [0, 0, 30, 0, 18, 0, -4, -8], [0, 0, 17, 0, 25, 0, -17, -8], [0, 0, 0, 0, -8, 0, 12, 11], [12, -9, 0, 10, 0, 0, 0, 0], [0, 0, 1, 0, 3, 0, 3, -1], [0, 0, 20, 0, -21, 0, -12, 4], [0, 0, -5, 0, 33, 0, -7, -1], [-4, -8, 0, -2, 0, -14, 0, 0], [0, -48, 0, -4, 0, 2, 0, 0], [18, -21, 0, 3, 0, 10, 0, 0], [-18, -3, 0, -1, 0, 12, 0, 0], [0, 0, 18, 0, 14, 0, 0, 3], [0, 0, -43, 0, -17, 0, 3, -1], [0, 0, 2, 0, -7, 0, 2, -18], [0, 38, 0, -1, 0, 5, 0, 0], [0, 0, -11, 0, 27, 0, 3, 5], [0, 0, -12, 0, -1, 0, 6, 14], [-18, -11, 0, -7, 0, 6, 0, 0], [28, 22, 0, -2, 0, 3, 0, 0], [-26, 21, 0, -2, 0, -2, 0, 0], [0, 0, -8, 0, -3, 0, -11, -2], [0, 0, -30, 0, 22, 0, -2, -3], [38, 10, 0, -4, 0, -2, 0, 0], [0, 0, 20, 0, -5, 0, 18, -2], [0, 0, -38, 0, -2, 0, 6, -8], [-12, 14, 0, 10, 0, 1, 0, 0], [0, 0, -25, 0, 3, 0, -7, 10], [10, 17, 0, 5, 0, 4, 0, 0], [26, 0, 0, -6, 0, -10, 0, 0], [18, 23, 0, 3, 0, -4, 0, 0], [44, 14, 0, -11, 0, 0, 0, 0], [18, 12, 0, 9, 0, 0, 0, 0], [0, 0, 6, 0, 30, 0, 8, 4], [-10, 24, 0, 1, 0, -6, 0, 0], [-2, -7, 0, 12, 0, 10, 0, 0], [0, 0, -2, 0, -15, 0, -3, -14], [0, 0, 13, 0, -6, 0, 14, 0], [-4, 30, 0, 9, 0, 6, 0, 0], [2, -4, 0, 4, 0, -8, 0, 0], [-6, -1, 0, -3, 0, 0, 0, 0], [0, 0, -22, 0, 15, 0, 19, -6], [0, 0, 0, 0, 16, 0, 14, -6], [6, 11, 0, 7, 0, 10, 0, 0], [0, 0, -23, 0, 19, 0, -9, 1], [0, 0, -30, 0, 1, 0, -4, 8], [0, 0, -2, 0, 10, 0, -16, 7], [32, -10, 0, -1, 0, 5, 0, 0], [0, 0, 10, 0, -31, 0, -1, 8], [-8, -2, 0, -18, 0, -7, 0, 0], [-18, 0, 0, 8, 0, -4, 0, 0], [-18, 8, 0, -10, 0, 2, 0, 0], [28, 32, 0, 4, 0, 0, 0, 0], [18, 0, 0, -4, 0, 0, 0, 0], [0, 0, -17, 0, -4, 0, -15, 1], [20, 26, 0, -11, 0, 1, 0, 0], [40, -8, 0, 0, 0, -4, 0, 0], [28, -20, 0, 7, 0, 7, 0, 0], [0, 0, 19, 0, -21, 0, 5, -2], [12, 30, 0, -2, 0, 2, 0, 0], [0, 0, 11, 0, 4, 0, -9, -3], [0, 0, -9, 0, -33, 0, 17, 12], [0, 0, 27, 0, 23, 0, 5, -6], [28, -16, 0, -3, 0, -1, 0, 0], [6, -21, 0, -10, 0, -2, 0, 0], [0, 0, 42, 0, -9, 0, -5, 6], [0, 0, 4, 0, 12, 0, -14, -3], [-22, -8, 0, 10, 0, -2, 0, 0], [0, 0, -11, 0, -25, 0, -1, -5], [-20, -26, 0, 5, 0, 2, 0, 0], [0, 0, -13, 0, -23, 0, 5, -3], [-54, -8, 0, -4, 0, 8, 0, 0], [-4, -22, 0, -1, 0, -14, 0, 0], [30, 4, 0, 0, 0, -8, 0, 0], [-40, 10, 0, 3, 0, -5, 0, 0], [0, 0, 15, 0, -13, 0, 17, -8], [0, 0, 8, 0, -19, 0, 4, 4], [0, 0, -16, 0, -9, 0, 3, -6], [0, 0, 12, 0, -27, 