
// Make newform 80.6.n.d in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_80_n();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_80_n_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_80_6_n_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_80_6_n_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 := [579398322543528055824, 0, 12044868290803250652, 0, 824105698447750789, 0, 14779507781220031, 0, 133816049059481, 0, 881545050522, 0, 5522961902, 0, 25993614, 0, 109637, 0, 271, 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, 0, 0, 0, 0], [-59796125630963059484035166165480637448609841792711444396589450967058527355119355424, 0, -4694568982569663806549064471932609969040735625526239944457222378533864358844700032, 0, -73614286430072414530431753660035155608759248821305759309557363125303938993793100, 0, -335310271155398505096641366509820275833033651493727600702225465758601184635239, 0, -3278016091113470405134441776508163090092543595849759389960296599985660582156, 0, -9237036172073909176672032966501542630592021952043346717918699668388264508, 0, -77739791695821221383548835088518683952445325930572822979313112249812926, 0, 12203048923274079101852228086462556785400965909229586981163211171204, 0, -823738617909189420141037064086996144862371928996999202111974981216, 0, 1254178201903086938983596921272517853931781219766635397959349945, 0], [0, 270653852870948730415923446110967787053901407919555138816, 0, 7790599041755809878147583197352567438110371075388086098, 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    Rf_basisdens := [1, 1121593795036841302747795238795074935160995345367722863674317311680848775917882360, 1619470712350716559357342343605013280292390449171456207120, 43880532227038497775504643639916556768555016272899234762491950656056432376154426992480, 570446918951500471081560367318915237991215211547690051912395358528733620890007550902240, 570446918951500471081560367318915237991215211547690051912395358528733620890007550902240, 570446918951500471081560367318915237991215211547690051912395358528733620890007550902240, 26997541333217662794877007504102301468410242316917527086857935133089376075861387361550312480, 26997541333217662794877007504102301468410242316917527086857935133089376075861387361550312480, 570446918951500471081560367318915237991215211547690051912395358528733620890007550902240, 570446918951500471081560367318915237991215211547690051912395358528733620890007550902240, 6749385333304415698719251876025575367102560579229381771714483783272344018965346840387578120, 