
// Make newform 61.2.g.a in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_61_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_61_g_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_61_2_g_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_61_2_g_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 := [1, 0, 34, 0, 229, 0, 558, 0, 624, 0, 361, 0, 111, 0, 17, 0, 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, 0, 0, 0], [-123, 293, -1533, 839, -5104, 2112, -6374, 3686, -3750, 3082, -1117, 1231, -162, 226, -9, 15], [-41, 79, -731, 1533, -2508, 5104, -2770, 6374, -1426, 3750, -387, 1117, -54, 162, -3, 9], [-15, 123, -217, 1533, -2596, 5104, -6258, 6374, -5674, 3750, -2333, 1117, -434, 162, -29, 9], [-41, -79, -731, -1533, -2508, -5104, -2770, -6374, -1426, -3750, -387, -1117, -54, -162, -3, -9], [15, 123, 217, 1533, 2596, 5104, 6258, 6374, 5674, 3750, 2333, 1117, 434, 162, 29, 9], [-123, 129, -37, -2217, 484, -10164, -82, -15402, -758, -10850, -501, -3837, -118, -650, -9, -41], [133, -114, 1817, -2186, 5896, -7392, 8258, -9602, 5670, -6004, 1983, -1926, 334, -302, 21, -18], [-49, -85, -835, -1897, -1606, -7634, -634, -10888, 338, -7300, 293, -2477, 68, -406, 5, -25], [-31, -3, -865, 1875, -4048, 7634, -5858, 10888, -3884, 7300, -1303, 2477, -212, 406, -13, 25], [79, -663, 1533, -6881, 5104, -20108, 6374, -24158, 3750, -14414, 1117, -4497, 162, -694, 9, -41], [-49, 85, -835, 1897, -1606, 7634, -634, 10888, 338, 7300, 293, 2477, 68, 406, 5, 25], [-117, 1057, -795, 8659, -2112, 22924, -3686, 26342, -3082, 15394, -1231, 4755, -226, 730, -15, 43], [79, 663, 1533, 6881, 5104, 20108, 6374, 24158, 3750, 14414, 1117, 4497, 162, 694, 9, 41], [-117, -1057, -795, -8659, -2112, -22924, -3686, -26342, -3082, -15394, -1231, -4755, -226, -730, -15, -43], [-133, -158, -1817, -2186, -5896, -7392, -8258, -9602, -5670, -6004, -1983, -1926, -334, -302, -21, -18]];
    Rf_basisdens := [1, 88, 88, 88, 88, 88, 88, 44, 44, 44, 88, 44, 88, 88, 88, 44];
    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_61_g();"
function MakeCharacter_61_g()
    N := 61;
    order := 10;
    char_gens := [2];
    v := [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_61_g_Hecke();"
