
// Make newform 3969.2.a.bh in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3969_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_3969_2_a_bh();".
// 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_3969_2_a_bh();".
// 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 := [-7, 60, 96, -736, -234, 1280, -94, -620, 125, 92, -22, -4, 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], [-10235919, -75709700, 88708281, 217572487, -141596603, -115990022, 63630698, 14881733, -9010433, -178989, 377504, -29600], [-14106727, -16532156, 316226486, 273870783, -454973007, -156257935, 191938557, 9534853, -26580896, 2049118, 1106428, -153653], [-14106727, -25405094, 316226486, 273870783, -454973007, -156257935, 191938557, 9534853, -26580896, 2049118, 1106428, -153653], [4252571, 246718636, -349019797, -356348877, 585403491, 81347402, -273622386, 30456613, 39941737, -7584581, -1719752, 377200], [-37815883, -79593119, 201819119, 62503395, -351832803, 68245598, 183651234, -50779916, -28246487, 8078080, 1253056, -352181], [-16670305, 4887547, 10654406, -203137878, -237915981, 258297167, 170701317, -96673697, -28741478, 12444820, 1331248, -491816], [-16913407, -24935219, 102871706, 248276754, 67595349, -259040476, -89740203, 83640367, 16809322, -9883901, -814814, 371053], [-5512040, -4887547, -6217937, 203137878, 237915981, -258297167, -170701317, 96673697, 28741478, -12444820, -1331248, 491816], [-13564244, 4474145, 258774249, -36274042, -462927506, 115238431, 226224623, -63418903, -33756309, 9575126, 1474988, -410574], [-100038029, 154189073, 1630110649, -642900054, -3146428860, 1130271061, 1605699294, -533002183, -243920686, 76422230, 10762397, -3199606], [-18960010, 52483887, 299857215, -158897277, -579246461, 228532297, 297294947, -102681496, -45298156, 14514229, 2002445, -604072]];
    Rf_basisdens := [1, 5915292, 8872938, 8872938, 17745876, 8872938, 8872938, 5915292, 4436469, 2957646, 17745876, 2957646];
    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_3969_a();"
function MakeCharacter_3969_a()
    N := 3969;
    order := 1;
    char_gens := [2108, 3727];
    v := [1, 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;

function MakeCharacter_3969_a_Hecke(Kf)
    return MakeCharacter_3969_a();