570446918951500471081560367318915237991215211547690051912395358528733620890007550902240, 20594622537908943325626914763241537788961266363285532782210928349135808039946005, 20594622537908943325626914763241537788961266363285532782210928349135808039946005, 2249795111101471899573083958675191789034186859743127257238161261090781339655115613462526040, 2249795111101471899573083958675191789034186859743127257238161261090781339655115613462526040, 30700933544867999885005444230771847941952032037542163142399274861025113462631727370840, 570446918951500471081560367318915237991215211547690051912395358528733620890007550902240, 6749385333304415698719251876025575367102560579229381771714483783272344018965346840387578120];
    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_80_n();"
function MakeCharacter_80_n()
    N := 80;
    order := 4;
    char_gens := [31, 21, 17];
    v := [2, 4, 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_80_n_Hecke();"
function MakeCharacter_80_n_Hecke(Kf)
    N := 80;
    order := 4;
    char_gens := [31, 21, 17];
    char_values := [[-1, 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, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 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 := 6;
    raw_aps := [[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], [-2, 0, 10, 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, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0], [39, 0, -39, 3, -1, -2, -1, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0], [-111, 0, -114, -6, 2, 6, -1, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, -2, 0], [0, 1, 0, 0, 0, 0, 0, -4, 4, 0, 0, 1, 0, 3, -2, -3, 5, 2, 0, 3], [0, 3, 0, 0, 0, 0, 0, 61, 0, 0, 0, -2, 0, 2, 1, 3, 2, -1, 0, 4], [17, 0, 963, -23, 0, -33, -3, 0, 0, 17, -10, 0, -10, 0, 0, 0, 0, 0, -2, 0], [0, 7, 0, 0, 0, 0, 0, -95, -95, 0, 0, -3, 0, 4, 5, 21, 23, 1, 0, 1], [2227, 0, 2220, -11, -14, -34, -18, 0, 0, -11, 20, 0, 0, 0, 0, 0, 0, 0, 14, 0], [-361, 0, -20, 19, -11, 31, -20, 0, 0, 37, -35, 0, 35, 0, 0, 0, 0, 0, 0, 0], [0, -8, 0, 0, 0, 0, 0, 321, 0, 0, 0, 2, 0, -37, -1, 12, -2, 6, 0, -4], [0, -23, 0, 0, 0, 0, 0, 0, 101, 0, 0, -8, 0, -2, -21, -4, 7, -19, 0, -4], [9106, 0, -9206, -58, -25, 167, -8, 0, 0, -92, 0, 0, -43, 0, 0, 0, 0, 0, -25, 0], [0, -11, 0, 0, 0, 0, 0, 48, -48, 0, 0, -11, 0, 31, -42, 59, -81, -26, 0, -33], [-1964, 0, -47, -232, 8, -3, -47, 0, 0, -80, -25, 0, 25, 0, 0, 0, 0, 0, 0, 0], [0, -42, 0, 0, 0, 0, 0, 0, -1083, 0, 0, -32, 0, -8, -84, -16, -30, -26, 