function MakeCharacter_61_g_Hecke(Kf)
    N := 61;
    order := 10;
    char_gens := [2];
    char_values := [[0, -1, 0, 0, 0, 0, 0, 0, 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 := [[0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, -1, 1, 0, -2, -2, 1, 1, -1, -1, -2, 2, -2, -1, 0, 1], [1, 0, 1, 0, 0, -1, 0, 0, -1, -1, -1, 0, -2, 0, 0, 0], [0, 0, -1, -2, 0, 2, -1, 0, 0, 2, 2, -1, 3, 1, 2, -1], [1, 2, -1, 1, 1, 0, 1, -1, 1, -1, 0, -1, 0, 0, -2, -1], [1, 0, 1, 1, 1, 0, -1, 0, -1, -1, 2, -1, 1, 2, 1, 0], [1, 0, -4, 5, 5, 4, -3, -3, 3, 2, 4, -3, 1, 1, -3, 1], [-2, -2, 1, -1, -3, -1, 1, 2, 0, 0, -2, 2, 1, -4, 0, -1], [-1, 0, 1, -2, -1, -2, 1, 1, -1, 0, 0, 1, -1, 1, 1, 1], [2, 4, -1, 2, 1, 3, -1, -1, 2, 1, 4, -2, 2, 2, 0, -1], [-2, -1, -1, 1, 0, 0, 1, 0, 0, -2, -2, 2, -2, 0, 0, 1], [2, 3, 1, 3, 2, -2, 1, -3, 0, 0, 1, -2, -4, 0, -3, 1], [0, 0, 0, -2, 0, 0, -2, -1, 2, 1, -1, -1, 2, 1, 0, 0], [-2, 0, -5, 2, 4, 7, -2, -4, 6, 3, 9, -4, 8, 3, 0, -3], [3, 0, 3, 1, 3, -1, 0, 0, -2, 0, -2, -2, -2, -2, -2, 0], [-3, 0, 1, -3, -5, -5, 2, 3, 1, 0, -5, 4, -2, -1, -1, 0], [1, 1, 0, -2, -2, -2, 0, 3, -1, -3, -2, 0, 0, -1, 2, -1], [-6, -2, 4, -6, -5, 0, 0, 3, -1, 0, -4, 3, 2, 1, 5, 0], [-3, 1, -3, 1, 0, 2, 0, 0, 2, 2, 5, 0, 0, 2, 0, 0], [-3, 0, 1, 0, -4, -5, 1, 0, -2, -1, -2, 1, -3, 3, -1, 1], [-3, -3, -1, -1, 2, 0, -1, -2, 2, 4, 0, -2, 1, -1, 0, 0], [2, -3, 2, 4, -1, -3, 2, 3, -1, -2, -4, 5, -6, -5, -2, 2], [1, 1, 1, 1, -1, 0, 1, 0, -4, -2, -2, 2, -4, -3, 2, 2], [4, -6, 7, 0, 0, -1, 2, 4, -3, -4, -4, 0, 0, -2, 1, 2], [0, -5, 0, -2, -1, -2, 0, 2, 0, 0, -1, 0, 0, -5, 2, 0], [1, 2, -2, 3, 2, 7, -4, -4, 2, 4, 6, -2, 7, 4, 1, -4], [-3, -3, 3, -8, -6, -3, 2, 6, -2, 2, -4, 4, 3, 2, 4, -1], [11, 6, 1, 1, 3, 2, -8, -4, 4, 3, 13, -2, 8, 12, 2, -5], [3, 0, 2, 0, 2, 1, -1, 0, -4, -1, 0, -4, -3, 0, -3, 0], [-3, 0, 0, 1, -5, -2, 5, -3, 1, 0, -2, 1, -3, -1, -5, 1], [1, -2, 5, -5, -6, -6, 1, 1, -5, -1, -5, 1, -6, -1, -1, 1], [-1, 0, -4, 1, 6, 4, -2, -2, 1, 1, 1, -2, 2, 2, 2, -2], [-7, -2, -2, 3, 4, 0, 0, 2, -2, -2, -2, 0, 0, -9, -2, 2], [-5, -7, 6, -7, -4, -4, 3, 3, -4, -2, -13, 3, -2, -13, 3, 1], [-2, 1, 5, -6, -10, -7, 10, 7, -7, -5, -7, 