
// Make newform 60.6.i.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_60_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_60_i_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_60_6_i_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_60_6_i_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 := [4896482487496796160000, -600638984600463360000, 36839424090977280000, -5202205084237824000, 1195920778734496000, -195554595639827200, 17753999958800000, 369253960052800, 244333990200544, -23647959230160, 238998177000, -60314430972, 12872525097, -161859774, 1160778, -661518, 117610, -382, 2, -2, 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], [-178498454576363546027751035712272753534769937773460152118272195952621755623704325547516206945280000, 2344566227555230949778128172337458429929652781728387920880908766270605451995636893399794172416000, 9983688984871570686352241418657483491892725701499855875497249607740415966909165030683076126182400, -66449226291272003652735194184435816141594340276907263716036179575965112169577059172501263257600, -33286770367156308590833687111338378532135391884561950098749415748772160015817970508878649616000, 11122349578953740389863561181394952731524461612660723812833128976885967027572119117852850235840, 1648265431379056610409958063742713208183566667670151063740290744447661627367277449408661228192, 42433787693287718606600220006577001673899495573920489317142087767696772461708466624075828816, -13311233970317911026561111798531060765518759382463428737293908839005543887658553214125091560, 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    Rf_basisdens := [1, 653558551108780157520429793806612041002517413901473683840599093678404587449214602262638152915968000, 4249554438710796577595341571487742900232432665098448926811812704335953315675304272916889869691363174103024882318347373869196288000, 2698129802356061319108153378722376444592020739745046937658293780530764009952574141534533250597690904192396750678315792932823040, 354129536559233048132945130957311908352702722091537410567651058694662776306275356076407489140946931175252073526528947822433024000, 4249554438710796577595341571487742900232432665098448926811812704335953315675304272916889869691363174103024882318347373869196288000, 4249554438710796577595341571487742900232432665098448926811812704335953315675304272916889869691363174103024882318347373869196288000, 2124777219355398288797670785743871450116216332549224463405906352167976657837652136458444934845681587051512441159173686934598144000, 2698129802356061319108153378722376444592020739745046937658293780530764009952574141534533250597690904192396750678315792932823040, 96580782697972649490803217533812338641646196934055657427541197825817120810802369839020224311167344865977838234507894860663552000, 472172715412310730843926841276415877803603629455383214090201411592883701741700474768543318854595908233669431368705263763244032000, 