0, -16], [0, -35, 0, 0, 0, 0, 0, 525, 525, 0, 0, 15, 0, 67, 62, 5, -5, -5, 0, -5], [13451, 0, -13144, -374, 39, -354, 221, 0, 0, 86, 0, 0, 57, 0, 0, 0, 0, 0, 39, 0], [0, -3, 0, 0, 0, 0, 0, 160, -160, 0, 0, -3, 0, 163, -166, -224, 218, 21, 0, -9], [0, 9, 0, 0, 0, 0, 0, -147, 0, 0, 0, -16, 0, -364, 8, 400, 16, 7, 0, 32], [82, 0, 28686, -394, 0, 156, 54, 0, 0, 82, 130, 0, 130, 0, 0, 0, 0, 0, -46, 0], [18544, 0, 18029, -198, -99, 1468, 218, 0, 0, 733, -117, 0, 0, 0, 0, 0, 0, 0, 99, 0], [-25552, 0, -264, -578, 42, 518, -264, 0, 0, -262, 210, 0, -210, 0, 0, 0, 0, 0, 0, 0], [0, -11, 0, 0, 0, 0, 0, -4219, 0, 0, 0, -76, 0, -599, 38, -115, 76, 87, 0, 152], [0, 86, 0, 0, 0, 0, 0, 0, -2795, 0, 0, 76, 0, 19, -603, 38, 782, 48, 0, 38], [-106, 0, 48307, 384, 0, -781, -521, 0, 0, -106, 155, 0, 155, 0, 0, 0, 0, 0, -194, 0], [63330, 0, -63112, 386, -105, -684, -143, 0, 0, 361, 0, 0, 156, 0, 0, 0, 0, 0, -105, 0], [0, -72, 0, 0, 0, 0, 0, 0, 3513, 0, 0, -192, 0, -48, -494, -96, 771, 24, 0, -96], [0, 46, 0, 0, 0, 0, 0, -3535, -3535, 0, 0, 141, 0, 312, 265, -407, -501, -47, 0, -47], [10937, 0, 11795, 1154, 80, -564, 376, 0, 0, -482, -56, 0, 0, 0, 0, 0, 0, 0, -80, 0], [0, -82, 0, 0, 0, 0, 0, -6784, 6784, 0, 0, -82, 0, 430, -512, -557, 393, 223, 0, -246], [-1115, 0, 7052, 2022, 0, 1177, 96, 0, 0, -1115, -130, 0, -130, 0, 0, 0, 0, 0, -242, 0], [0, -600, 0, 0, 0, 0, 0, -5243, -5243, 0, 0, -252, 0, 511, 595, -1193, -1025, 84, 0, 84], [42570, 0, 43525, 1530, 264, 1135, 839, 0, 0, -116, 264, 0, 0, 0, 0, 0, 0, 0, -264, 0], [0, -688, 0, 0, 0, 0, 0, 3521, 0, 0, 0, -8, 0, -1142, 4, -304, 8, 696, 0, 16], [0, -297, 0, 0, 0, 0, 0, 0, 8869, 0, 0, 88, 0, 22, -869, 44, -1295, -341, 0, 44], [23812, 0, -23013, -608, -338, -1183, 753, 0, 0, 46, 0, 0, -1190, 0, 0, 0, 0, 0, -338, 0], [0, -26, 0, 0, 0, 0, 0, -7776, 7776, 0, 0, -26, 0, 228, -254, -867, 815, -747, 0, -78], [-157767, 0, 1105, 2544, -56, -2014, 1105, 0, 0, 1187, -935, 0, 935, 0, 0, 0, 0, 0, 0, 0], [0, 820, 0, 0, 0, 0, 0, 6009, 6009, 0, 0, 396, 0, -34, -166, -1061, -1325, -132, 0, -132], [-3334, 0, 5433, 1400, 526, -2980, 692, 0, 0, 1407, 0, 0, -851, 0, 0, 0, 0, 0, 526, 0], [169650, 0, 169724, 1019, -637, 206, 308, 0, 0, 234, 67, 0, 0, 0, 0, 0, 0, 0, 637, 0], [0, 99, 0, 0, 0, 0, 0, 3120, -3120, 0, 0, 99, 0, 185, -86, -138, 336, 1557, 0, 297], [0, 292, 0, 0, 0, 0, 0, -3459, -3459, 0, 0, -939, 0, 58, 371, 437, 1063, 313, 0, 313], [0, -380, 0, 0, 0, 0, 0, -20711, 0, 0, 0, -490, 0, 2785, 245, -3719, 490, 870, 0, 980], [0, 569, 0, 0, 0, 0, 0, 0, -18895, 0, 0, 684, 