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, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, -1], [0, 1, 0, -1, 0, 0, -1, 0, 0, 0, 1, 0], [0, -1, 0, 1, -1, 0, 1, 1, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0], [-3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, -1, 0, 0, 0, 0, 0, 0, -1, 0, 2], [0, 1, 0, 0, -1, 0, 0, -2, 0, 0, 1, 0], [1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1], [0, 1, 0, -2, 2, 0, -2, 0, 0, 0, 0, 0], [-1, 0, 3, 0, 0, 0, 0, 0, 1, 2, 0, -1], [0, -1, 0, 0, 0, 0, 1, -1, 0, 0, 1, 0], [-2, 0, -1, 0, 0, 1, 0, 0, 0, -2, 0, 1], [0, -3, 0, 4, -1, 0, 0, -1, 0, 0, -1, 0], [0, 2, 0, -1, 1, 0, 2, 1, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, -2], [-5, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, -2], [0, -1, 0, -2, 0, 0, -1, 1, 0, 0, 1, 0], [-2, 0, 2, 0, 0, -1, 0, 0, -3, 1, 0, 0], [0, 0, 0, 0, 0, 0, 1, -3, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, -1, -2, 0, 0, -2, 0], [0, -1, 0, 4, 1, 0, -3, -1, 0, 0, -1, 0], [0, -1, 0, 5, -2, 0, 0, -3, 0, 0, 1, 0], [0, 0, 0, -1, 1, 0, 2, -1, 0, 0, 0, 0], [-4, 0, -4, 0, 0, -1, 0, 0, 4, 0, 0, 1], [0, 0, -4, 0, 0, -1, 0, 0, 1, 2, 0, 1], [0, 0, -2, 0, 0, 1, 0, 0, 0, 0, 0, 0], [-4, 0, 1, 0, 0, 1, 0, 0, -3, -1, 0, 0], [0, -2, 0, 1, 2, 0, -1, 3, 0, 0, 1, 0], [2, 0, -3, 0, 0, -3, 0, 0, 5, -2, 0, 0], [0, 0, 0, -3, -1, 0, -4, -1, 0, 0, -1, 0], [-9, 0, 1, 0, 0, -2, 0, 0, -1, -1, 0, 1], [-2, 0, -1, 0, 0, 2, 0, 0, 2, 0, 0, 2], [0, -2, 0, -1, 1, 0, 0, 1, 0, 0, -1, 0], [-2, 0, 4, 0, 0, -2, 0, 0, -1, -1, 0, -3], [0, 0, 0, -4, 1, 0, 2, 4, 0, 0, -1, 0], [0, -1, 0, -1, 1, 0, -4, 0, 0, 0, 1, 0], [-8, 0, 2, 0, 0, -2, 0, 0, 1, -1, 0, 0], [0, -1, 0, 4, -3, 0, 1, 0, 0, 0, -1, 0], [-7, 0, -3, 0, 0, 3, 0, 0, 3, -1, 0, 2], [2, 0, -1, 0, 0, -1, 0, 0, -2, -2, 0, 0], [-2, 0, 1, 0, 0, 2, 0, 0, 0, 1, 0, 1], [0, 0, 0, 5, -1, 0, 4, 1, 0, 0, 0, 0], [-7, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, -3], [0, 5, 0, -3, 0, 0, -2, 0, 0, 0, -1, 0], [0, -4, 0, 0, 0, 0, 6, 5, 0, 0, 1, 0], [0, -5, 0, 1, -4, 0, 4, 2, 0, 0, -3, 0], [-5, 0, 1, 0, 0, 2, 0, 0, 1, 2, 0, -3], [-9, 0, 4, 0, 0, -1, 0, 0, 4, -2, 0, 2], [0, 1, 0, 5, -2, 0, 4, -7, 0, 0, -2, 0], [0, 4, 0, -6, 0, 0, -4, 1, 0, 0, -2, 0], [0, 5, 0, 2, 1, 0, -3, 0, 0, 0, 2, 0], [-4, 0, -2, 0, 0, 4, 0, 0, -1, 1, 0, 0], [0, 4, 0, 0, 0, 0, -8, -5, 0, 0, 5, 0], [0, -5, 0, -1, 0, 0, -1, 6, 0, 0, 4, 0], [-3, 0, -6, 0, 0, -3, 0, 0, 1, -2, 0, 1], [-5, 0, 0, 0, 0, -1, 0, 0, 1, 2, 0, -3], [0, -6, 0, 3, 0, 0, 3, 0, 0, 0, 3, 0], [0, 2, 0, -5, -4, 0, -2, -5, 0, 0, 1, 0], [0, 3, 0, -11, 3, 0, -4, 5, 0, 0, 4, 0], [0, 0, 0, -2, -1, 0, 1, 2, 0, 0, 2, 0], [0, 2, 0, -7, -4, 0, -2, -3, 0, 0, -3, 0], [-5, 0, -5, 0, 0, -3, 0, 0, 5, 0, 0, 2], [-6, 0, 0, 0, 0, 3, 0, 0, -3, 4, 0, -2], [9, 0, -7, 0, 0, -1, 0, 0, 4, -1, 0, -2], [-5, 0, 2, 0, 0, -2, 0, 0, 1, -1, 0, 3], [0, -1, 0, -11, 1, 0, -5, 0, 0, 0, 4, 0], [0, -7, 0, 6, 0, 0, -2, -1, 0, 0, -4, 0], [-15, 0, 10, 0, 0, -2, 0, 0, -3, 3, 0, 0], [0, 9, 0, -10, 5, 0, -3, 0, 0, 0, 1, 0], [-11, 0, 1, 0, 0, 4, 