4249554438710796577595341571487742900232432665098448926811812704335953315675304272916889869691363174103024882318347373869196288000, 472172715412310730843926841276415877803603629455383214090201411592883701741700474768543318854595908233669431368705263763244032000, 708259073118466096265890261914623816705405444183074821135302117389325552612550712152814978281893862350504147053057895644866048000, 874507999337937619351030889932502079537213839233029355436864130904310009922138870156495697111796718770108051609684755968000, 386323130791890597963212870135249354566584787736222629710164791303268483243209479356080897244669379463911352938031579442654208000, 4249554438710796577595341571487742900232432665098448926811812704335953315675304272916889869691363174103024882318347373869196288000, 437253999668968809675515444966251039768606919616514677718432065452155004961069435078247848555898359385054025804842377984000, 15653626006003962888144049394555350369679181378406754413169425598725210798451209574635079107654492178392812589139148288000, 4249554438710796577595341571487742900232432665098448926811812704335953315675304272916889869691363174103024882318347373869196288000];
    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_60_i();"
function MakeCharacter_60_i()
    N := 60;
    order := 4;
    char_gens := [31, 41, 37];
    v := [4, 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_60_i_Hecke();"
function MakeCharacter_60_i_Hecke(Kf)
    N := 60;
    order := 4;
    char_gens := [31, 41, 37];
    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, 1, 0, 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, -1, 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], [4, 4, 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, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [53, -53, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 1, 0, 0, 0, -1], [0, 0, -5, 0, -1, 1, 0, 4, 0, 0, 0, -1, 0, 1, -2, -2, -1, 1, -1, -1], [0, -120, -4, -5, 1, -1, 4, 0, 5, 1, -1, 0, 1, 0, 0, -2, -1, 0, -1, 0], [-3, 3, -2, 0, 1, -9, -21, -9, 0, 1, 0, -1, 3, 1, 2, -2, 1, 4, 4, -1], [0, 6, 23, 0, 0, -31, -26, 15, 0, 7, -3, -3, 3, -1, 0, 2, 0, 0, -5, 1], [-252, 0, -66, -10, 2, 0, -66, 3, -10, 0, -7, 3, -7, 1, -10, 0, 2, 3, 0, 1], [-29, -29, 94, 0, -5, 5, 0, -5, -43, -5, 7, -5, 0, 4, 19, 19, -4, -5, 5, -5], [42, 0, 182, 0, 7, 105, 216, -58, 0, 21, -21, -6, -21, 0, 14, 0, 7, -9, 0, 0], [-1169, 1169, 0, 26, -5, 5, 257, 5, 0, -5, 0, 5, 23, 5, -10, 10, 5, 5, 5, 5], [-21, -21, -207, 0, 2, 118, 24, 132, 0, 28, 21, -22, 0, -2, 4, 4, 2, -17, 17, 2], [-54, 54, -43, 0, -3, -83, -482, -218, 0, 46, 0, -19, 54, -3, -6, 6, -3, -27, -27, 3], [0, 102, 425, 0, 0, -192, -429, 170, 0, 40, -51, -34, 51, -2, 0, 4, 0, 0, 29, 2], [2824, 0, -847, -35, -2, 0, -847, -11, -35, 0, -90, -11, -90, -9, 26, 0, -2, -11, 0, -9], [-3977, -3977, 637, 0, 25, -25, 0, 25, 64, 25, 83, 25, 0, -15, -90, -90, 15, 25, -25, 25], [330, 0, 1371, 0, -40, 170, 1427, -150, 0, 34, -165, -62, -165, 0, -80, 0, -40, 44, 