0, 171, 163, 342, 1812, 227, 0, 342], [1152, 0, -326658, -6223, 0, 5807, 2672, 0, 0, 1152, 1615, 0, 1615, 0, 0, 0, 0, 0, 368, 0], [-47036, 0, 41873, 3912, 498, 7562, -3262, 0, 0, -1901, 0, 0, 1209, 0, 0, 0, 0, 0, 498, 0], [0, 213, 0, 0, 0, 0, 0, -9812, 9812, 0, 0, 213, 0, -1332, 1545, 1436, -1010, -589, 0, 639], [394235, 0, -732, 2127, -183, 3243, -732, 0, 0, 897, 1065, 0, -1065, 0, 0, 0, 0, 0, 0, 0], [0, 209, 0, 0, 0, 0, 0, 1901, 1901, 0, 0, 438, 0, -1138, -1284, 503, 211, -146, 0, -146], [-199408, 0, -207434, -11480, -1361, -1270, -4815, 0, 0, 3211, -1516, 0, 0, 0, 0, 0, 0, 0, 1361, 0], [0, 22, 0, 0, 0, 0, 0, 3793, 0, 0, 0, 352, 0, 5148, -176, 3221, -352, -374, 0, -704], [662, 0, -393563, 834, 0, -1171, -1647, 0, 0, 662, 2785, 0, 2785, 0, 0, 0, 0, 0, 3296, 0], [0, -3223, 0, 0, 0, 0, 0, 2181, 2181, 0, 0, -1419, 0, -1199, -726, 3457, 4403, 473, 0, 473], [116090, 0, 118807, 8562, -1014, 4613, 4831, 0, 0, 2114, -3432, 0, 0, 0, 0, 0, 0, 0, 1014, 0], [856689, 0, 818, -13671, 559, -8129, 818, 0, 0, -7437, 385, 0, -385, 0, 0, 0, 0, 0, 0, 0], [0, -1459, 0, 0, 0, 0, 0, -18467, 0, 0, 0, 286, 0, 5579, -143, -3872, -286, 1173, 0, -572], [-669091, 0, 668772, -14605, 494, 6070, 4938, 0, 0, -5257, 0, 0, -34, 0, 0, 0, 0, 0, 494, 0], [0, -1608, 0, 0, 0, 0, 0, 0, -5083, 0, 0, -1628, 0, -407, 7259, -814, 3344, -794, 0, -814], [0, -1094, 0, 0, 0, 0, 0, -55007, -55007, 0, 0, 468, 0, -1560, -1716, 2651, 2339, -156, 0, -156], [338284, 0, -339974, -19162, -2119, 6538, 5277, 0, 0, -6967, 0, 0, -478, 0, 0, 0, 0, 0, -2119, 0], [311521, 0, 309925, -5768, 3449, 2583, -723, 0, 0, 873, -1889, 0, 0, 0, 0, 0, 0, 0, -3449, 0], [0, 544, 0, 0, 0, 0, 0, -35791, -35791, 0, 0, -615, 0, -2704, -2499, -3239, -2829, 205, 0, 205], [-964241, 0, -975563, -10800, -2334, 13080, -1812, 0, 0, 9510, -1794, 0, 0, 0, 0, 0, 0, 0, 2334, 0], [0, 1425, 0, 0, 0, 0, 0, 0, 83177, 0, 0, 940, 0, 235, -925, 470, 5894, 955, 0, 470], [7026, 0, -741244, -8693, 0, -24503, -8366, 0, 0, 7026, -745, 0, -745, 0, 0, 0, 0, 0, -3752, 0], [69969, 0, -58597, 11822, -2542, -23718, 1568, 0, 0, 9804, 0, 0, -7696, 0, 0, 0, 0, 0, -2542, 0], [0, 422, 0, 0, 0, 0, 0, -35684, 35684, 0, 0, 422, 0, -2187, 2609, -10810, 11654, 290, 0, 1266], [0, 2988, 0, 0, 0, 0, 0, 0, -33351, 0, 0, 1968, 0, 492, 1926, 984, -657, 2004, 0, 984], [-805102, 0, 808013, 1318, 1594, 173, 4401, 0, 0, -1490, 0, 0, -10016, 0, 0, 0, 0, 0, 1594, 0], [0, 670, 0, 0, 0, 0, 0, 81004, -81004, 0, 0, 670, 0, 1936, -1266, -5029, 6369, -2719, 0, 2010], [0, 684, 0, 0, 0, 0, 0, 6329, 0, 0, 0, -1266, 