
// Make newform 8004.2.a.h in Magma, downloaded from the LMFDB on 17 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_8004_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_8004_2_a_h();".
// 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_8004_2_a_h();".
// 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 := [-5832, -5184, 15102, 7832, -15445, -3259, 7388, 65, -1652, 180, 158, -27, -5, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [30728510682372, -13353849263538, -68386010563462, 31079563327595, 35348665957787, -19340319330322, -6261574688221, 4668264761068, 232155583236, -452564017120, 29382303177, 14190017491, -1690006925], [-34093521863244, -37940862111138, 64700494527334, 70597924476775, -51594415603229, -45096431232206, 22030722743971, 11607589617668, -4749857805276, -1195536481460, 446589014949, 39779102519, -13787660437], [-52850941100892, -54048711679398, 95990682979838, 92492719714457, -72779616014599, -54320468285350, 28390816616465, 13514485061380, -5688540158004, -1374176976448, 513140254587, 45388236577, -15536082095], [-557666645400, 3910054440014, 3766957425458, -8069925791881, -2654434165681, 5236246246106, 247986187723, -1367979642768, 129313852652, 141573490824, -23117172527, -4656709097, 905712243], [-2052309240216, -4524879364430, 2710161806296, 7364886658123, -2214476607911, -4279468555196, 1201415865197, 1055875622376, -307208588408, -106568815746, 31563088001, 3494592671, -1019705775], [-11343802257396, -15333631200502, 16842794267506, 23603889525985, -10949657689331, -12416428807394, 4165302849813, 2816864826292, -855454394380, -269326838404, 79092677443, 8549688593, -2437175963], [34162657809876, 45137834103882, -67107507868658, -73851918539075, 51248521715941, 40596989990194, -19504667463587, -9404404673716, 3757635764364, 908769962656, -328022585121, -29112317203, 9734759813], [-28467417835332, -40098763498014, 45149229350024, 58070744996285, -31080010349149, -29695433860456, 11434971721055, 6604463634076, -2206830144924, -623292257566, 194857600527, 19644634477, -5847934169], [-119261517833700, -170502012658314, 228971110289870, 290823676955747, -181800066250657, -165869322951934, 73867437754463, 39743603527756, -15115618592820, -3931489120324, 1373816531673, 127653995371, -41709748121], [106105578701832, 