0, 0, -3, 3, 0, -2], [-1, 0, -7, 0, 0, -4, 0, 0, 6, -4, 0, 1], [0, 0, 0, -3, 0, 0, 1, 6, 0, 0, -5, 0], [-1, 0, 10, 0, 0, 2, 0, 0, -5, 0, 0, -2], [0, 6, 0, 3, 2, 0, -5, -8, 0, 0, -3, 0], [-4, 0, 11, 0, 0, 3, 0, 0, -1, -2, 0, 3], [0, -1, 0, 1, 1, 0, 6, 8, 0, 0, -4, 0], [0, -5, 0, 6, 0, 0, 1, 1, 0, 0, -2, 0], [-1, 0, -8, 0, 0, -2, 0, 0, 5, 3, 0, 1], [-7, 0, -5, 0, 0, 2, 0, 0, 0, -1, 0, 2], [0, 7, 0, 2, 4, 0, -1, 0, 0, 0, -3, 0], [0, 0, 0, 3, -2, 0, 1, -5, 0, 0, 1, 0], [-2, 0, 8, 0, 0, -1, 0, 0, -10, 0, 0, 2], [-8, 0, -6, 0, 0, 0, 0, 0, -1, 0, 0, 2], [6, 0, -2, 0, 0, 4, 0, 0, 1, -2, 0, -1], [0, 5, 0, -13, 1, 0, -4, 3, 0, 0, 2, 0], [-1, 0, -7, 0, 0, -4, 0, 0, -2, -1, 0, 1], [0, -3, 0, 13, -1, 0, -1, -4, 0, 0, -5, 0], [0, 4, 0, -10, 1, 0, -8, -1, 0, 0, -1, 0], [-1, 0, -5, 0, 0, 4, 0, 0, 2, -1, 0, 6], [-12, 0, -7, 0, 0, 1, 0, 0, -3, -3, 0, 3], [9, 0, -6, 0, 0, 3, 0, 0, 2, -4, 0, -1], [0, -1, 0, 10, 2, 0, 5, 4, 0, 0, 2, 0], [0, -2, 0, 11, -5, 0, 5, -1, 0, 0, 4, 0], [0, 1, 0, -8, 2, 0, 6, 6, 0, 0, -1, 0], [0, -5, 0, 5, -3, 0, -1, -3, 0, 0, 1, 0], [0, 0, 0, 0, 0, 3, 0, 0, 2, 0, 0, 9], [-6, 0, -3, 0, 0, -6, 0, 0, 0, 2, 0, -1], [9, 0, -1, 0, 0, 0, 0, 0, -8, -2, 0, -1], [0, -3, 0, -11, 2, 0, 3, 2, 0, 0, 2, 0], [-7, 0, -4, 0, 0, 1, 0, 0, -6, 2, 0, -4], [6, 0, -3, 0, 0, 3, 0, 0, -4, -1, 0, -1], [0, 5, 0, 5, 0, 0, -3, -1, 0, 0, -6, 0], [0, 7, 0, 6, 0, 0, -1, -8, 0, 0, -6, 0], [0, -7, 0, 11, 4, 0, 7, 3, 0, 0, 5, 0], [-9, 0, -10, 0, 0, 1, 0, 0, 0, -6, 0, 3], [0, -9, 0, 4, 0, 0, 5, 2, 0, 0, -2, 0], [0, -6, 0, 14, -1, 0, 5, -4, 0, 0, -7, 0], [-3, 0, -2, 0, 0, 1, 0, 0, 8, 0, 0, 4], [0, 0, 3, 0, 0, -5, 0, 0, -1, 5, 0, -2], [0, -9, 0, 7, -4, 0, 8, 5, 0, 0, 2, 0], [11, 0, 3, 0, 0, 0, 0, 0, -7, 2, 0, 2], [14, 0, -9, 0, 0, 2, 0, 0, 0, -7, 0, 2], [0, 2, 0, 17, -2, 0, 8, -5, 0, 0, 3, 0], [0, -5, 0, -7, -2, 0, -6, -1, 0, 0, -3, 0], [13, 0, -20, 0, 0, -3, 0, 0, 5, -3, 0, 8], [-14, 0, -2, 0, 0, 3, 0, 0, -3, 1, 0, 4], [0, 8, 0, 1, 0, 0, -1, -11, 0, 0, -5, 0], [-2, 0, 11, 0, 0, -1, 0, 0, -5, 0, 0, -1], [0, -2, 0, 15, 0, 0, 1, 7, 0, 0, 1, 0], [6, 0, 2, 0, 0, -2, 0, 0, -2, -5, 0, -1], [0, -1, 0, -5, -2, 0, 4, 12, 0, 0, 4, 0], [-9, 0, 4, 0, 0, 0, 0, 0, 0, 6, 0, -3], [5, 0, -6, 0, 0, 0, 0, 0, 4, -8, 0, 1], [0, -2, 0, 0, 0, 0, -7, 7, 0, 0, 1, 0], [0, -1, 0, -6, 3, 0, -1, 4, 0, 0, -11, 0], [0, 1, 0, 7, 2, 0, 3, 4, 0, 0, -5, 0], [13, 0, -1, 0, 0, 2, 0, 0, 0, -7, 0, 4], [-15, 0, 4, 0, 0, -1, 0, 0, -9, 0, 0, -3], [-5, 0, 8, 0, 0, -7, 0, 0, 3, -2, 0, 0], [0, 0, 6, 0, 0, 0, 0, 0, -1, 1, 0, -11], [0, -4, 0, 22, 2, 0, 2, -2, 0, 0, -2, 0], [0, -4, 0, -1, 7, 0, -11, 3, 0, 0, -2, 0], [0, -5, 0, 3, 6, 0, 4, 4, 0, 0, 1, 0], [0, 2, 0, 3, -8, 