0, 6161, 633, -5479, 1266, 582, 0, 2532], [-9770, 0, -766101, 25838, 0, 4383, -3921, 0, 0, -9770, 2455, 0, 2455, 0, 0, 0, 0, 0, 4832, 0], [1599032, 0, 1573822, -41679, 1053, 5234, -15416, 0, 0, 9794, 9, 0, 0, 0, 0, 0, 0, 0, -1053, 0], [358475, 0, -4041, 37360, -2520, 24530, -4041, 0, 0, 22433, -3465, 0, 3465, 0, 0, 0, 0, 0, 0, 0], [-15386, 0, 431497, 21865, 0, 25175, 3837, 0, 0, -15386, 2115, 0, 2115, 0, 0, 0, 0, 0, -2955, 0], [0, 30, 0, 0, 0, 0, 0, -26516, 26516, 0, 0, 30, 0, 2192, -2162, -2859, 2919, 3631, 0, 90], [-1589077, 0, -10458, -33178, 5242, 13723, -10458, 0, 0, -10285, -2150, 0, 2150, 0, 0, 0, 0, 0, 0, 0], [0, -1167, 0, 0, 0, 0, 0, 7597, 7597, 0, 0, 321, 0, 6357, 6250, -13519, -13733, -107, 0, -107], [1620361, 0, -1613492, -3940, 5873, -5030, 2835, 0, 0, 4034, 0, 0, 8177, 0, 0, 0, 0, 0, 5873, 0], [0, -484, 0, 0, 0, 0, 0, -33564, 33564, 0, 0, -484, 0, 5411, -5895, 12362, -13330, -2768, 0, -1452], [0, -3055, 0, 0, 0, 0, 0, -2067, 0, 0, 0, 1050, 0, -9085, -525, -22892, -1050, 2005, 0, -2100], [2873, 0, 2076726, -5777, 0, -6867, -4902, 0, 0, 2873, 5810, 0, 5810, 0, 0, 0, 0, 0, 877, 0], [-1764680, 0, -1730996, 55274, -6817, -26634, 14773, 0, 0, -18911, 3232, 0, 0, 0, 0, 0, 0, 0, 6817, 0], [-2941678, 0, 7510, -2042, -2302, -19358, 7510, 0, 0, -7516, 5480, 0, -5480, 0, 0, 0, 0, 0, 0, 0], [0, 4963, 0, 0, 0, 0, 0, -77219, 0, 0, 0, 3068, 0, -27213, -1534, 9535, -3068, -8031, 0, -6136], [0, -3995, 0, 0, 0, 0, 0, 0, -133991, 0, 0, -5240, 0, -1310, -16990, -2620, -8970, -1375, 0, -2620], [0, -1780, 0, 0, 0, 0, 0, -133308, 133308, 0, 0, -1780, 0, 1769, -3549, -8492, 4932, -3682, 0, -5340], [0, -5832, 0, 0, 0, 0, 0, 0, -143159, 0, 0, 2508, 0, 627, -8889, 1254, 5407, -7086, 0, 1254], [0, -696, 0, 0, 0, 0, 0, -16999, -16999, 0, 0, 933, 0, 14080, 13769, 873, 251, -311, 0, -311], [0, -156, 0, 0, 0, 0, 0, -89092, 89092, 0, 0, -156, 0, 11840, -11996, 8129, -8441, -7071, 0, -468], [0, -9211, 0, 0, 0, 0, 0, 53381, 0, 0, 0, -1336, 0, -10419, 668, 35317, 1336, 10547, 0, 2672], [290, 0, -1540396, -14677, 0, 7813, 74, 0, 0, 290, 7955, 0, 7955, 0, 0, 0, 0, 0, -6068, 0], [-512370, 0, 13086, 11682, 6302, -18342, 13086, 0, 0, -2932, 4460, 0, -4460, 0, 0, 0, 0, 0, 0, 0], [0, 1971, 0, 0, 0, 0, 0, 245493, 0, 0, 0, 216, 0, -30326, -108, 20310, -216, -2187, 0, -432], [-3271672, 0, -17920, 36262, 2242, 65358, -17920, 0, 0, 26906, 370, 0, -370, 0, 0, 0, 0, 0, 0, 0], [0, 3585, 0, 0, 0, 0, 0, 0, -33695, 0, 0, 11620, 0, 2905, -15055, 5810, -3872, -2225, 0, 5810], [4627185, 0, 4629014, 15509, -6450, 