0, 0], [-2739, 2739, 0, -62, 4, -30, 1428, -30, 0, 30, 0, -30, 166, -30, 86, -86, -30, -30, -30, -4], [0, -13358, -943, 80, 45, -13, 943, 0, -80, 13, -144, 0, 144, 32, 0, 38, -45, 0, -13, -32], [-249, 249, 14, 0, -1, -261, -2193, 69, 0, -13, 0, -53, 249, -1, -2, 2, -1, 66, 66, 1], [0, 396, 1659, 0, 0, -131, -1739, -225, 0, -9, -198, 45, 198, 44, 0, -88, 0, 0, -28, -44], [-7335, -7335, 1996, 0, -20, 20, 0, -20, 282, -20, 238, -20, 0, -14, 46, 46, 14, -20, 20, -20], [570, 0, 2727, 0, 55, -260, 2624, 310, 0, -52, -285, 106, -285, 0, 110, 0, 55, -93, 0, 0], [-28176, 28176, 0, -125, 30, 50, 3164, 50, 0, -50, 0, 50, 376, 50, -180, 180, 50, 50, 50, -30], [-141, -141, -1029, 0, 23, 27, 10, -717, 0, -125, 141, 13, 0, -23, 46, 46, 23, -98, 98, 23], [0, -24326, -2493, 340, -64, 45, 2493, 0, -340, -45, -233, 0, 233, -19, 0, 52, 64, 0, 45, 19], [-252, 252, 70, 0, -21, 1084, -2573, 329, 0, -49, 0, 200, 252, -21, -42, 42, -21, -79, -79, 21], [-37736, -37736, 440, 0, -55, 55, 0, -55, -551, -55, -18, -55, 0, 73, 238, 238, -73, -55, 55, -55], [612, 0, 3164, 0, 62, -1430, 2840, 252, 0, -286, -306, 100, -306, 0, 124, 0, 62, -101, 0, 0], [-186, -186, -1959, 0, -118, -1032, -230, -1053, 0, -305, 186, 112, 0, 118, -236, -236, -118, -132, 132, -118], [0, -70240, 824, -465, -85, -91, -824, 0, 465, 91, 217, 0, -217, -176, 0, -534, 85, 0, -91, 176], [0, 504, 2287, 0, 0, 271, -2694, -2290, 0, -67, -252, 458, 252, 16, 0, -32, 0, 0, 376, -16], [92730, 0, -799, 455, 148, 0, -799, -23, 455, 0, 72, -23, 72, -171, -250, 0, 148, -23, 0, -171], [-78523, -78523, 1728, 0, 0, 0, 0, 0, 289, 0, 29, 0, 0, 163, 163, 163, -163, 0, 0, 0], [-125893, 125893, 0, -894, 208, -40, -1759, -40, 0, 40, 0, -40, 17, -40, -88, 88, -40, -40, -40, -208], [603, 603, 5689, 0, 8, -88, -16, -1142, 0, -222, -603, 24, 0, -8, 16, 16, 8, 477, -477, 8], [936, -936, 42, 0, -146, 339, 7743, 64, 0, 104, 0, -49, -936, -146, -292, 292, -146, 66, 66, 146], [0, -1986, -9383, 0, 0, 36, 9691, 1710, 0, -172, 993, -342, -993, 206, 0, -412, 0, 0, -263, -206], [99650, 0, 10850, -180, -198, 0, 10850, -26, -180, 0, 1022, -26, 1022, 172, 448, 0, -198, -26, 0, 172], [-1554, 0, -6617, 0, 26, -1620, -5833, -5514, 0, -324, 777, -1082, 777, 0, 52, 0, 26, -202, 0, 0], [-231485, 231485, 0, 3158, 79, -95, -15278, -95, 0, 95, 0, -95, -1608, -95, 206, -206, -95, -95, -95, -79], [54, 54, 824, 0, 277, -1622, -269, 4127, 0, 1047, -54, 546, 0, -277, 554, 554, 277, -487, 487, 277], [0, -220852, 14314, -1235, -257, 163, -14314, 0, 1235, -163, 1775, 0, -1775, -94, 0, 138, 257, 0, 163, 94], [183482, 0, 10403, -1995, -616, 0, 10403, -299, -1995, 0, 706, -299, 706, 317, 1830, 0, -616, -299, 0, 317], [-408000, 408000, 0, -2721, -555, -5, -8816, -5, 0, 5, 0, -5, -1534, -5, 570, -570, -5, -5, -5, 555], [2187, 2187, 20383, 0, 172, 11228, 2280, 112, 0, 160, -2187, -2108, 0, -172, 344, 344, 172, 78, -78, 172], [0, -414864, -1606, 4485, -411, 206, 1606, 0, -4485, -206, 141, 0, -141, -205, 0, 2, 411, 0, 