166083004109142, -167464172157692, -262240045314851, 118462643729539, 144611937263500, -48102446520869, -33996477904156, 10288162455228, 3320160658246, -972434668869, -106595365123, 30303637499], [-70761852857268, -104540651148630, 120560040508930, 162763438597669, -88709659600319, -89115795792098, 35331329478001, 20921586976244, -7277849991516, -2046823086404, 668494464807, 65987984381, -20460244711]];
    Rf_basisdens := [1, 1, 1564335784584, 1564335784584, 1564335784584, 86907543588, 86907543588, 173815087176, 521445261528, 260722630764, 1564335784584, 782167892292, 521445261528];
    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_8004_a();"
function MakeCharacter_8004_a()
    N := 8004;
    order := 1;
    char_gens := [4003, 2669, 3133, 553];
    v := [1, 1, 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_8004_a_Hecke( : Kf := Kf)
    return MakeCharacter_8004_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]];
    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], [-1, 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], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, -1], [-1, 0, 0, 0, 1, 1, 0, 0, 0, -1, 0, -1, 0], [-1, 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], [-2, 1, 0, -1, -1, 0, 0, 0, -1, 0, 0, 0, 1], [1, 2, -1, 0, -1, -1, 1, -2, 1, 1, 1, 0, -1], [1, -1, 0, 0, 1, 0, -1, 0, 0, -1, 0, -1, 1], [0, -1, 1, 1, 1, 0, 0, 1, -1, 0, 0, 0, 0], [0, 1, 0, -1, 1, 1, 0, 0, 1, -1, -1, -1, 1], [0, 2, 0, -1, -1, 0, 0, -1, 0, 0, 0, -1, 1], [-1, 0, -1, 0, -1, 0, 0, 1, -1, 1, 0, 1, 1], [1, 0, 0, 0, 0, 0, -2, 2, -1, 0, -1, 1, 0], [-2, -1, 0, 1, 0, -1, -1, -1, 1, -1, 1, -1, 1], [0, 1, 0, -1, -1, 0, 0, -1, -1, 0, 1, 1, 2], [1, 1, 0, 1, -1, -1, 1, -1, 0, 0, 0, 0, -1], [-1, 3, -1, -1, 0, 0, 2, -1, 0, 3, 0, 1, -2], [-1, 1, 0, 0, 0, -1, 0, -1, 1, -1, -1, 0, 1], [2, 0, 0, 1, 0, 1, 2, 1, 0, 1, 0, 0, 0], [2, 0, 1, 1, -2, -1, -1, 0, -1, 0, 2, 1, 1], [3, 2, -1, -1, 1, 0, 0, -1, -1, 1, 0, 0, 0], [-2, 2, 0, 0, 1, 1, 1, 0, 2, 1, 0, 1, -1], [3, -2, 2, 0, 3, 2, 0, 2, -1, -1, -2, 1, 0], [2, 2, 0, 0, 1, 1, 1, 0, 0, -1, -1, 1, -1], [3, 2, -2, 0, 0, -1, 3, -2, -1, 2, 1, 0, -1], [-1, 4, -1, 0, -1, -2, 0, -2, -2, 2, 2, 0, -2], [2, 1, -1, 1, -2, -1, 1, -2, 2, 2, 1, 0, -2], [1, -2, 1, 0, 2, 0, -1, 1, -1, -3, -2, -1, 2], [2, -1, 1, 1, 0, 0, 1, 1, -2, 1, -1, 1, -1], [3, 1, -1, -2, 0, 