0, 1, 0, 0, 0, -5, 0], [0, 4, 0, -9, 0, 0, 3, -5, 0, 0, -2, 0], [14, 0, -10, 0, 0, -2, 0, 0, 3, 2, 0, -4], [0, -13, 0, 6, -4, 0, 5, 7, 0, 0, 2, 0], [5, 0, 6, 0, 0, 1, 0, 0, 3, 11, 0, -1], [11, 0, 8, 0, 0, 5, 0, 0, -4, 9, 0, 2], [-11, 0, -5, 0, 0, 1, 0, 0, 6, -2, 0, 1], [0, 2, 0, -13, 1, 0, 1, -3, 0, 0, -5, 0], [0, 3, 0, 1, -4, 0, 7, 2, 0, 0, 1, 0], [0, -4, 0, 11, -7, 0, 8, 1, 0, 0, -7, 0], [0, -9, 0, -2, 2, 0, -7, 5, 0, 0, 4, 0], [0, 5, 0, -16, -2, 0, 5, 4, 0, 0, 9, 0], [-9, 0, -4, 0, 0, 5, 0, 0, -3, 0, 0, 3], [11, 0, -2, 0, 0, 1, 0, 0, -7, 3, 0, -2], [0, -12, 0, -7, -2, 0, 3, -2, 0, 0, -1, 0], [-7, 0, 1, 0, 0, -5, 0, 0, 9, -2, 0, 5], [0, 3, 0, 11, -2, 0, -1, -17, 0, 0, 2, 0], [-4, 0, -6, 0, 0, -3, 0, 0, 6, -3, 0, -6], [-13, 0, 5, 0, 0, -1, 0, 0, 8, 4, 0, -3], [-2, 0, 3, 0, 0, -3, 0, 0, 1, 11, 0, -4], [0, -7, 0, 16, -1, 0, -4, 7, 0, 0, -1, 0], [0, 5, 0, 17, -3, 0, 4, -9, 0, 0, 8, 0], [0, -6, 0, 2, 4, 0, 3, -2, 0, 0, -7, 0], [22, 0, -13, 0, 0, 2, 0, 0, 10, -7, 0, 6], [8, 0, 6, 0, 0, -5, 0, 0, 2, -8, 0, 6], [-8, 0, 18, 0, 0, 0, 0, 0, 1, 3, 0, -3], [0, -1, 0, 18, -9, 0, 16, -1, 0, 0, -6, 0], [35, 0, -10, 0, 0, 2, 0, 0, 6, -7, 0, 5], [0, 5, 0, -6, 6, 0, 8, 5, 0, 0, 6, 0], [-10, 0, 14, 0, 0, 2, 0, 0, -1, 3, 0, -4], [0, -10, 0, 1, 0, 0, -11, 6, 0, 0, -4, 0], [6, 0, -4, 0, 0, -4, 0, 0, -1, 6, 0, 2], [0, 12, 0, 0, 3, 0, -6, -15, 0, 0, -7, 0], [-17, 0, 9, 0, 0, -5, 0, 0, 2, 0, 0, -1], [0, 0, 0, 5, -9, 0, 1, 4, 0, 0, -4, 0], [-19, 0, -6, 0, 0, -5, 0, 0, -8, 0, 0, -8], [-14, 0, 20, 0, 0, -4, 0, 0, -2, 9, 0, -4], [0, -8, 0, 4, 3, 0, -1, 6, 0, 0, -5, 0], [0, 12, 0, -14, -1, 0, -2, 8, 0, 0, 4, 0], [27, 0, 3, 0, 0, -3, 0, 0, -5, -3, 0, 0], [3, 0, 5, 0, 0, 6, 0, 0, 5, 2, 0, 1], [0, -1, 0, -19, 4, 0, 4, 3, 0, 0, -2, 0], [0, 1, 0, -4, 9, 0, -2, -4, 0, 0, -1, 0], [-31, 0, 5, 0, 0, -1, 0, 0, -2, 0, 0, 2], [0, -4, 0, 8, -5, 0, 3, -3, 0, 0, 5, 0], [5, 0, 18, 0, 0, -3, 0, 0, -10, 0, 0, -3], [0, 1, 0, 6, 0, 0, -10, -5, 0, 0, 5, 0], [-10, 0, 3, 0, 0, -2, 0, 0, -6, -1, 0, 2], [0, 4, 0, 0, 3, 0, -1, 7, 0, 0, 7, 0], [16, 0, 1, 0, 0, -2, 0, 0, -9, -4, 0, 6], [0, 2, 0, -10, 2, 0, -8, -3, 0, 0, 0, 0], [-9, 0, -1, 0, 0, 2, 0, 0, 5, 1, 0, 3], [0, 16, 0, -20, -1, 0, -7, -3, 0, 0, 5, 0], [0, 6, 0, -5, 2, 0, 0, -3, 0, 0, 7, 0], [6, 0, -5, 0, 0, -9, 0, 0, 1, 10, 0, -4], [6, 0, -3, 0, 0, 12, 0, 0, 4, 4, 0, -2], [0, 5, 0, 16, -4, 0, 8, -2, 0, 0, -8, 0], [-4, 0, 5, 0, 0, -4, 0, 0, -1, 4, 0, -12], [0, 8, 0, 8, -5, 0, 0, -9, 0, 0, 12, 0], [13, 0, 6, 0, 0, -3, 0, 0, -7, -11, 0, 7], [7, 0, 0, 0, 0, 0, 0, 0, -11, 5, 0, -1], [0, -9, 0, -10, 1, 0, 9, 13, 0, 0, 12, 0], [0, 8, 0, -26, 5, 0, -5, 10, 0, 0, 5, 0], [2, 0, 2, 0, 0, -1, 0, 