29876, 7230, 0, 0, 5401, 18294, 0, 0, 0, 0, 0, 0, 0, 6450, 0], [0, 3966, 0, 0, 0, 0, 0, -244823, 0, 0, 0, -3844, 0, 2544, 1922, 15584, 3844, -122, 0, 7688], [1910, 0, 2402891, 15995, 0, -37655, -16585, 0, 0, 1910, -2575, 0, -2575, 0, 0, 0, 0, 0, 655, 0], [0, 16677, 0, 0, 0, 0, 0, -158671, -158671, 0, 0, 7128, 0, -1108, -3484, -4465, -9217, -2376, 0, -2376], [-1920338, 0, -1929834, -46460, 8833, -73990, -28131, 0, 0, -18635, -17422, 0, 0, 0, 0, 0, 0, 0, -8833, 0], [0, 315, 0, 0, 0, 0, 0, 0, -54903, 0, 0, -5600, 0, -1400, -30870, -2800, 49356, 3115, 0, -2800], [2758541, 0, -2745205, 20644, -14728, -41456, -56, 0, 0, 13392, 0, 0, -21868, 0, 0, 0, 0, 0, -14728, 0], [0, 1553, 0, 0, 0, 0, 0, 215464, -215464, 0, 0, 1553, 0, -3563, 5116, -18950, 22056, 3555, 0, 4659], [5294703, 0, -28556, -83571, 3019, 27011, -28556, 0, 0, -21895, -11225, 0, 11225, 0, 0, 0, 0, 0, 0, 0], [0, 1761, 0, 0, 0, 0, 0, 0, 280085, 0, 0, 2216, 0, 554, 10397, 1108, 13379, 653, 0, 1108], [432021, 0, -433215, 39249, -8687, -23926, -17627, 0, 0, 16433, 0, 0, 3751, 0, 0, 0, 0, 0, -8687, 0], [435597, 0, 407096, -48280, -7122, -13750, -26901, 0, 0, 1600, 17073, 0, 0, 0, 0, 0, 0, 0, 7122, 0], [0, 633, 0, 0, 0, 0, 0, 119608, -119608, 0, 0, 633, 0, 12649, -12016, -209, 1475, -14574, 0, 1899], [0, 1954, 0, 0, 0, 0, 0, -179215, -179215, 0, 0, 12084, 0, -21404, -25432, 40987, 32931, -4028, 0, -4028], [-1207171, 0, -29910, -50249, 19281, 59399, -29910, 0, 0, -637, -19065, 0, 19065, 0, 0, 0, 0, 0, 0, 0], [0, -8489, 0, 0, 0, 0, 0, -52651, 0, 0, 0, 716, 0, -4696, -358, 41264, -716, 7773, 0, -1432], [0, -10100, 0, 0, 0, 0, 0, 0, 539481, 0, 0, -1140, 0, -285, -32375, -570, 215, -9530, 0, -570], [-4718908, 0, 4775128, -106716, -8609, -61601, 59448, 0, 0, -3228, 0, 0, -24017, 0, 0, 0, 0, 0, -8609, 0], [0, -1930, 0, 0, 0, 0, 0, -116748, 116748, 0, 0, -1930, 0, -23808, 21878, 79113, -82973, -13617, 0, -5790], [1963337, 0, -32712, 33028, -37972, 60947, -32712, 0, 0, 32245, 1250, 0, -1250, 0, 0, 0, 0, 0, 0, 0], [3906533, 0, -3960996, -65230, 1495, 126800, 16379, 0, 0, -70842, 0, 0, -36875, 0, 0, 0, 0, 0, 1495, 0], [4763222, 0, 4780228, 68377, -15747, 29348, 35624, 0, 0, 18618, -27765, 0, 0, 0, 0, 0, 0, 0, 15747, 0], [0, -4564, 0, 0, 0, 0, 0, 47761, 0, 0, 0, 2966, 0, 53879, -1483, 38160, -2966, 1598, 0, -5932], [0, -26690, 0, 0, 0, 0, 0, 42197, 42197, 0, 0, -7257, 0, -15494, -13075, 29137, 33975, 2419, 0, 2419], [-3442268, 0, 38521, -76546, 27714, -104559, 38521, 0, 0, -59836, 4605, 0, -4605, 0, 0, 0, 0, 0, 0, 0], [54193, 0, 2246543, -98525, 