0, 0, -3, -2, 1, 1, 0, -1], [-4, 0, 2, -1, 1, 0, -1, -1, 0, -2, -1, -1, 1], [2, -2, 2, 0, -2, 0, -1, 2, 0, 0, -1, 2, 4], [-2, 2, 0, 0, 1, 1, 1, -1, 1, -2, -1, -1, 0], [-2, 2, -1, 0, 0, 0, 0, -1, 2, 0, 0, -4, -3], [1, -3, 0, 3, -1, 0, -1, 3, 1, 0, -2, 0, 1], [4, -3, 1, 0, 0, 0, -1, -1, 0, -1, 1, 0, 0], [3, 1, 1, -1, -1, 1, -1, -2, 1, -3, 1, -3, 0], [-1, -1, 0, -1, -2, 0, -2, 2, 1, 0, 0, 2, 1], [4, 2, -2, 1, 1, 1, -1, -1, 1, 2, 1, -1, -3], [4, 0, -2, 0, -2, 0, 2, -1, -3, 1, 0, -1, -1], [-1, -1, -2, -1, 1, 1, -1, 1, 0, -2, -1, -2, -1], [1, -3, 1, 3, -1, 1, -1, 1, 2, 2, 1, 3, 1], [-3, -1, 0, 0, 1, 1, -1, 1, 0, -4, 0, -3, 0], [-1, -3, 1, 1, 4, 1, -2, 1, -1, -3, -2, -3, 0], [5, 2, 1, -3, -2, 1, -1, 1, 0, -1, -1, 0, 1], [-3, -4, 3, 3, 3, 1, -4, 2, 0, -5, -1, -2, -1], [2, 0, 0, 0, 1, 0, 1, -3, 1, -1, 2, -3, -2], [5, -1, -2, -1, 4, 1, 0, 0, 0, 0, 1, -2, -1], [5, -5, 0, 3, 1, -1, 0, 3, -1, 0, 0, 1, 0], [7, 4, 0, -1, -2, -1, 3, -2, -3, 4, 1, 1, -4], [3, -4, 1, -2, 0, 0, 0, -1, -1, -1, 1, -1, 4], [9, -4, 0, 1, 2, -2, 1, 1, -1, 1, -1, -1, -1], [1, 4, -4, -3, -1, -2, 2, -4, -2, 0, 1, -1, -1], [3, 0, -2, -2, -1, -3, 2, -3, -1, 2, 2, -1, -1], [4, 2, 1, 0, -1, 2, 2, 1, 0, 0, 1, 4, 0], [4, 2, -1, 1, 3, 2, 2, -2, 0, 1, 0, -3, -3], [5, -2, 0, 1, -1, 0, 0, 0, -1, 1, 0, 2, -1], [-4, 3, -2, -2, 0, 0, -1, -4, -1, 0, 0, 0, 3], [3, -3, 3, 1, 1, 0, 1, 1, 1, -2, -1, -2, 1], [3, -1, 1, -1, -1, 0, -1, -1, -3, 0, -1, 0, -1], [3, 1, -1, -5, 0, -1, 0, -2, -1, -1, -1, 1, 3], [3, 2, -1, 3, 1, 0, 2, 0, 3, 1, -1, 0, -1], [2, -2, 1, 1, -1, 0, -1, 0, -1, -1, 1, -2, 0], [4, -1, -1, 0, 0, 0, -1, -1, -2, 1, 1, -2, -2], [1, -4, 1, 0, 1, -1, -2, 2, -1, -2, -2, -2, 0], [4, 4, -1, -3, 0, 2, 0, -4, 2, 0, 3, -3, 1], [5, 2, -1, 0, -4, -1, 0, -1, 0, 5, 3, 1, 0], [1, 0, 1, 0, 0, 0, -1, 2, -4, 0, -2, 0, 1], [3, -1, -2, 2, 2, -2, 0, 0, 1, 0, 1, 1, 0], [-2, -3, 3, -1, 1, 1, -2, 6, -1, -3, -4, 1, 3], [-1, 5, -3, -2, -3, -2, 3, -4, -2, 1, 1, 0, -1], [2, 1, -2, 0, -4, -2, -1, -2, 1, 4, 2, 3, 3], [-1, -2, 1, 0, -2, 2, -1, 0, 0, -2, 2, 0, 1], [5, 0, -2, -1, -3, -2, -1, 0, -2, 1, 4, 3, 4], [5, -7, 2, 1, 3, 4, -3, 2, 0, -5, 2, -1, 2], [1, -3, 0, -4, 0, 0, 0, -2, 3, 0, 1, 1, 6], [8, 5, 0, -1, -3, 0, 0, -2, 0, 3, 2, 1, -3], [8, -5, 2, 1, -4, -1, -3, -1, 1, 1, 2, 1, 3], [1, -3, -1, -1, -2, -2, -1, -3, 2, -1, 1, -3, 4], [-5, 0, 4, 0, 0, 