9, -10, 2, 2, 9], [9, 18, -4, 4, 4, 2, 2, -7, 1, -2, 8, -1, -2, 6, -2, -7], [2, 2, -6, 5, 5, 5, -5, -1, 3, 1, 14, 0, 7, 7, 2, 2], [0, 0, 6, -8, -2, -2, 0, -1, -4, -2, -7, -2, 0, 2, -3, 4], [0, -7, 2, 4, 0, -4, 0, 2, -6, -3, -14, 7, -10, -4, -1, 6], [4, -5, 2, 3, -3, -6, 0, 4, 0, -3, -2, 5, -6, -6, -1, 3], [8, 13, -4, 2, 6, 2, 3, -7, -3, 1, 6, -2, -7, 2, -3, -2], [-5, 1, -12, 1, 1, 6, 0, -9, 8, 9, 8, 0, 5, 4, -1, -2], [4, 0, 5, 11, -3, -11, 4, 2, -8, -8, -14, 9, -20, -5, -7, 8], [1, -5, 8, -4, 0, -5, 4, 5, -11, -5, -9, 6, -14, -2, -1, 5], [-2, 0, 4, -7, -4, 1, -2, 4, -1, -2, 1, -1, 5, 1, 5, -4], [0, -1, -10, 5, 8, 11, -5, -2, 10, 5, 2, -11, 16, -1, 3, -5], [3, -5, 1, -1, -1, 3, -6, -3, 6, 3, 4, 0, 6, 2, 3, -3], [-1, -2, -3, -2, 3, -4, 2, 7, -3, -2, -5, 3, -10, -1, 6, 7], [-5, 2, -4, -11, -6, -1, 5, 5, 6, 10, 3, -8, 9, -6, 5, -10], [5, 2, 1, 2, 2, -5, 7, 6, -6, -7, -6, 1, -7, -1, -5, 5], [0, 0, 3, 1, -3, -7, 8, 1, -4, -8, -6, 4, -14, -10, -2, 1], [-5, 3, -4, -7, 0, 2, -1, -1, -2, 6, -1, -1, 4, 1, 1, -1], [-1, -1, -1, -1, 1, -1, -1, 0, 6, 2, 2, -2, 5, 2, -4, -3], [8, 1, 4, 7, 9, -3, -1, 0, 0, -5, -3, -3, -6, -3, -9, -1], [1, 0, 1, -1, 5, 7, -2, -2, 0, -2, 8, 0, 5, 8, 5, 2], [11, 11, -7, -1, 13, 11, -6, -5, 7, 7, 12, -12, 11, 8, 0, -4], [0, 7, -8, 4, 8, 8, -12, -11, 12, 6, 22, -6, 12, 13, -5, 0], [7, 4, 0, 3, 11, 3, -12, -7, 7, 8, 17, -9, 12, 6, -2, -6], [-1, 3, -1, 3, 3, 4, -3, -1, 3, 1, 10, 0, 6, 5, 2, 1], [16, 12, 0, 9, 3, 1, -1, -4, 0, -9, 3, -1, -8, 2, -5, -1], [-4, -4, 12, -10, -14, -18, 4, 4, -10, -4, -8, 8, -8, 6, 6, 4], [5, 8, 7, -4, -7, -4, 5, 0, 0, -3, 0, -1, -5, 1, -1, -3], [-7, 7, 1, -2, -2, 5, 3, -4, 9, 4, 10, 0, 10, 5, 5, -4], [-3, -5, -3, 8, 5, -3, -1, -1, -7, -4, -2, 6, -10, 4, -1, 14], [-1, -1, -1, 1, -9, -7, 1, 0, 2, 2, 2, 0, 3, -1, 0, -2], [-6, -7, 0, 3, 2, 4, -8, -4, 4, 0, -3, -2, 8, -8, 2, -8], [-12, -8, -8, -6, -6, 9, -1, -3, 4, 3, -2, 0, 10, -1, 1, 5], [0, 0, -4, 2, 4, -7, 9, 8, -10, -9, -14, 10, -21, -4, 3, 8], [-5, -5, 1, 1, -3, -2, 6, 6, -8, -6, -12, 12, -14, 0, 2, 8], [14, 7, -7, 17, 10, 1, -8, -8, 4, -2, 10, -3, 1, 15, -8, 1], [-15, 0, -3, -21, -13, 2, 9, 5, -2, 9, -7, -2, 8, -7, 8, -5], [0, -5, 13, 5, -13, -15, 2, 