206, 205], [2760, -2760, -1474, 0, 119, 2364, 24867, -7251, 0, 1355, 0, 568, -2760, 119, 238, -238, 119, 741, 741, -119], [0, -5244, -23690, 0, 0, -3276, 23438, 1760, 0, 860, 2622, -352, -2622, -256, 0, 512, 0, 0, 246, 256], [296308, 0, 25355, 2340, 613, 0, 25355, 121, 2340, 0, 3363, 121, 3363, -492, -1468, 0, 613, 121, 0, -492], [-1416, 0, -7038, 0, 54, 3870, -5482, -3856, 0, 774, 708, -728, 708, 0, 108, 0, 54, 593, 0, 0], [-612, -612, -8957, 0, -415, -7425, -1568, 8530, 0, 1374, 612, 1153, 0, 415, -830, -830, -415, 895, -895, -415], [69, -69, 1298, 0, 171, -13439, 4027, 6661, 0, -1469, 0, -2551, -69, 171, 342, -342, 171, 384, 384, -171], [0, 2724, 12011, 0, 0, -6889, -14586, -6500, 0, 1789, -1362, 1300, 1362, -514, 0, 1028, 0, 0, -1346, 514], [495730, 0, -3983, 5215, 1108, 0, -3983, 773, 5215, 0, 236, 773, 236, -335, -3762, 0, 1108, 773, 0, -335], [-243635, -243635, 18872, 0, -280, 280, 0, -280, 1535, -280, 2475, -280, 0, -347, 493, 493, 347, -280, 280, -280], [-5166, 0, -21344, 0, 299, -6525, -21424, -5826, 0, -1305, 2583, -926, 2583, 0, 598, 0, 299, 3495, 0, 0], [-508769, 508769, 0, 7462, 616, -240, 6969, -240, 0, 240, 0, -240, 1417, -240, 104, -104, -240, -240, -240, -616], [636, -636, -1191, 0, 608, 12778, 5722, -5347, 0, 583, 0, 3042, -636, 608, 1216, -1216, 608, -468, -468, -608], [-612597, -612597, -20071, 0, 170, -170, 0, 170, -600, 170, -2403, 170, 0, 154, -356, -356, -154, 170, -170, 170], [7122, 0, 32947, 0, 12, -4370, 29451, 13122, 0, -874, -3561, 2634, -3561, 0, 24, 0, 12, 32, 0, 0], [-760469, 760469, 0, -4390, 1399, 85, -9050, 85, 0, -85, 0, 85, 384, 85, -1654, 1654, 85, 85, 85, -1399], [-5610, -5610, -44365, 0, 681, 10019, 2140, -16324, 0, -2720, 5610, -1459, 0, -681, 1362, 1362, 681, -531, 531, 681], [424994, 0, -47237, -2435, -1168, 0, -47237, 17, -2435, 0, -6426, 17, -6426, 1185, 2302, 0, -1168, 17, 0, 1185], [-254439, -254439, 47084, 0, 500, -500, 0, 500, -1626, 500, 6410, 500, 0, -1234, -2734, -2734, 1234, 500, -500, 500], [1431, 1431, 18423, 0, 1392, -16552, -3032, 1512, 0, 1416, -1431, 4424, 0, -1392, 2784, 2784, 1392, 78, -78, 1392], [0, -573308, -67928, -3135, -1575, 1432, 67928, 0, 3135, -1432, -6609, 0, 6609, -143, 0, 2578, 1575, 0, 1432, 143], [-1194, 1194, 4994, 0, 358, -16247, -5993, 25328, 0, -5352, 0, -2963, 1194, 358, 716, -716, 358, -1368, -1368, -358], [0, -2388, -5512, 0, 0, 13168, 4352, -20150, 0, -1688, 1194, 4030, -1194, -1182, 0, 2364, 0, 0, 974, 1182], [-631434, -631434, -2258, 0, -170, 170, 0, -170, -10357, -170, -894, -170, 0, 662, 1172, 1172, -662, -170, 170, -170], [-903117, 903117, 0, 4255, -2180, 65, 45218, 65, 0, -65, 0, 65, 2837, 65, 1985, -1985, 65, 65, 65, 2180], [0, -280752, 10344, -3245, 503, 1533, -10344, 0, 3245, -1533, -35, 0, 35, 2036, 0, 7138, -503, 0, 1533, -2036], [-8181, 8181, 220, 0, -98, 30492, -80139, 1002, 0, -122, 0, 6020, 8181, -98, -196, 196, -98, 1323, 1323, 98], [0, 18012, 88291, 0, 0, 29563, -76192, 30330, 0, -5431, -9006, -6066, 