2, -3, 2, 2, -4, -3, -2, 1], [3, -3, 2, 1, 0, 1, -4, 4, -2, -2, -1, 2, 1], [8, 4, 2, 0, 0, 0, 0, -1, 1, -3, 0, 2, -1], [-1, 2, -2, 0, -4, -2, -1, 1, -1, -1, -1, 1, 0], [10, 2, -2, -1, 2, 0, 0, -4, 2, 0, 2, -4, -1], [-8, 1, 1, -1, 2, 1, 1, 1, 1, 1, -1, 5, -1], [6, 5, -2, 1, 5, 0, 4, -2, 0, -1, 0, -5, -9], [0, 0, 2, 0, 1, -1, -2, 1, -2, -2, -3, 1, 1], [-2, 0, -3, 0, 3, 0, -1, 0, 1, -1, -2, -5, -3], [3, 0, 4, -3, -2, 0, 1, 2, -2, 0, -3, 3, 4], [-5, 5, -4, -3, 2, 2, 2, -2, 1, -2, -1, -4, 3], [-1, 4, -2, 1, 1, 1, 1, 0, 2, 5, -1, 0, -3], [1, -2, 2, 4, 3, 2, 0, -1, 2, -2, 4, -4, -3], [9, -2, 0, 0, 1, 3, 1, -2, 4, 1, -1, -5, -2], [0, 3, -1, -1, 1, 1, 0, -1, 0, 0, -3, -2, -4], [6, 3, -3, -4, 0, -2, 1, -8, -1, 2, 3, -3, -1], [1, 1, 1, 0, -2, 1, 2, 1, 1, -1, 0, 4, 1], [6, -3, -2, 1, -3, -1, 1, 1, -1, 6, 1, 5, -1], [-3, 0, -1, 2, 4, 3, -2, -1, 1, -5, 0, -2, 1], [6, 3, -1, -2, 5, 1, -2, 0, 1, -3, -2, 0, -2], [-1, -1, -1, -1, -3, -4, 1, -5, 3, 2, 4, -2, -3], [3, 7, -4, -2, -2, -1, -2, -3, -1, 3, 3, 4, -1], [10, 1, 0, 1, 0, 3, -1, 0, -2, 0, 2, -4, -6], [2, -1, 1, 1, -3, 1, -2, 0, -1, -1, 0, 5, 5], [16, -5, 2, 2, 2, 0, 1, 3, 0, -5, -2, 0, 0], [-4, -1, 2, 0, 1, 3, 0, -1, 2, -2, 1, -5, 1], [-4, 3, 0, 2, 3, 0, 2, -2, 0, -1, -3, -4, 0], [9, -6, 2, 5, -1, -2, 0, 3, 2, 2, -4, 3, 0], [1, -9, 1, 2, 1, 2, -3, 4, 4, -1, 1, -2, 1], [-14, 2, -2, -3, -7, -2, 1, -3, 0, 0, -1, -1, 5], [5, 0, 1, 5, 1, 4, 1, -1, 1, 2, 3, -2, -5], [8, -5, 3, 4, 1, 0, -2, 4, 2, -3, 0, 2, 0], [1, 6, -4, 0, -2, -2, 2, -2, -4, 6, 6, 0, -5], [1, 7, -5, -5, 1, 1, 3, -4, -2, 0, 2, -2, -4], [11, -3, -1, 3, 0, 2, 2, 5, 0, 3, 0, 0, 0], [0, -2, 0, 0, -3, 0, 2, -5, 2, 0, 2, -2, 0], [14, -2, 0, -1, 0, 0, 0, 1, -3, -3, -4, 0, 2], [0, -11, -1, 0, 1, 1, -4, 5, -2, -4, -4, -1, 5], [8, 4, 2, -2, -1, 0, 3, 1, 2, 0, 1, -4, -2], [-7, 4, 0, 1, -1, 1, 0, 1, 0, -1, -1, 0, 0], [-5, 3, 0, 2, 1, -1, 3, -2, -1, 1, 0, -4, -3], [1, -6, 4, 3, -3, -2, -4, 5, 0, -4, -3, 3, 4], [4, 5, -3, -1, -5, 0, 4, -6, 4, 7, 4, 5, -1], [2, -2, 5, -2, 0, 4, -4, 1, 3, -5, -3, -4, 3], [-3, -1, -2, -4, 4, 2, 0, 4, -3, 0, -3, -1, 2], [2, 0, -3, -4, -2, -1, 2, -4, -1, 4, 0, -2, 2], [1, 0, -1, -2, -5, 0, -2, -2, -2, 0, 2, 4, 2], [-3, 0, 0, -3, 0, 2, -1, 1, -1, -1, -4, -2, -1], [0, 7, 3, -4, 0, 1, -1, -2, 0, -2, -1, 1, 1], [-1, -3, -3, 0, 2, 2, 1, 1, -1, 1, 0, -5, -4], [2, 2, -1, 