6, 0, -11, -11, 8, -7, -4, -5, 2], [11, 0, 3, -16, -13, -1, 1, 8, 3, 1, -3, 3, 6, -3, 6, -8], [6, -4, 0, 4, 3, -3, 3, -2, 0, 6, 1, -7, -7, 3, -12, 3], [0, 0, -3, 8, 3, 1, -3, -3, -3, -3, -8, -3, -5, -8, -5, 3], [-3, 2, -3, 4, 1, 3, -2, -2, 4, 1, 8, -11, 12, -9, -2, -13], [12, 8, -1, 2, 5, 11, -2, -2, 0, 1, 9, -6, 3, 16, -2, 0], [-2, -2, 0, -9, 5, 12, -12, -2, 0, 10, 10, -13, 21, 0, 10, -12], [-3, 1, -7, -1, -1, 2, 11, -1, -2, 1, 2, 0, -1, 1, -3, 4], [6, 5, -6, 21, 10, 9, -1, -8, 4, -1, 3, -2, -1, -3, -6, 1], [5, -4, 5, 11, 5, -4, 4, -5, -15, -9, -4, 3, -22, 1, -8, 9], [-4, -5, 9, -12, 1, 2, 0, 0, 2, 3, -5, 3, -4, -5, 0, 5], [3, -11, -1, 5, 10, 13, -13, -5, 9, 7, 19, -9, 20, 9, 13, -9], [14, 10, -3, -7, 2, 5, 1, 1, 5, 9, 8, -14, 11, 12, 1, -8], [4, 8, 12, -4, -8, -10, 6, 5, -5, -10, -3, 5, -6, -1, 0, 0], [0, 1, 9, -2, -7, -2, 9, -2, -6, -3, -15, 6, -9, -5, -5, -3], [-4, -4, -1, -7, 1, -8, 6, 10, -1, -2, -12, 12, -13, -7, -1, 11], [-12, -8, 11, -5, -9, -4, -7, 3, 0, -1, -4, -1, 12, -6, 8, -7], [-1, -8, 0, 4, 0, 0, -1, 2, -2, -8, -13, 3, 1, -6, 1, -4], [0, 3, -2, 5, -3, -8, 0, -7, 0, 0, -2, 11, -11, 3, -7, 0], [7, 14, 3, -4, -3, -7, 3, 4, -6, -3, 3, 6, -2, 5, -4, 4], [2, -1, -14, -7, 7, 9, -4, -4, 4, 7, 13, -4, 11, 4, 4, -4], [5, 0, 1, 21, 17, 5, -10, -8, -3, -10, -6, -3, -16, -6, -16, 8], [-7, -7, -3, -15, -5, 11, -4, 9, 7, 17, 8, -8, 29, -2, 10, -12], [-2, -4, -12, 3, 12, 17, -14, -3, 14, 14, 19, -14, 24, 5, 4, -3], [-13, 5, -3, -4, -11, -7, 12, 7, -7, -10, -10, 16, -12, -7, 9, 4], [-2, -4, 2, 0, -2, 4, -4, -5, 9, 4, 2, -9, 17, 2, -9, -5], [2, 14, -9, 8, 8, -6, 0, 3, 0, -3, 4, 0, -3, 2, 3, -6], [-7, 1, -12, 7, 7, 7, -7, -5, 10, 5, 8, 0, 12, 4, 5, -3], [11, 5, -10, 22, 24, 23, -6, -15, 5, -1, 15, -10, 4, 0, -5, 1], [-5, -11, 6, 4, 4, 3, 4, 0, -5, 0, -10, 0, -2, -5, -5, 8], [-2, 14, -9, -6, 0, 8, -10, -5, 13, 8, 9, -14, 26, 8, 9, -8], [11, 0, 9, -10, -13, -10, -2, 11, 3, -2, 15, 3, 10, 15, 10, -11], [0, -19, -8, 17, 9, 4, -8, -4, 2, 1, -4, 6, -5, -12, -3, 6], [-12, -12, -1, 9, -3, 6, -1, -3, 2, -1, 2, -2, 5, -4, -3, -6], [5, -3, 12, -15, -15, -8, 17, 5, -12, -5, -14, 0, -15, -7, -7, 4], [3, 9, 5, -12, -19, -24, 10, 10, -4, -3, -4, 20, -11, 20, 10, 