9006, -602, 0, 1204, 0, 0, -3146, 602], [-211745, -211745, 77190, 0, 375, -375, 0, 375, -8511, 375, 7951, 375, 0, 584, -541, -541, -584, 375, -375, 375], [3228, 0, 10454, 0, -1582, 4540, 7096, 19748, 0, 908, -1614, 2684, -1614, 0, -3164, 0, -1582, 3062, 0, 0], [0, -770910, -68783, -2595, 1133, -1529, 68783, 0, 2595, 1529, -8096, 0, 8096, -396, 0, -3850, -1133, 0, -1529, 396], [0, -2862, -23115, 0, 0, -35648, 10499, -26020, 0, 6000, 1431, 5204, -1431, 1412, 0, -2824, 0, 0, -2213, -1412], [-339312, 0, -35371, 16855, 296, 0, -35371, 601, 16855, 0, -3968, 601, -3968, 305, -1794, 0, 296, 601, 0, 305], [13602, 0, 67037, 0, 3142, 2280, 62881, 26202, 0, 456, -6801, 7754, -6801, 0, 6284, 0, 3142, 7240, 0, 0], [-72371, 72371, 0, -3494, 2224, -450, -27936, -450, 0, 450, 0, -450, -830, -450, -874, 874, -450, -450, -450, -2224], [0, 165792, 23124, -2665, -1419, -3723, -23124, 0, 2665, 3723, 5643, 0, -5643, -5142, 0, -17730, 1419, 0, -3723, 5142], [-8769, 8769, -1328, 0, 1032, -49548, -64677, -5608, 0, 296, 0, -9084, 8769, 1032, 2064, -2064, 1032, 838, 838, -1032], [0, 13266, 50427, 0, 0, -19081, -52273, 12330, 0, 1833, -6633, -2466, 6633, 2479, 0, -4958, 0, 0, 8455, -2479], [781247, 781247, 47906, 0, -865, 865, 0, -865, 7366, -865, -208, -865, 0, 5627, 8222, 8222, -5627, -865, 865, -865], [918, 0, 7613, 0, -247, -19880, 16984, -66982, 0, -3976, -459, -13594, -459, 0, -494, 0, -247, -1275, 0, 0], [616742, -616742, 0, -1383, 3221, 1895, 23990, 1895, 0, -1895, 0, 1895, 5676, 1895, -8906, 8906, 1895, 1895, 1895, -3221], [2511, 2511, 15843, 0, 655, -11565, -2182, 47535, 0, 10031, -2511, 2837, 0, -655, 1310, 1310, 655, 1090, -1090, 655], [0, -14340, -58566, 0, 0, 10900, 68378, 36440, 0, -804, 7170, -7288, -7170, -1720, 0, 3440, 0, 0, -5862, 1720], [-375432, -375432, -74020, 0, -2695, 2695, 0, -2695, 4533, -2695, -7386, -2695, 0, -539, 7546, 7546, 539, -2695, 2695, -2695], [-2388, 0, -22840, 0, 1742, 77890, -11548, 23172, 0, 15578, 1194, 6028, 1194, 0, 3484, 0, 1742, 83, 0, 0], [0, 1478012, 65806, 18465, 475, 2641, -65806, 0, -18465, -2641, 5663, 0, -5663, 3116, 0, 11514, -475, 0, 2641, -3116], [3423, -3423, -5928, 0, 3476, 40526, 37301, -26164, 0, 2452, 0, 10886, -3423, 3476, 6952, -6952, 3476, -4981, -4981, -3476], [0, 3342, 16839, 0, 0, -27047, -22606, -5065, 0, 8031, -1671, 1013, 1671, -3277, 0, 6554, 0, 0, 2871, 3277], [-29616, 0, -141182, 0, -616, 33390, -131680, -14736, 0, 6678, 14808, -3440, 14808, 0, -1232, 0, -616, -9144, 0, 0], [2248933, -2248933, 0, -21352, -3337, 345, -24441, 345, 0, -345, 0, 345, -6091, 345, 2302, -2302, 345, 345, 345, 3337], [-1967822, 0, 78506, 7290, 2124, 0, 78506, 3094, 7290, 0, 9128, 3094, 9128, 970, -10436, 0, 2124, 3094, 0, 970], [94151, 94151, -134451, 0, 3330, -3330, 0, 3330, -31508, 3330, -15383, 3330, 0, 74, -9916, -9916, -74, 3330, -3330, 3330], [3090, 3090, 26600, 0, 213, 6562, 1355, 10523, 0, 2275, -3090, -1142, 0, -213, 426, 426, 213, 7037, -7037, 