0, -3, -1, 1, 4, -2, 3, 0, 5, 3], [5, -7, 5, 3, 5, 3, -6, 6, 4, -7, -3, -2, -3], [2, 2, 2, 0, -2, -1, 0, -3, 0, -3, -6, -3, 1], [-2, -6, 2, 0, 1, -2, 1, 3, 0, 0, -2, 4, 4], [11, 4, 2, 0, 1, -1, 2, -1, -1, 2, 2, 0, -5], [4, 0, 5, 0, -1, 3, -3, 0, 3, -4, -3, -4, 1], [-8, -7, 1, 2, 1, 1, 0, 4, 3, -1, 0, 2, 0], [9, 5, 2, 0, -2, -1, 4, -4, -4, 4, 5, -3, -2], [-3, -5, 3, -1, 5, 2, -1, -2, 2, -3, 1, -2, 4], [1, 3, 3, -1, 5, 0, 0, -1, 0, -6, -5, -3, -2], [8, 3, -3, -2, 0, 0, 3, -2, -5, 2, 1, 3, -1], [-5, 6, -6, -4, -1, -2, 2, -2, -1, 5, 3, -3, -6], [5, 3, -2, 5, 1, -2, 1, -2, 0, -1, 6, -7, -2], [-2, 4, -3, 3, 4, -3, 4, -1, -4, -1, -1, -2, -4], [-11, 10, -3, 1, -1, 0, 4, -4, 3, 1, -1, 2, -1], [-1, 8, -1, 2, -3, -2, 0, 4, -1, 3, -1, 4, 0], [-3, -7, 1, 3, 0, -1, 0, 2, 2, 0, 4, 0, -1], [2, 7, -2, 1, 3, -2, 4, 0, -2, 5, 1, -1, -7], [1, -9, 3, 4, 5, 1, -3, 8, -7, -1, -1, 4, 1], [-2, 1, 2, 2, 2, 3, -3, 2, 0, -2, 4, -5, -5], [3, -8, -3, 0, -2, -4, -3, -3, 1, -3, 6, -3, 0], [5, 4, -3, -2, 0, -3, 2, -3, -3, -1, -2, 0, 1], [2, -2, 4, 1, 1, 2, -4, 3, -1, -4, 1, -2, -6], [1, 8, -6, 2, -7, -9, 0, -4, -2, 7, 6, 3, -4], [7, -5, 0, 0, -5, -2, -1, 3, -3, 4, 4, 5, -4], [2, -2, -2, 0, -1, -5, 2, -3, -2, 6, 3, -1, 1], [4, 0, 1, 1, 5, 3, 1, -2, -2, 1, -4, 0, -2], [4, 4, -3, -4, 3, -2, 2, 1, -2, 6, -3, 4, 0], [-9, 0, -2, 0, -5, 1, 0, 1, 5, 4, 0, 6, 1], [6, 1, -1, -4, 1, 0, -1, -2, 7, -3, -2, -3, 1], [-12, 4, -2, -2, -3, -5, 1, -8, -2, 1, 2, -1, -1], [14, -5, 1, -2, 1, -1, 0, -3, -2, -4, -1, -7, -1], [0, 4, -3, 3, 5, -1, -1, -2, -2, 5, 6, -2, -8], [-1, 1, 1, 0, -1, 4, 3, 2, 0, 1, 1, 0, 7], [11, -6, 2, -1, -1, -2, 0, -3, 2, -2, -4, -5, -6], [-1, 8, -1, -6, -1, 2, -2, -5, 0, -4, -1, 0, -1], [6, 3, 3, -3, 1, 3, 4, 1, 2, 0, -2, 0, -4], [6, 1, 0, -2, -6, 1, 1, 5, -5, 5, -2, 6, 4], [-3, -1, 3, 3, 8, 4, 2, 4, -1, 0, -6, -2, 1], [-22, -8, 4, 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-2, 5, 2, 3, 0, -5, 2, 1, -7, -4, -1, 11], [9, 1, -2, 6, 3, -3, 6, -7, 1, 4, 10, 0, -3], [4, 4, 4, 0, -2, -3, 6, -2, -2, 1, 1, 5, -8], [6, -5, 0, 15, 0, -2, 4, -1, -1, 3, 4, -2, -18], [9, 8, -3, -15, -1, 1, 3, -8, -4, -9, -2, -4, 5], [-8, -10, 8, -1, 7, 10, -9, 10, 3, -9, -3, -8, 14], [30, 9, -2, 3, 4, 4, 7, -2, -3, 4, 4, -6, -12], [8, 4, -4, 4, 7, 7, 1, 1, 4, -1, 0, -13, -6], [18, -3, -3, 0, -2, -2, 6, -3, -1, 3, 3, -3, -5], [10, -5, -3, 