2], [-10, -11, 6, -11, -12, -4, 9, 9, -10, -4, -24, 20, -17, -8, 9, 14], [0, 0, -7, 1, 7, 11, -10, -19, 19, 10, 18, -19, 22, 2, -2, -19], [25, 12, 17, 11, -9, -13, 5, 3, -2, -15, 8, 4, -18, -4, -5, 4], [7, 7, -5, 14, 6, 1, 3, -2, -2, -8, -6, 6, -19, 13, -6, 11], [2, 2, -3, 4, 6, 12, 1, -1, -1, -3, -2, 2, 2, 1, -2, -5], [-28, -13, -10, -13, -7, 13, -3, 3, 0, 16, -4, -6, 16, -14, 7, -3], [7, 7, 17, -5, -10, -13, -1, 7, -11, -5, -9, 3, -12, -3, 1, 3], [-2, -4, 5, -2, -5, -8, 6, 8, -9, -6, 8, 9, -9, -24, -3, 8], [14, 10, 3, 3, 4, -4, 1, 7, 0, -8, -3, 6, -4, -7, -5, 1], [0, -10, 1, 5, 5, -2, 0, -1, -6, 1, -14, 0, -9, -7, -7, 4], [-2, -14, 5, 3, -6, -7, 2, -1, 0, -6, -3, 5, -5, 16, 1, 2], [11, 13, -6, 1, 8, 6, -3, -10, 2, -2, -2, -5, -1, -4, -5, 2], [14, 8, 0, 4, 2, -5, 1, 10, -10, -11, 6, 12, -1, 15, 2, 9], [-10, 2, -1, 10, -2, -5, 6, -3, -1, -3, -8, 7, -12, 6, -10, 3], [1, -1, -1, 2, 2, -2, 11, 0, -5, 0, -4, 0, -7, -2, -5, 3], [13, -1, -4, 3, 7, 7, -7, -7, 15, 8, -8, -13, 13, -15, -7, -13], [-7, -3, -21, 4, 13, 10, -1, -4, 4, 15, 9, -5, 1, 3, -1, 7], [0, 0, 3, -6, 7, 10, -7, 8, 0, 0, 3, -13, 13, 7, 8, 7], [-28, -23, 0, 4, -1, -6, 6, 2, 0, -4, -20, 6, -10, -7, -6, 6], [-22, -11, 3, -6, -19, -9, 13, 13, -17, -10, -19, 19, -13, -15, 13, 7], [-15, 0, -4, 22, 10, 0, -8, -7, 6, -8, -11, 6, -13, -11, -13, 7], [18, 10, -4, 11, 17, 10, -4, -8, 2, 1, 21, -6, 1, 6, 4, -6], [-1, 0, -6, 0, -4, 6, -4, -1, -5, -4, 17, -5, 6, 17, 6, 1], [-22, -14, 11, -11, -20, -9, -2, 13, 0, -4, -13, 13, 13, -10, 13, -2], [2, 4, -20, 8, 20, 16, -8, -12, 19, 8, 23, -19, 27, -3, -11, -12], [-6, -6, 7, -18, -22, -13, 6, 19, -11, 7, -12, 12, -3, 3, 18, 10], [0, 20, -4, 2, -5, -5, 1, -7, 8, 4, 2, 7, -3, 15, -3, -9], [-8, -16, -6, -2, 6, 14, -16, -2, 5, 16, 16, -5, 27, 0, 5, -2], [10, 3, -1, -7, 16, 5, -4, 3, 0, 10, 9, -7, 11, 1, 1, -4], [0, 2, -6, 4, 5, 11, -5, -9, 12, 6, 30, -7, 13, 1, -3, -7], [15, 0, -12, 13, 14, -2, -6, -6, 3, 5, -1, -7, -1, 9, -6, 9], [-13, -17, 12, 0, -20, -16, 8, 16, 0, -12, -25, 12, -8, -12, -8, 12], [0, 11, -4, -14, 3, 14, -4, -2, 16, 8, 19, -24, 32, 7, 6, -12], [-2, 0, 1, -10, -8, 4, -1, 0, -2, 12, 13, -14, 25, 4, 14, -12], [23, 11, 1, 7, 7, -12, 7, -3, -7, 3, 