
// Make newform 1856.4.a.bl in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1856_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_1856_4_a_bl();".
// 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_1856_4_a_bl();".
// 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 := [14751072, 1083456, -38912400, -7979328, 7106592, 989168, -509640, -37368, 16022, 520, -217, -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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-130880883178269802368, 1372623293185654194000, 2144506383015197261040, -48747936398114755248, -285345134208428931464, 2999195858668965360, 11676011560120460220, -137831479565874128, -180796617966497194, 2346639969921367, 902081932283552, -8262965802853], [-24411272474786102064, -75826481215462680096, -84417525336207643200, -17382422610327413592, 9573935910987249320, 2864020741416137100, -383841222389299668, -122407881172320082, 6266643552994801, 1932206317362965, -37676872998341, -9923410960919], [-117083380965640227072, -35995190424411812400, 232114178944772669808, 88907411951978970000, -25927326058279232456, -10565197552015638384, 1060546087442001084, 414949071090163264, -17621476139832082, -6304114806233345, 107796402154664, 31658966007779], [-370754794707176290608, 748881004783427192880, 708488448739440793200, 116862826758660754872, -69652022198234858144, -22288763754574032900, 2570196807407968200, 978773849787088462, -40189407869822629, -15567651816457448, 244066653715127, 79714148233592], [69134902374085319520, -38552590668837849264, -134903291594073354192, -41224085724211265728, 15025065960084577032, 5431079678282637528, -609409510740200140, -219921611144934476, 10051587748607228, 3389509146986425, -61050049383782, -17168595976539], [-138387215003650031664, -20376639934106823360, 301522966561976516960, 90319284922766926216, -34723972146274052712, -10179601450871767060, 1392617848807915940, 388935303766329166, -22362014136328147, -5831092367414829, 128904048786731, 29213196068751], [388242946812251435328, 1115638517533144322160, -2884904342187036830640, -1612551093240424302192, 313925490878338825864, 188854338533045876160, -12863573580235804380, -7377811995239846072, 219758966810908694, 111817069614277393, -1443129095982772, -558861445914547], [2643687924078247535232, 722997879981424302960, -5427714657682065409200, -2097932237598579735888, 589736302397573718376, 244455104103922443120, -23562360702441730860, -9487533645764059328, 387228589046785106, 143279267639749357, -2368265974363168, -717031054281463], [-1250532375411231023472, -329947139403080685360, 3153902333494178895120, 1476510308933712790968, -335334630008995508816, -175903275840468652020, 13506476427425555040, 6873952630563023158, -226853177728184641, -104058321231612902, 1450712508774743, 520026572887418], [-1718936033766509266848, -276545945273205263760, 4328285506266624032400, 1600970623934459622112, -483056994237602318984, -187857599434843482120, 19449207468102971500, 7290092592720633012, -318934469441125344, -110012287537477233, 1927048190173762, 550701841585907]];
    Rf_basisdens := [1, 1, 18803225825160531840, 1880322582516053184, 3760645165032106368, 9401612912580265920, 1253548388344035456, 1044623656953362880, 18803225825160531840, 18803225825160531840, 9401612912580265920, 6267741941720177280];
    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_1856_a();"
function MakeCharacter_1856_a()
    N := 1856;
    order := 1;
    char_gens := [639, 581, 321];
    v := [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_1856_a_Hecke(Kf)
    return MakeCharacter_1856_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 := 4;