0, 6, 10], [0, 0, 22, 0, 15, 0, 9, -4], [6, -12, 0, 10, 0, 6, 0, 0], [0, 0, 4, 0, 16, 0, 20, -2], [-36, -16, 0, 11, 0, -1, 0, 0], [-8, -4, 0, 9, 0, -7, 0, 0], [-16, 0, 0, 6, 0, 0, 0, 0], [0, 0, 22, 0, -14, 0, -10, 3], [0, 0, -13, 0, 18, 0, -4, -12], [0, 0, 16, 0, -16, 0, 8, -11], [-30, -34, 0, 6, 0, -3, 0, 0], [-18, -11, 0, 12, 0, 6, 0, 0], [0, 0, 11, 0, 16, 0, -21, -11], [0, 0, 23, 0, -9, 0, -19, 4], [28, 18, 0, 5, 0, -8, 0, 0], [0, 0, -1, 0, 3, 0, 1, -10], [-40, 21, 0, 2, 0, 4, 0, 0], [-32, 20, 0, 5, 0, 3, 0, 0], [0, 0, 3, 0, -28, 0, 17, 5], [0, 0, 4, 0, 0, 0, -4, 11], [-14, -41, 0, 7, 0, -2, 0, 0], [0, 0, -5, 0, -4, 0, -2, -16], [32, -26, 0, -1, 0, 5, 0, 0], [0, 0, 6, 0, 30, 0, 0, 6], [0, 0, 13, 0, 12, 0, 9, 11], [0, 0, 16, 0, -1, 0, -11, 16], [-8, 40, 0, -4, 0, -8, 0, 0], [-32, -8, 0, 4, 0, -4, 0, 0], [56, 16, 0, 1, 0, 1, 0, 0], [-32, -28, 0, 9, 0, -7, 0, 0], [-2, -2, 0, -4, 0, -1, 0, 0], [0, 0, -7, 0, -38, 0, 16, 12], [30, 1, 0, 3, 0, -10, 0, 0], [0, 0, -23, 0, -21, 0, 15, 7], [0, 0, 30, 0, -18, 0, 4, -3], [-10, 9, 0, -8, 0, -10, 0, 0], [0, 0, -27, 0, -8, 0, 5, -13], [0, 0, 1, 0, -1, 0, 31, 1], [-12, 6, 0, 8, 0, 9, 0, 0], [0, 0, 10, 0, 33, 0, -9, -4], [26, 6, 0, 10, 0, -19, 0, 0], [-18, -9, 0, 7, 0, -2, 0, 0], [0, 0, -1, 0, -32, 0, -6, -8], [0, 0, -17, 0, -3, 0, 21, -3], [-12, -14, 0, -7, 0, 3, 0, 0], [0, 0, -25, 0, 0, 0, 16, 0], [0, 0, -19, 0, 34, 0, 5, -1], [0, 0, -21, 0, 9, 0, 1, -11], [6, 7, 0, 3, 0, -20, 0, 0], [-8, 36, 0, 1, 0, 1, 0, 0], [18, -10, 0, 8, 0, 1, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7056_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7056_2_b_x();".
// 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_7056_2_b_x(:prec:=8)
    chi := MakeCharacter_7056_b();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 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_7056_2_b_x();".
// 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_7056_2_b_x( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7056_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![196, 0, 640, 0, 284, 0, 32, 0, 1]>,<11,R![16, 0, 608, 0, 392, 0, 40, 0, 1]>,<13,R![98, 0, 20, 0, 1]>,<17,R![6724, 0, 4160, 0, 860, 0, 64, 0, 1]>,<19,R![-56, -96, -40, 0, 1]>,<31,R![-824, 160, 56, -16, 1]>,<53,R![784, -32, -88, 8, 1]>],Snew);
    return Vf;
end function;