0, -2, -6, 0, 1], [0, 1, 0, 17, 4, 0, 4, 8, 0, 0, 1, 0], [0, 2, 0, -7, 6, 0, 0, 11, 0, 0, -3, 0], [0, 14, 0, 0, 9, 0, -6, -10, 0, 0, 6, 0], [0, 16, 0, 10, 5, 0, -5, -6, 0, 0, 2, 0], [0, -7, 0, -3, -6, 0, 3, 8, 0, 0, -8, 0], [0, 14, 0, -17, 0, 0, -2, 0, 0, 0, 2, 0], [19, 0, -5, 0, 0, 0, 0, 0, -8, 5, 0, -3], [10, 0, 4, 0, 0, 2, 0, 0, -4, -3, 0, -7], [0, 1, 0, 7, 6, 0, 0, -2, 0, 0, -6, 0], [-1, 0, -3, 0, 0, 3, 0, 0, -3, -1, 0, -7], [0, 5, 0, -14, 8, 0, 6, 1, 0, 0, 1, 0], [11, 0, -4, 0, 0, 5, 0, 0, 6, -7, 0, 4], [0, -1, 0, 4, -4, 0, 0, 10, 0, 0, 4, 0], [0, 10, 0, -8, 2, 0, 0, 3, 0, 0, 12, 0], [0, -5, 0, 21, -5, 0, 1, -1, 0, 0, 1, 0], [-14, 0, 21, 0, 0, -3, 0, 0, -10, 3, 0, -6], [0, -1, 0, -6, 3, 0, -2, 8, 0, 0, -11, 0], [-3, 0, -14, 0, 0, 3, 0, 0, 7, 7, 0, 8], [16, 0, 19, 0, 0, -6, 0, 0, -6, -2, 0, -8], [2, 0, -5, 0, 0, -2, 0, 0, 10, 1, 0, -7], [0, -11, 0, 21, -5, 0, -1, -6, 0, 0, -6, 0], [-7, 0, -14, 0, 0, -7, 0, 0, 2, -8, 0, 3], [16, 0, 15, 0, 0, 0, 0, 0, 1, 4, 0, -8], [0, 7, 0, 11, 7, 0, -14, -10, 0, 0, 2, 0], [6, 0, -6, 0, 0, 0, 0, 0, 3, 5, 0, -1], [0, 5, 0, -15, -6, 0, 1, 2, 0, 0, 6, 0], [-26, 0, 15, 0, 0, 6, 0, 0, -18, 4, 0, -5], [0, 2, 0, -6, 9, 0, -11, 0, 0, 0, 2, 0], [-18, 0, 7, 0, 0, 2, 0, 0, 5, -4, 0, -2], [-28, 0, 9, 0, 0, 0, 0, 0, 3, -4, 0, -4], [0, 7, 0, -8, 1, 0, -7, 0, 0, 0, 6, 0], [-10, 0, 2, 0, 0, 2, 0, 0, -4, -2, 0, -6], [9, 0, 0, 0, 0, -2, 0, 0, 9, 3, 0, -9], [0, 7, 0, -18, 11, 0, 2, 6, 0, 0, 8, 0], [0, -6, 0, 13, -10, 0, 9, 2, 0, 0, -7, 0], [0, -12, 0, -4, -2, 0, 2, 8, 0, 0, -6, 0], [2, 0, 3, 0, 0, -5, 0, 0, 4, -6, 0, 1], [7, 0, -13, 0, 0, 11, 0, 0, 9, -2, 0, 1], [0, 6, 0, 2, 1, 0, -1, 1, 0, 0, -7, 0], [-27, 0, 9, 0, 0, -1, 0, 0, -3, 6, 0, 0], [0, -19, 0, 7, -13, 0, -2, -3, 0, 0, -9, 0], [0, 5, 0, -2, 4, 0, -11, 0, 0, 0, 4, 0], [-29, 0, 13, 0, 0, -5, 0, 0, -2, 1, 0, 2], [0, -5, 0, 7, -4, 0, 8, -3, 0, 0, -10, 0], [0, -5, 0, -17, -1, 0, 11, 9, 0, 0, -4, 0], [0, -4, 0, 6, 0, 0, 11, -5, 0, 0, -17, 0], [0, -7, 0, -3, 6, 0, 6, 2, 0, 0, -10, 0], [12, 0, -4, 0, 0, -4, 0, 0, -1, -1, 0, -2], [4, 0, -14, 0, 0, 0, 0, 0, -14, -1, 0, -6], [17, 0, -13, 0, 0, 8, 0, 0, 9, -11, 0, 9], [0, -6, 0, 14, -8, 0, 2, -2, 0, 0, -4, 0], [-12, 0, 3, 0, 0, -2, 0, 0, 10, -2, 0, 11], [0, -17, 0, 32, -7, 0, 10, -13, 0, 0, -6, 0], [0, 11, 0, 14, -2, 0, 0, -3, 0, 0, -5, 0], [4, 0, -4, 0, 0, 9, 0, 0, -1, -9, 0, 5], [-8, 0, 26, 0, 0, -1, 0, 0, -1, 10, 0, -3], [0, 0, 0, 21, -3, 0, -5, -11, 0, 0, -7, 0], [0, 5, 0, -27, 3, 0, 10, 8, 0, 0, 5, 0], [0, 5, 0, 16, 0, 0, -3, -10, 0, 0, 9, 0], [-11, 0, 15, 0, 0, -3, 0, 0, 4, -1, 0, 5], [-12, 0, 22, 0, 0, 9, 0, 0, -11, -2, 0, 2], [0, 19, 0, -6, 6, 0, -4, -6, 0, 0, -2, 