3, -8, 1, -1, 1, -3, 7, 9, -1, -7], [5, -1, 2, 9, 5, -1, 3, 2, -1, -3, 0, -5, -4], [4, -4, 4, -3, -3, 4, -3, 4, 3, 1, -5, 3, 4], [12, 7, -1, -4, -1, 2, 10, -2, 1, 6, -5, 1, -4], [-6, 14, 0, -5, -10, 1, 1, -3, 3, -3, 0, -3, 13], [5, 2, 2, -4, -3, 4, -3, 7, -4, -4, -9, -4, 9], [0, 7, 5, 5, 4, 0, 1, 0, 3, -6, -1, 3, -4], [-21, -10, 3, 4, -2, 3, 0, 1, -5, 3, 4, 2, -5], [19, 15, -2, -1, 9, 6, 5, 0, 2, -7, -4, -9, -10], [1, -1, -1, -3, 2, -2, 0, -1, -2, -5, -4, 2, 2], [-2, 6, -1, 5, -9, -6, 5, -9, -6, 2, 6, 4, -7], [11, 4, 0, 1, -1, 3, 0, -8, 2, 3, 4, -2, -3], [0, 0, 2, 1, 5, 6, 1, 10, -5, -5, -9, -10, -4], [5, 13, -3, -1, 5, -3, 3, 6, -5, 3, -1, 3, -6], [2, 23, -10, -4, -7, -4, 6, -6, 2, 13, 1, 2, 0], [9, 8, 0, -1, -7, 0, 4, -4, 5, 7, -4, 7, -3], [-15, -16, 8, 3, 8, 13, -13, 11, 2, -11, -5, -4, 7], [3, -13, 2, 9, 4, 3, -6, 0, 8, 0, -3, -10, -7], [-1, 7, -6, 0, -8, -4, 3, -5, -7, 11, 7, 9, -10], [20, -1, 1, 1, -9, -6, -3, -2, -6, 7, 3, 7, 1], [14, 4, -1, -2, -9, -5, -3, -5, 5, 8, 0, 0, 6], [5, 7, -4, 2, -1, -5, 1, 1, -2, 10, 2, 5, -18], [-13, -1, 3, 0, 5, 0, 3, -1, -7, -8, 3, 1, -2], [7, 10, -7, -1, -10, -8, 3, -8, -2, 12, 8, 13, -12], [2, -11, 0, -4, -3, -2, 2, -6, 4, 5, 4, 12, 10], [19, 17, -4, -13, 0, 1, 6, -4, -4, 4, 5, 3, -5], [-15, 7, 0, 4, -7, -5, -1, -12, 9, 3, 6, -6, 3], [6, -24, 6, 2, 12, 7, -9, 12, -4, -18, -8, -1, 7], [6, 5, -9, -3, -4, -3, 1, -5, 7, 7, 2, -7, -3], [-5, -13, 9, 2, 0, 0, -3, 4, 0, -4, -6, 2, 5], [1, -13, 6, -10, 2, 2, -4, -1, 0, -15, -7, -2, 17], [9, 19, -1, -2, -5, 1, 7, -8, 5, 3, -1, 2, -11], [0, -10, 5, -2, 4, 3, -10, 4, 7, -12, -8, -2, 2], [-13, -1, 1, 1, 5, 1, -1, 5, 0, 2, -9, 3, 5], [25, -5, 4, 10, 8, 3, 0, 8, 3, -7, -4, 0, -10], [13, -4, -3, -6, 2, -5, -4, 0, -1, -4, 3, -2, 7], [2, -4, 4, -1, 4, 2, 0, 3, 3, -3, -4, -5, 4], [1, 1, -2, -4, -4, 3, 2, 5, -5, -3, -5, 8, -3], [13, 12, -2, -3, -9, -7, -6, -4, 0, -1, -2, -2, -5], [-4, 13, -6, -6, -5, 1, 0, -8, -1, 3, 5, 8, -4], [6, -4, 0, -8, 0, 0, 2, -1, 5, 3, -1, 1, 7], [-21, -8, 7, -2, 5, 5, -5, 12, -3, -9, -7, 4, -1], [-2, 7, 1, 1, -3, -6, -4, -2, 0, 3, 4, 1, -1], [25, -17, 5, -3, 5, -2, -7, 3, -7, -12, -3, -8, 5], [-38, -7, -2, 10, 6, -1, 1, -3, 2, -4, -1, -9, -4], [17, -4, 6, 5, 6, 2, -6, 8, -5, -4, -1, -8, 3], [15, 1, 7, -2, -2, 1, 6, -4, -2, 0, 6, -5, -6], [2, -5, -1, 4, 0, 1, -6, 10, 