0, -39835, 7389, 0, 0, 54193, -420, 0, -420, 0, 0, 0, 0, 0, 16830, 0], [0, -3123, 0, 0, 0, 0, 0, 137788, -137788, 0, 0, -3123, 0, -38392, 35269, -40332, 34086, 2683, 0, -9369], [0, 342, 0, 0, 0, 0, 0, 0, 37841, 0, 0, -10668, 0, -2667, 71209, -5334, 30671, 5676, 0, -5334], [-4716544, 0, 4690873, 62640, -1724, 785, -48833, 0, 0, 23162, 0, 0, 56464, 0, 0, 0, 0, 0, -1724, 0], [0, 44, 0, 0, 0, 0, 0, -520536, 520536, 0, 0, 44, 0, -26624, 26668, -13927, 14015, -677, 0, 132], [0, 27908, 0, 0, 0, 0, 0, 368073, 0, 0, 0, 3678, 0, 84497, -1839, -1829, -3678, -31586, 0, -7356], [0, -708, 0, 0, 0, 0, 0, 459441, 459441, 0, 0, -12564, 0, -25234, -21046, -15693, -7317, 4188, 0, 4188], [-2214256, 0, -2255913, -105530, 13924, 4645, -49949, 0, 0, -8292, 57254, 0, 0, 0, 0, 0, 0, 0, -13924, 0], [-473015, 0, -30519, 73616, 4776, 131494, -30519, 0, 0, 38455, 27225, 0, -27225, 0, 0, 0, 0, 0, 0, 0], [-35923, 0, -1970179, 13586, 0, 138706, 53889, 0, 0, -35923, -4995, 0, -4995, 0, 0, 0, 0, 0, -9366, 0], [-1579923, 0, 1501241, 88895, -43009, 55500, -58855, 0, 0, -19827, 0, 0, -34021, 0, 0, 0, 0, 0, -43009, 0], [0, 6648, 0, 0, 0, 0, 0, 0, -542723, 0, 0, 2168, 0, 542, 72346, 1084, 44146, 5564, 0, 1084], [-1481601, 0, -1404939, 69684, 13683, -77409, 6705, 0, 0, -69957, 42117, 0, 0, 0, 0, 0, 0, 0, -13683, 0], [96736, 0, -4704468, -185398, 0, -72258, 1424, 0, 0, 96736, 21630, 0, 21630, 0, 0, 0, 0, 0, 31128, 0], [0, 12897, 0, 0, 0, 0, 0, 99717, 99717, 0, 0, -19170, 0, -41716, -35326, -85377, -72597, 6390, 0, 6390], [-6054832, 0, -15983, -41138, 13682, 42733, -15983, 0, 0, -22240, 19325, 0, -19325, 0, 0, 0, 0, 0, 0, 0], [0, -5361, 0, 0, 0, 0, 0, 868925, 0, 0, 0, -6456, 0, 44141, 3228, 20765, 6456, 11817, 0, 12912], [0, 8853, 0, 0, 0, 0, 0, 0, -210983, 0, 0, 31388, 0, 7847, -86069, 15694, -47978, -6841, 0, 15694], [50070, 0, 1293269, -28418, 0, -130043, -29759, 0, 0, 50070, -20455, 0, -20455, 0, 0, 0, 0, 0, 21508, 0], [0, -814, 0, 0, 0, 0, 0, 496680, -496680, 0, 0, -814, 0, -71024, 70210, -8782, 7154, 7348, 0, -2442], [-1510511, 0, 1625905, 150311, 38487, -189984, 2317, 0, 0, 113077, 0, 0, -3381, 0, 0, 0, 0, 0, 38487, 0], [5376333, 0, 5429535, -19584, 19694, -145026, -53092, 0, 0, -106294, 100960, 0, 0, 0, 0, 0, 0, 0, -19694, 0], [0, 1135, 0, 0, 0, 0, 0, -128464, 128464, 0, 0, 1135, 0, 8753, -7618, -9381, 11651, 63344, 0, 3405], [0, -3411, 0, 0, 0, 0, 0, -459899, 0, 0, 0, 1954, 0, 96986, -977, 10281, -1954, 1457, 0, -3908], [-7162610, 0, -7244399, -65464, -26266, 56529, -9941, 0, 0, 71848, -50960, 0, 0, 