0], [0, -6, 0, 7, 1, 0, 20, 14, 0, 0, 3, 0], [0, 12, 0, 8, -2, 0, -8, -2, 0, 0, -6, 0], [0, -10, 0, -2, -2, 0, 6, 8, 0, 0, -16, 0], [-28, 0, 14, 0, 0, -4, 0, 0, -9, 14, 0, -7], [0, 4, 0, -21, 0, 0, -5, 4, 0, 0, -3, 0], [-3, 0, -1, 0, 0, 5, 0, 0, -1, 3, 0, 5], [14, 0, -21, 0, 0, 5, 0, 0, 0, -1, 0, 8], [0, 14, 0, 2, 4, 0, -22, -11, 0, 0, 12, 0], [-5, 0, -4, 0, 0, -4, 0, 0, 9, -4, 0, 10], [0, 2, 0, -11, -7, 0, 3, 2, 0, 0, 13, 0], [8, 0, -14, 0, 0, -8, 0, 0, 1, 4, 0, -4], [0, -9, 0, 18, -11, 0, 15, -3, 0, 0, -1, 0], [0, -8, 0, 3, 3, 0, -8, 2, 0, 0, -13, 0], [-9, 0, 1, 0, 0, -2, 0, 0, 6, 6, 0, -6], [12, 0, -38, 0, 0, 1, 0, 0, 5, -11, 0, 8], [17, 0, 2, 0, 0, -7, 0, 0, 10, -1, 0, 4], [0, 12, 0, -11, 8, 0, -5, 2, 0, 0, 5, 0], [0, -9, 0, 15, 0, 0, 7, 12, 0, 0, 5, 0], [-19, 0, 1, 0, 0, -5, 0, 0, 11, -2, 0, 2], [0, 11, 0, -18, 6, 0, 4, -1, 0, 0, 2, 0], [0, -4, 0, -16, 0, 0, 3, 17, 0, 0, -3, 0], [0, -15, 0, 13, -9, 0, 9, 3, 0, 0, 12, 0], [-29, 0, 4, 0, 0, -7, 0, 0, -7, 6, 0, -10], [-10, 0, 24, 0, 0, 3, 0, 0, -15, 5, 0, -4], [23, 0, -5, 0, 0, -3, 0, 0, -4, 4, 0, -9], [0, -18, 0, -8, -4, 0, 16, 13, 0, 0, -9, 0], [0, 9, 0, -14, -1, 0, 0, -13, 0, 0, 10, 0], [29, 0, -13, 0, 0, 1, 0, 0, 10, -3, 0, -5], [0, -7, 0, -13, -5, 0, 15, 12, 0, 0, 12, 0], [12, 0, 4, 0, 0, 10, 0, 0, -7, -4, 0, -5], [0, -11, 0, 4, -4, 0, 11, 15, 0, 0, -5, 0], [-4, 0, -3, 0, 0, 6, 0, 0, 0, 10, 0, -5], [0, 2, 0, 1, -4, 0, -1, -2, 0, 0, 15, 0], [0, 11, 0, -9, -5, 0, -13, -11, 0, 0, -9, 0], [0, 7, 0, 3, -9, 0, 13, -8, 0, 0, -4, 0], [0, 15, 0, -5, -1, 0, 13, -16, 0, 0, -1, 0], [0, 0, 0, 5, -3, 0, 13, 4, 0, 0, 5, 0], [0, -4, 0, 5, -11, 0, 0, -4, 0, 0, -7, 0], [-10, 0, 19, 0, 0, 1, 0, 0, 2, -2, 0, 3], [-20, 0, 1, 0, 0, 7, 0, 0, -6, 2, 0, -6], [-5, 0, 2, 0, 0, -2, 0, 0, -7, 6, 0, -4], [-1, 0, 34, 0, 0, 4, 0, 0, -12, 5, 0, -3], [-24, 0, 7, 0, 0, -5, 0, 0, 2, 9, 0, -5], [12, 0, 11, 0, 0, 5, 0, 0, 2, 2, 0, -11], [0, -3, 0, 1, 7, 0, 2, -13, 0, 0, 3, 0], [-35, 0, -2, 0, 0, -1, 0, 0, 6, 1, 0, 7], [18, 0, -27, 0, 0, 0, 0, 0, 11, -7, 0, 8], [0, 7, 0, -31, 7, 0, -10, -4, 0, 0, -1, 0], [23, 0, -12, 0, 0, 3, 0, 0, 13, -9, 0, 4], [-29, 0, 1, 0, 0, -4, 0, 0, 7, -4, 0, 3], [17, 0, 0, 0, 0, -9, 0, 0, 7, 8, 0, -1], [2, 0, 20, 0, 0, 1, 0, 0, 15, 11, 0, -1], [0, 14, 0, 19, 0, 0, -6, -1, 0, 0, 3, 0], [0, -8, 0, 17, -2, 0, -10, 8, 0, 0, -6, 0], [-48, 0, 15, 0, 0, -1, 0, 0, -3, 12, 0, -6], [0, -18, 0, 11, 3, 0, 13, 10, 0, 0, 5, 0], [-7, 0, 8, 0, 0, -2, 0, 0, -2, -6, 0, 1], [0, 9, 0, 2, -6, 0, 5, -7, 0, 0, 4, 0], [-14, 0, 8, 0, 0, -1, 0, 0, -9, -2, 0, 3], [0, 3, 0, -10, 7, 0, -7, 13, 0, 0, -1, 0], [6, 0, 22, 0, 0, 7, 0, 0, -8, 0, 0, -5], [10, 0, 25, 0, 0, -7, 0, 