213], [15249, -15249, -14106, 0, -733, 44867, 125189, -71263, 0, 14839, 0, 8387, -15249, -733, -1466, 1466, -733, 4223, 4223, 733], [0, -21078, -86581, 0, 0, 64923, 115479, 81710, 0, -14699, 10539, -16342, -10539, 2143, 0, -4286, 0, 0, 6307, -2143], [-64442, 0, 128269, 4295, -4348, 0, 128269, -5857, 4295, 0, 13158, -5857, 13158, -1509, 20410, 0, -4348, -5857, 0, -1509], [2394679, 2394679, -92434, 0, 1895, -1895, 0, 1895, -9786, 1895, -3348, 1895, 0, -7133, -12818, -12818, 7133, 1895, -1895, 1895], [9021, 9021, 93649, 0, 3054, 45846, 9780, 1834, 0, 2810, -9021, -6726, 0, -3054, 6108, 6108, 3054, 16121, -16121, 3054], [17694, -17694, 1580, 0, -1146, 159, 154401, 6754, 0, -434, 0, -885, -17694, -1146, -2292, 2292, -1146, -804, -804, 1146], [0, -13596, -74328, 0, 0, 15296, 59256, -83290, 0, -8952, 6798, 16658, -6798, 7366, 0, -14732, 0, 0, 1642, -7366], [-3739008, 0, 204367, -28565, 418, 0, 204367, -4537, -28565, 0, 25646, -4537, 25646, -4955, 8238, 0, 418, -4537, 0, -4955], [93054, 93054, -77210, 0, 1855, -1855, 0, 1855, 13899, 1855, -15108, 1855, 0, 6323, 758, 758, -6323, 1855, -1855, 1855], [3468653, -3468653, 0, -22049, 4243, -4260, -68088, -4260, 0, 4260, 0, -4260, -2849, -4260, 8537, -8537, -4260, -4260, -4260, -4243], [8844, 8844, 85929, 0, 515, -53185, -10534, -20850, 0, -3758, -8844, 11049, 0, -515, 1030, 1030, 515, -3475, 3475, 515], [0, 4247492, 62978, -31715, 1469, -145, -62978, 0, 31715, 145, 5593, 0, -5593, 1324, 0, 2358, -1469, 0, -145, -1324], [1131268, 0, 132170, 12570, 8718, 0, 132170, 7201, 12570, 0, 19403, 7201, 19403, -1517, -31838, 0, 8718, 7201, 0, -1517], [-39654, 0, -138710, 0, 9111, -107555, -146224, -60874, 0, -21511, 19827, -4886, 19827, 0, 18222, 0, 9111, -2429, 0, 0], [5186821, -5186821, 0, 2168, -1611, -2205, -53397, -2205, 0, 2205, 0, -2205, -7299, -2205, 8226, -8226, -2205, -2205, -2205, 1611], [12165, 12165, 123551, 0, -643, -106247, -21378, -83833, 0, -17281, -12165, 20735, 0, 643, -1286, -1286, -643, 148, -148, -643], [-5304, 5304, 6454, 0, 2670, 51655, -46853, 34940, 0, -9124, 0, 12467, 5304, 2670, 5340, -5340, 2670, 9570, 9570, -2670], [0, 6822, 46421, 0, 0, 19628, -29257, 60830, 0, 364, -3411, -12166, 3411, -5362, 0, 10724, 0, 0, 18645, 5362], [-4761800, 0, -94883, 10505, -6042, 0, -94883, -7687, 10505, 0, -12314, -7687, -12314, -1645, 27458, 0, -6042, -7687, 0, -1645], [1080721, -1080721, 0, -3612, -7776, 2900, 7688, 2900, 0, -2900, 0, 2900, -7244, 2900, -924, 924, 2900, 2900, 2900, 7776], [-4728, -4728, -30617, 0, -1842, 34842, 6600, -98357, 0, -21145, 4728, -8442, 0, 1842, -3684, -3684, -1842, -2958, 2958, -1842], [20769, -20769, 40, 0, -760, 18040, 180121, -560, 0, 720, 0, 3000, -20769, -760, -1520, 1520, -760, -13565, -13565, 760], [-2451186, 0, 38053, 3675, 5052, 0, 38053, 10757, 3675, 0, 3304, 10757, 3304, 5705, -31618, 0, 5052, 10757, 0, 5705], [53016, 0, 234107, 0, -884, 13485, 229880, 33736, 0, 2697, -26508, 6040, -26508, 0, -1768, 0, -884, 7796, 0, 0], [0, 