0, -6, -2, -1, 3], [-15, 8, -2, -8, -3, -2, -10, -9, 2, -6, 2, -10, 1], [20, -9, 1, 2, 4, 8, -6, 9, 3, -3, -3, -3, 3], [-1, 11, 1, -7, -7, 3, 3, -4, 3, 1, 3, -5, 12], [18, -10, 6, 0, 7, 13, -3, 3, 9, -12, -1, -3, 2], [3, -4, 2, 4, 10, 5, 3, 7, 8, -7, -1, 0, -6], [7, 20, -10, -8, -9, -6, 2, -13, -10, 10, 4, 4, -1], [9, 10, -6, -7, -3, -6, 6, -13, 10, 6, -1, 1, -4], [4, 0, 0, -4, 8, -3, -4, -5, -2, -1, -2, -1, 3], [-11, -9, -1, -3, -4, -4, -10, -5, 0, -9, 0, -3, 4], [7, 7, -11, -2, 6, 1, 2, -2, -2, 2, -6, -1, -10], [11, 13, -9, -5, -12, -5, 4, -12, 3, 13, 5, 3, 3], [5, -13, 7, -5, -2, 7, -12, 10, 2, -14, 2, 2, 21], [-13, 15, -8, -4, 3, -3, 5, 0, -5, 3, 4, -5, -11], [-25, -3, 0, 5, -5, -1, -3, 9, -5, 5, 1, 7, -1], [-37, 6, -1, -3, 5, -1, 0, -1, -3, -8, -5, -7, -4], [19, 3, -4, -9, -3, -4, -5, -4, -2, 9, 10, 1, 0], [8, 2, 1, 0, -3, -2, -2, -3, 8, -2, -9, 2, 10], [9, -4, 6, 1, 13, 8, 2, 4, 9, -7, -5, -4, -7], [17, -9, 3, 1, 1, 1, -3, 10, -5, -9, -1, 1, 8], [-17, 5, 2, -1, -4, 7, 2, -4, 0, -2, -7, 0, 3], [1, -1, -3, 5, -3, 0, 7, 2, -8, 9, 3, 13, -16], [12, -8, 6, -2, 7, 10, -5, 2, 7, -11, -9, -9, -7], [-10, -12, 1, 5, 3, 4, -5, 5, 4, 8, 3, -5, -1], [3, -8, 5, 8, 3, -2, -5, 0, 4, 5, 5, 0, -11], [-7, 2, -1, -9, 1, 4, -11, 12, -5, -3, -4, 4, 9], [-3, -2, -2, -7, -8, 1, -9, -5, 6, -9, 1, 0, 5], [1, -8, 5, 2, -2, 3, 3, -9, 12, -7, 8, -7, 4], [-13, 7, -1, 4, 5, -5, -3, -3, -9, -1, 4, 2, -4], [-22, -9, 3, 10, 2, 5, 7, 9, -1, 3, -9, 4, -8], [7, 2, -12, 3, 2, -2, 3, -2, 2, 10, 1, 1, -14], [-8, -1, 4, 6, 2, 7, 1, 0, 0, -4, -6, -3, 3], [-8, 1, 2, -9, -5, -2, 6, -1, -9, 4, 1, 2, 10], [25, -1, 4, 4, -2, -4, 2, 4, 3, 4, 7, 9, -2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8004_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8004_2_a_h();".
// 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_8004_2_a_h(:prec:=13)
    chi := MakeCharacter_8004_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_8004_2_a_h();".
// 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_8004_2_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8004_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-5832, -5184, 15102, 7832, -15445, -3259, 7388, 65, -1652, 180, 158, -27, -5, 1]>,<7,R![-16, -520, -2576, 1988, 18608, -10466, -14742, 2079, 3318, 20, -283, -23, 8, 1]>],Snew);
    return Vf;
end function;