0, -2, 5, 0, -6], [0, -3, 0, -1, 3, 0, -13, -1, 0, 0, -2, 0], [13, 0, -11, 0, 0, 10, 0, 0, -2, -3, 0, 7], [0, 11, 0, 5, 11, 0, -12, -13, 0, 0, 10, 0], [-15, 0, 1, 0, 0, -2, 0, 0, -1, 2, 0, 10], [19, 0, 9, 0, 0, 12, 0, 0, -2, 8, 0, 0], [-12, 0, 2, 0, 0, 2, 0, 0, 6, -6, 0, 2], [-15, 0, -6, 0, 0, 2, 0, 0, 6, -6, 0, 0], [0, 15, 0, 1, 2, 0, -1, -16, 0, 0, -1, 0], [0, 7, 0, -12, 0, 0, 4, 7, 0, 0, -7, 0], [-3, 0, 13, 0, 0, -5, 0, 0, -9, 10, 0, -7], [0, -21, 0, 18, -6, 0, -9, -3, 0, 0, -8, 0], [33, 0, -7, 0, 0, -1, 0, 0, -3, -1, 0, -2], [0, -12, 0, -11, -1, 0, -8, 5, 0, 0, 1, 0], [0, -2, 0, -17, 8, 0, -7, 15, 0, 0, 14, 0], [0, 11, 0, -13, 6, 0, -4, 7, 0, 0, -11, 0], [13, 0, 12, 0, 0, -6, 0, 0, -4, -12, 0, 0], [2, 0, 9, 0, 0, -1, 0, 0, 5, 7, 0, -15], [0, 3, 0, -26, -1, 0, -9, 8, 0, 0, 5, 0], [-26, 0, 5, 0, 0, 2, 0, 0, -6, -10, 0, -2], [-3, 0, -19, 0, 0, 5, 0, 0, -5, 1, 0, 8], [5, 0, -24, 0, 0, -9, 0, 0, 9, -1, 0, 2], [0, 1, 0, -23, 12, 0, -5, 2, 0, 0, 2, 0], [27, 0, -2, 0, 0, 10, 0, 0, 0, 0, 0, -5], [0, 11, 0, 7, 8, 0, -8, -8, 0, 0, -9, 0], [0, -11, 0, 12, 6, 0, 16, 7, 0, 0, 1, 0], [0, 1, 0, -23, 12, 0, -10, -9, 0, 0, 1, 0], [17, 0, -18, 0, 0, 0, 0, 0, 22, -4, 0, 5], [22, 0, -25, 0, 0, 1, 0, 0, 12, -11, 0, -2], [0, 0, 0, 7, 5, 0, -15, -3, 0, 0, -2, 0], [33, 0, -1, 0, 0, 8, 0, 0, 4, 3, 0, -1], [0, 11, 0, -7, -5, 0, -5, -15, 0, 0, 5, 0], [0, 4, 0, -19, 1, 0, 16, 11, 0, 0, 6, 0], [0, 13, 0, -20, 2, 0, -15, -18, 0, 0, -2, 0], [8, 0, -6, 0, 0, -3, 0, 0, 14, 1, 0, -9], [1, 0, 12, 0, 0, -11, 0, 0, -5, -1, 0, -9], [-2, 0, 18, 0, 0, -15, 0, 0, -16, 11, 0, -7], [0, 2, 0, -3, 6, 0, -17, -1, 0, 0, 1, 0], [-13, 0, -1, 0, 0, -13, 0, 0, -3, -4, 0, 5], [11, 0, 0, 0, 0, 3, 0, 0, -15, 11, 0, -1], [0, 13, 0, 2, -7, 0, -7, -12, 0, 0, -4, 0], [-12, 0, 6, 0, 0, -3, 0, 0, 0, 9, 0, -3], [0, 9, 0, -28, 10, 0, -11, 7, 0, 0, 10, 0], [18, 0, 7, 0, 0, 1, 0, 0, 6, -3, 0, 4], [-17, 0, -3, 0, 0, 3, 0, 0, 8, -8, 0, 16], [-19, 0, -2, 0, 0, 6, 0, 0, -7, -5, 0, 6], [0, 9, 0, 13, 5, 0, 5, 11, 0, 0, -13, 0], [9, 0, 7, 0, 0, 9, 0, 0, -14, -2, 0, -1], [22, 0, -20, 0, 0, -7, 0, 0, 4, -4, 0, 12], [0, -9, 0, -14, -10, 0, 7, -5, 0, 0, -15, 0], [19, 0, -19, 0, 0, -4, 0, 0, 3, 2, 0, -5], [0, -2, 0, -1, -8, 0, -1, -16, 0, 0, 7, 0], [-20, 0, -11, 0, 0, -15, 0, 0, 9, -8, 0, 1], [0, 12, 0, 7, -3, 0, -4, 5, 0, 0, -5, 0], [0, 7, 0, 7, 5, 0, -18, -12, 0, 0, -9, 0], [0, 27, 0, -28, -1, 0, -12, -12, 0, 0, -2, 0], [0, 2, 0, 4, 2, 0, -8, -17, 0, 0, -7, 0], [37, 0, -11, 0, 0, 6, 0, 0, 1, 5, 0, 12], [-44, 0, 9, 0, 0, 6, 0, 0, -2, 7, 0, -8], [36, 0, -9, 0, 0, 7, 0, 0, 12, 0, 0, 12], [0, -9, 0, 7, 4, 0, -10, 14, 0, 0, -3, 0], [0, -10, 0, 17, -2, 0, 11, 0, 0, 