10, 0, -22, 5, -10, -5], [-2, -7, 7, -7, -14, -2, -5, 0, 0, 14, 6, -12, 12, 0, 0, -5], [-4, -19, -13, 7, 22, 13, -15, -4, 4, 21, 13, -11, 15, 6, -7, 13], [0, -4, 35, -37, -24, -23, 20, 20, -30, -15, -34, 6, -21, -9, 5, 10], [3, 23, -1, -1, -2, 3, 8, -6, 6, 3, 12, -14, -8, -8, -8, -9], [8, 0, 1, -2, -5, -3, -1, 3, 7, -1, -1, 7, 1, -1, 1, -3], [6, 6, 0, 6, 10, -3, -5, -13, 0, -3, 10, -10, -6, -5, -3, 3], [-17, -13, -2, -5, -4, 8, -9, -7, 10, 10, 3, -8, 21, 8, 9, -8], [0, 22, -3, 0, 2, 5, -10, 6, 11, -3, 8, -2, 16, 2, 10, -2], [17, 7, -5, 15, 9, 0, -5, -5, -4, -5, 6, -3, -1, 19, -5, 5], [17, -9, -18, 13, 12, 21, -18, -7, 7, 7, 9, -4, 18, 20, 3, -7], [-7, -14, 18, -4, -18, -6, 2, 9, -9, -2, -17, 9, -9, -1, 5, 9], [-15, 11, 7, -13, -13, -7, 9, -3, -1, 3, -18, 0, -11, -9, -4, -6], [2, -5, 13, -19, -26, -34, 12, 15, -11, -8, -30, 22, -28, -16, -7, 8], [8, 0, 1, -10, 4, 11, -22, 6, 7, 11, 26, -8, 32, 8, 22, -8], [-8, -15, -7, -1, -1, 4, -4, -2, 11, -2, 1, -3, 10, 3, 4, -3], [17, 0, -5, 9, 7, -1, 4, -6, 1, 4, 16, 1, -20, 16, -20, 6], [-28, -28, -2, -8, 2, 5, 1, 6, -6, 4, -2, 2, -1, -6, 10, 5], [2, -2, 11, 5, 5, -5, 0, 11, -17, -11, -10, 0, -11, -5, -6, 12], [7, 14, 7, -5, -7, -4, -1, 7, -9, 1, -7, 9, -5, 23, 7, 7], [4, 4, -12, 19, 30, 29, -18, -18, 17, 11, 34, -25, 13, 7, -18, -1], [-12, -3, 4, 16, -4, -9, 22, -8, -6, -8, -26, 7, -29, -7, -22, 7], [6, 7, 0, 12, 14, -15, 20, 10, -22, -22, -38, 15, -49, -15, -20, 15], [-19, 0, -4, 1, 2, 4, 1, -3, 6, 1, -10, 6, 7, -10, 7, 3], [-32, -9, -2, -1, 1, 4, -2, -2, 0, -2, -13, 1, 3, -17, 6, -2], [-2, -3, 20, -11, -14, -21, 18, 9, -9, -13, -25, 6, -18, -9, -3, 5], [1, 2, -19, 3, 19, 16, -13, -6, 16, 13, 23, -16, 18, 5, 8, -6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_61_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_61_2_g_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_61_2_g_a(:prec:=16)
    chi := MakeCharacter_61_g();
    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_61_2_g_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_61_2_g_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_61_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([],Snew);
    return Vf;
end function;