0, 0, 0, 0, 0, 26266, 0], [-11870945, 0, 94952, 130877, -15493, -180697, 94952, 0, 0, 11225, 13475, 0, -13475, 0, 0, 0, 0, 0, 0, 0], [0, 12978, 0, 0, 0, 0, 0, -984935, 0, 0, 0, 4428, 0, -18938, -2214, -37088, -4428, -17406, 0, -8856], [0, 32052, 0, 0, 0, 0, 0, 0, -100935, 0, 0, 3012, 0, 753, -92571, 1506, -75225, 30546, 0, 1506], [0, -17724, 0, 0, 0, 0, 0, 0, 886261, 0, 0, -19024, 0, -4756, 73702, -9512, -43808, -8212, 0, -9512], [0, -14749, 0, 0, 0, 0, 0, -103751, -103751, 0, 0, 48297, 0, 12636, -3463, -6763, -38961, -16099, 0, -16099], [0, -10135, 0, 0, 0, 0, 0, 55040, -55040, 0, 0, -10135, 0, 20637, -30772, -46430, 26160, 20455, 0, -30405], [-29055, 0, -1966060, 2533, 0, 199123, 95744, 0, 0, -29055, -21420, 0, -21420, 0, 0, 0, 0, 0, 18747, 0], [-8431901, 0, -8292152, 328236, 8976, 325314, 197463, 0, 0, 57714, 3447, 0, 0, 0, 0, 0, 0, 0, -8976, 0], [5266667, 0, -40009, -264032, -25432, -61578, -40009, 0, 0, -107859, -8305, 0, 8305, 0, 0, 0, 0, 0, 0, 0], [0, -31368, 0, 0, 0, 0, 0, 0, 358917, 0, 0, -17228, 0, -4307, -135111, -8614, -39480, -22754, 0, -8614], [-10265636, 0, 10085000, 296766, -9665, 209246, -142361, 0, 0, -38275, 0, 0, -59984, 0, 0, 0, 0, 0, -9665, 0], [0, -47049, 0, 0, 0, 0, 0, 0, 370829, 0, 0, -1244, 0, -311, 44552, -622, 135695, -46427, 0, -622], [0, -560, 0, 0, 0, 0, 0, -5355, -5355, 0, 0, 4095, 0, 111494, 110129, -148055, -150785, -1365, 0, -1365], [20980876, 0, 20903505, -107868, 43895, 128018, 13398, 0, 0, 90769, -110813, 0, 0, 0, 0, 0, 0, 0, -43895, 0], [0, 4572, 0, 0, 0, 0, 0, 512409, 0, 0, 0, 16572, 0, -11112, -8286, 120093, -16572, -21144, 0, -33144], [0, 10326, 0, 0, 0, 0, 0, -1030827, -1030827, 0, 0, -14292, 0, 15371, 20135, -48369, -38841, 4764, 0, 4764], [-1909225, 0, -2022741, -164666, -24989, 91771, -76139, 0, 0, 37377, 118145, 0, 0, 0, 0, 0, 0, 0, 24989, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_80_n_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_80_6_n_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_80_6_n_d(:prec:=20)
    chi := MakeCharacter_80_n();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 6));
    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_80_6_n_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_80_6_n_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_80_n();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<3,R![216763542247808434176, 0, 0, 0, 108964362492176302080, 0, 0, 0, 12246534533136960, 0, 0, 0, 297452922160, 0, 0, 0, 1095980, 0, 0, 0, 1]>],Snew);
    return Vf;
end function;