0, -18, 0], [-1, 0, 9, 0, 0, 0, 0, 0, -6, -5, 0, 4], [0, -14, 0, 31, -8, 0, 18, -8, 0, 0, -10, 0], [-19, 0, -2, 0, 0, 4, 0, 0, 5, -7, 0, 0], [0, 22, 0, -16, 1, 0, -6, -4, 0, 0, 0, 0], [16, 0, -6, 0, 0, 0, 0, 0, 6, -10, 0, 5], [0, 3, 0, -42, -3, 0, 4, 3, 0, 0, 11, 0], [-18, 0, 8, 0, 0, 6, 0, 0, 3, 0, 0, 6], [0, -13, 0, 7, 1, 0, 9, 11, 0, 0, -1, 0], [-28, 0, -23, 0, 0, -3, 0, 0, 3, -1, 0, -1], [26, 0, 4, 0, 0, 13, 0, 0, -6, 2, 0, 3], [0, -8, 0, -14, -6, 0, 7, 20, 0, 0, -7, 0], [0, -16, 0, 4, -7, 0, -9, -5, 0, 0, -9, 0], [-11, 0, 29, 0, 0, -4, 0, 0, -8, 4, 0, -15], [4, 0, 18, 0, 0, 8, 0, 0, -20, 11, 0, -3], [0, -1, 0, -11, 5, 0, 10, 0, 0, 0, 9, 0], [20, 0, 9, 0, 0, -8, 0, 0, -1, 13, 0, -3], [-19, 0, 1, 0, 0, -5, 0, 0, -12, -3, 0, -14], [0, 12, 0, -15, -6, 0, 12, 9, 0, 0, 7, 0], [0, 19, 0, -18, -3, 0, -12, 10, 0, 0, -3, 0], [0, -3, 0, 1, 1, 0, 4, 1, 0, 0, -11, 0], [-7, 0, 29, 0, 0, 5, 0, 0, -9, -5, 0, -3], [-21, 0, 25, 0, 0, 9, 0, 0, 2, 17, 0, -17], [0, -2, 0, 3, 8, 0, -13, 3, 0, 0, -10, 0], [0, -9, 0, -6, -9, 0, 7, 9, 0, 0, 3, 0], [0, 19, 0, -11, 1, 0, 2, 0, 0, 0, 6, 0], [7, 0, 11, 0, 0, -7, 0, 0, -15, -8, 0, -8], [22, 0, 7, 0, 0, 7, 0, 0, -20, -4, 0, -6], [23, 0, 0, 0, 0, 6, 0, 0, 11, 11, 0, -1], [-6, 0, 6, 0, 0, 3, 0, 0, -1, 24, 0, -12], [0, -1, 0, 33, -3, 0, 7, -4, 0, 0, 16, 0], [13, 0, -15, 0, 0, 2, 0, 0, 13, -7, 0, 0], [0, 15, 0, -29, -5, 0, -20, 1, 0, 0, -14, 0], [0, 7, 0, -7, 10, 0, 5, 8, 0, 0, -12, 0], [0, -1, 0, -24, 3, 0, -3, -7, 0, 0, 3, 0], [5, 0, 5, 0, 0, 5, 0, 0, -15, -1, 0, -7], [0, -3, 0, -32, -1, 0, 14, 11, 0, 0, 11, 0], [14, 0, -7, 0, 0, 15, 0, 0, -8, 3, 0, -2], [0, 15, 0, 32, 1, 0, -8, -8, 0, 0, 11, 0], [0, -10, 0, -12, -2, 0, -3, -4, 0, 0, 5, 0], [0, 21, 0, -8, 17, 0, -9, -4, 0, 0, 4, 0], [31, 0, 24, 0, 0, 6, 0, 0, -2, 2, 0, 9], [30, 0, 13, 0, 0, 4, 0, 0, -11, -8, 0, -1], [0, 11, 0, -6, 11, 0, 3, 5, 0, 0, -8, 0], [0, -6, 0, -25, -2, 0, 7, 22, 0, 0, -2, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3969_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3969_2_a_bh();".
// 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_3969_2_a_bh(:prec:=12)
    chi := MakeCharacter_3969_a();
    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_3969_2_a_bh();".
// 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_3969_2_a_bh( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3969_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 14, 10, -12, -7, 2, 1]>,<5,R![49, 0, -4470, 0, 5850, 0, -2580, 0, 465, 0, -36, 0, 1]>,<11,R![283, 160, -211, -134, 8, 10, 1]>,<13,R![502681, 0, -621030, 0, 211779, 0, -31500, 0, 2283, 0, -78, 0, 1]>],Snew);
    return Vf;
end function;