387268, -43220, 63555, 735, -9320, 43220, 0, -63555, 9320, -1395, 0, 1395, -8585, 0, -35810, -735, 0, -9320, 8585], [0, 36768, 198120, 0, 0, 115272, -184696, -52520, 0, -19560, -18384, 10504, 18384, -4368, 0, 8736, 0, 0, 22016, 4368], [1775876, 1775876, 330128, 0, -8515, 8515, 0, -8515, 18001, -8515, 45778, -8515, 0, -8151, 17394, 17394, 8151, -8515, 8515, -8515], [1373667, -1373667, 0, -3083, 13646, 5245, 210490, 5245, 0, -5245, 0, 5245, 36451, 5245, -29381, 29381, 5245, 5245, 5245, -13646], [0, 7822300, -242866, 43645, -2835, -3841, 242866, 0, -43645, 3841, -22443, 0, 22443, -6676, 0, -21034, 2835, 0, -3841, 6676], [-20517, 20517, 10128, 0, -3162, 48868, -207707, 47478, 0, -6966, 0, 7244, 20517, -3162, -6324, 6324, -3162, -6983, -6983, 3162], [463244, 0, -79182, 76850, -3098, 0, -79182, 4797, 76850, 0, -14561, 4797, -14561, 7895, -3398, 0, -3098, 4797, 0, 7895], [3290341, 3290341, 59488, 0, 880, -880, 0, 880, 132311, 880, -797, 880, 0, 7309, 4669, 4669, -7309, 880, -880, 880], [40068, 0, 147250, 0, -10962, 55660, 153208, 14908, 0, 11132, -20034, -5788, -20034, 0, -21924, 0, -10962, -24754, 0, 0], [0, 1703232, 115250, -56305, -13217, -1036, -115250, 0, 56305, 1036, 26483, 0, -26483, -14253, 0, -30578, 13217, 0, -1036, 14253], [-87498, 87498, -19858, 0, 6209, -18356, -757733, -93081, 0, 13649, 0, 1296, 87498, 6209, 12418, -12418, 6209, 15461, 15461, -6209], [-930771, -930771, 233067, 0, 2460, -2460, 0, 2460, 35242, 2460, 27739, 2460, 0, -2116, -9496, -9496, 2116, 2460, -2460, 2460], [7554, 7554, 54937, 0, 255, -64745, -12898, 70600, 0, 14324, -7554, 13153, 0, -255, 510, 510, 255, 31875, -31875, 255], [0, -44520, -220557, 0, 0, -129135, 206231, 54955, 0, 27867, 22260, -10991, -22260, -2550, 0, 5100, 0, 0, -11714, 2550], [9329242, 0, -169153, -24535, 1668, 0, -169153, -18077, -24535, 0, -7084, -18077, -7084, -19745, 32818, 0, 1668, -18077, 0, -19745], [152952, 0, 639019, 0, -19248, 53845, 636980, 44792, 0, 10769, -76476, -6440, -76476, 0, -38496, 0, -19248, 11952, 0, 0], [3854190, -3854190, 0, -157003, -4802, -11970, 401130, -11970, 0, 11970, 0, -11970, 41098, -11970, 40712, -40712, -11970, -11970, -11970, 4802], [-67563, -67563, -634207, 0, -1747, 110297, 21710, 94013, 0, 17405, 67563, -23457, 0, 1747, -3494, -3494, -1747, 7647, -7647, -1747], [0, -170290, -78571, 6690, 6950, -5461, 78571, 0, -6690, 5461, -13253, 0, 13253, 1489, 0, -7944, -6950, 0, -5461, -1489], [0, 94932, 409172, 0, 0, -46792, -424900, -27910, 0, 6208, -47466, 5582, 47466, 3938, 0, -7876, 0, 0, -64566, -3938], [-4305579, 4305579, 0, -51101, -5377, -6820, 620048, -6820, 0, 6820, 0, -6820, 64275, -6820, 25837, -25837, -6820, -6820, -6820, 5377], [32958, 32958, 318765, 0, 11366, -113136, -20354, 127971, 0, 34687, -32958, 31720, 0, -11366, 22732, 22732, 11366, 1524, -1524, 11366], [0, 5068608, -388344, -59425, -10225, -4719, 388344, 0, 59425, 4719, -30827, 0, 30827, -14944, 0, -39326, 10225, 0, -4719, 14944], [7419, -7419, 28832, 0, -4820, -46790, 55885, 139340, 0, -24012, 0, -13214, -7419, -4820, -9640, 9640, -4820, 9415, 9415, 4820], [5869469, 5869469, 384552, 0, 10440, -10440, 0, 10440, -56843, 10440, 37809, 10440, 0, 3759, -27561, -27561, -3759, 10440, -10440, 10440], [8430, 0, 24328, 0, 10445, 172485, 70456, -47710, 0, 34497, -4215, -1186, -4215, 0, 20890, 0, 10445, -7635, 0, 0], [7109239, -7109239, 0, 153390, -15922, -3230, -312443, -3230, 0, 3230, 0, -3230, -50279, -3230, 25612, -25612, -3230, -3230, -3230, 15922], [-29019, -29019, -197769, 0, 9537, 107533, 23414, -116733, 0, -15717, 29019, -13877, 0, -9537, 19074, 19074, 9537, 16048, -16048, 9537], [-1504279, -1504279, 239579, 0, -8155, 8155, 0, -8155, 14162, -8155, 43177, -8155, 0, -15651, 8814, 8814, 15651, -8155, 8155, -8155], [3150, 0, -35643, 0, -20340, 45690, -41171, 52990, 0, 9138, -1575, -5674, -1575, 0, -40680, 0, -20340, -24774, 0, 0], [0, 4932486, -423949, 13250, -10457, 5165, 423949, 0, -13250, -5165, -38944, 0, 38944, -5292, 0, -254, 10457, 0, 5165, 5292], [0, -114342, -501357, 0, 0, -13873, 465915, -170590, 0, 8577, 57171, 34118, -57171, -7253, 0, 14506, 0, 0, 32227, 7253], [-1382961, -1382961, -708682, 0, -8905, 8905, 0, -8905, -70640, -8905, -71274, -8905, 0, -6479, 20236, 20236, 6479, -8905, 8905, -8905], [124902, 0, 587855, 0, 10757, -21410, 574212, 57562, 0, -4282, -62451, 20118, -62451, 0, 21514, 0, 10757, -26903, 0, 0], [20979, 20979, 139687, 0, -2596, 61956, 11872, 191104, 0, 36144, -20979, -14468, 0, 2596, -5192, -5192, -2596, -36674, 36674, -2596], [55224, -55224, -11000, 0, -13361, 57594, 429381, -68361, 0, 24361, 0, 830, -55224, -13361, -26722, 26722, -13361, 2661, 2661, 13361], [-8695036, -8695036, 697680, 0, -4050, 4050, 0, -4050, -138991, -4050, 81772, -4050, 0, -3802, 8348, 8348, 3802, -4050, 4050, -4050], [-65028, 0, -246664, 0, 10182, -127990, -242396, -139148, 0, -25598, 32514, -19684, 32514, 0, 20364, 0, 10182, -357, 0, 0], [37230, 37230, 324867, 0, -9812, 84732, 14984, -155037, 0, -38857, -37230, -24796, 0, 9812, -19624, -19624, -9812, 15222, -15222, -9812], [92931, -92931, 6424, 0, 18146, 222376, 868117, 50266, 0, -24570, 0, 58992, -92931, 18146, 36292, -36292, 18146, 11769, 11769, -18146], [19573894, 0, 359811, -66195, 13716, 0, 359811, 13923, -66195, 0, 45960, 13923, 45960, 207, -55278, 0, 13716, 13923, 0, 207], [10014861, 10014861, -773676, 0, -5310, 5310, 0, -5310, -7223, -5310, -85203, -5310, 0, -171, 15759, 15759, 171, -5310, 5310, -5310]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_60_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_60_6_i_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_60_6_i_a(:prec:=20)
    chi := MakeCharacter_60_i();
    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_60_6_i_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_60_6_i_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_60_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([],Snew);
    return Vf;
end function;