    raw_aps := [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [-4, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [3, 2, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0], [3, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1], [2, 2, 1, 1, 0, 1, 0, -1, 1, 1, 1, 0], [12, -3, 1, 2, 1, 0, 1, -1, 1, -1, -1, -1], [-29, 0, 0, -1, -2, -1, 1, -1, 2, 1, 0, -1], [-29, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-33, 11, 1, 3, -4, 1, 0, 0, -4, 1, 0, -1], [32, -14, -1, 0, -8, -6, -8, 2, -2, 4, 1, -1], [0, 20, 4, 1, -2, -1, -4, -1, -1, 1, -4, -1], [34, 16, 5, 3, 8, 0, 1, 0, -8, 4, 4, 2], [-40, 16, -5, -1, 0, -6, -2, 6, 0, 6, 4, 0], [-52, -8, 1, 0, 0, 9, 2, -3, -4, 5, -1, 1], [100, 10, -7, -7, -10, 1, -2, -2, 4, -1, -8, -5], [115, -18, -2, -5, -8, -2, -3, 8, -2, -6, 4, 2], [133, 0, 6, 5, 4, -1, 0, 5, -2, 11, -2, -7], [-278, 34, 4, -6, 22, 0, -6, -4, 10, 2, -8, 0], [34, 26, 3, 2, 8, 4, 18, -10, -4, 6, 7, 11], [-155, 2, -2, 3, -4, -12, -11, 5, -2, 0, -4, -8], [309, 34, -3, -10, -14, -16, -14, 15, -6, 10, 4, -6], [54, -40, 14, -1, 8, 1, -6, -11, 1, -1, -6, 1], [72, -46, -13, -11, -12, -13, -4, -11, 23, 7, -9, -2], [44, 48, 6, -6, -10, -12, 6, -8, -6, 4, 2, -2], [-619, 44, 16, -7, 40, 19, 8, -3, -8, 11, 6, -3], [411, -12, 11, 2, 0, 2, 0, -9, 0, 6, 24, -4], [102, 36, -7, -16, 35, -25, -11, 19, 7, 15, -1, -8], [-98, -76, 8, -2, 22, 0, -32, -4, 16, 28, -16, -10], [-99, 1, -9, 39, -65, 7, -7, 8, -25, -8, -11, -12], [605, 11, 12, 32, -21, 26, 4, -18, -17, -5, -7, -15], [356, -42, -9, -12, 34, -12, -26, -6, -6, 6, 15, 21], [65, -40, -7, -22, 94, -18, -16, 1, -2, 12, -2, -18], [332, 8, 23, 14, 2, -23, 30, 7, -10, -7, 15, -31], [-262, 48, -8, -8, -62, -2, -7, 10, 22, 12, 20, 2], [745, 58, 2, 7, 54, 4, -33, 10, 10, -8, -8, 8], [-37, 33, -6, -8, 128, 2, 28, 6, -8, 18, -16, -10], [-759, 36, 8, 31, 6, 3, -64, 5, -10, 37, -2, -35], [148, 88, -14, -24, 80, 18, -28, 8, 8, 42, 8, -14], [202, 2, -26, -38, 30, 16, 30, 20, -26, 26, 10, -22], [851, -32, -20, -8, 31, 24, 4, 18, 16, -8, -22, 0], [-220, -143, -22, -13, -25, -15, 16, 43, -75, -74, -7, -8], [500, -74, 33, 29, 18, 15, 114, -31, 29, -1, -15, 10], [888, 0, -6, -50, -40, -28, -36, 90, -34, 16, 8, -36], [0, -96, -14, 12, -80, -28, -15, 92, -40, -4, 46, -54], [344, -79, 35, 14, -56, -8, 47, 31, -20, -28, 64, -22], [-525, -138, -8, -17, -44, -61, 6, 13, 16, 31, -48, 19], [1047, 18, 20, 111, -108, 13, 24, -51, 62, -23, -26, 27], [979, -12, 27, 33, -18, 42, 55, -96, 62, 4, -49, 51], [1146, -80, 31, -55, -8, 56, 16, -6, 27, -58, -17, -12], [269, -74, -42, 59, 34, -7, -61, -41, 50, 7, -66, 71], [1211, -64, -8, -7, -16, 61, 45, 103, -22, -49, 34, 8], [-520, -413, 31, -26, 16, 46, 39, -19, 38, 90, 4, 32], [376, -42, 20, -19, -73, 34, 13, -72, 34, -18, 22, 26], [214, -13, -10, 81, 0, 29, -59, -45, -2, 45, 20, -15], [575, -64, -3, -53, -88, 64, 91, -18, -26, -66, 17, 7], [146, -413, 40, -3, -132, 13, 17, -53, -4, 1, -8, -11], [2144, -16, 6, -10, -80, 4, 56, 22, -14, -40, 72, 60], [1695, 92, -60, -109, -40, -95, -49, 45, 62, 3, -68, 16], [-736, -494, -9, -3, -68, -25, -2, 70, -102, -89, 20, -23], [760, -196, 34, -36, -102, 16, 90, 56, -48, 88, 26, 32], [-1258, 242, 17, -33, 36, 100, 151, -2, 70, 20, 48, 94], [966, -107, 0, 65, -13, -23, -12, -111, 107, -110, -47, -40], [2201, -302, -51, 43, -149, -53, 106, -95, 19, -85, -31, -12], [157, 120, 13, -95, -44, 10, -81, -20, 78, 56, -103, 69], [-1560, -55, 21, 170, 124, -18, -119, -17, 86, 118, 8, 24], [902, -250, 3, 37, -58, 41, -58, 39, -61, 29, -1, 44], [438, -642, -2, -4, -118, -110, -108, 188, -70, 32, 20, -54], [-329, -228, -12, 20, 18, -18, 143, -46, -33, -38, 16, -77], [-663, -596, -13, -108, -130, -43, 183, 103, -71, -59, -1, -20], [481, 392, -44, 41, 248, 10, -57, -9, -88, 82, -12, 6], [1430, -215, 18, 11, -69, 119, -4, 5, -57, 8, 53, 30], [1922, 12, 79, 140, 259, 97, 21, -95, 95, 105, 85, -74], [-3334, -407, 31, 0, 165, -12, 17, -175, 55, -1, -93, 63], [-835, -376, 20, -151, 90, 43, 1, 13, 26, 33, 84, -9], [1434, -2, -11, 52, -178, 54, -234, -114, 22, 88, -9, -25], [-1031, 656, 6, -136, 307, -10, -96, -10, 8, 70, 58, -14], [-393, -132, 96, 91, -22, -51, -101, -57, -66, 39, -38, 96], [1512, -272, 12, -78, 160, 174, 28, -50, 68, 102, 128, 14], [811, -108, 16, -69, 310, 37, 54, -25, 60, -161, -54, 79], [3233, 632, -87, 51, -50, 28, -175, -18, 142, 86, -91, -31], [426, -148, 46, -10, 34, 22, -275, 170, -30, 172, 28, -66], [256, -50, 51, 150, -116, -98, 72, 20, -126, -36, 163, -7], [1073, 380, -22, -33, 20, -3, 337, -67, -64, 65, 150, 91], [-1260, -373, 5, -82, -309, 82, 69, -79, 209, 39, 25, -61], [-2074, 258, 7, 52, -154, -34, 234, -64, -56, -120, -41, -107], [2649, -16, -75, -102, -62, -1, -71, -125, 189, -137, -69, 154], [1168, 174, 1, 146, -216, -104, 106, 16, 52, -74, 171, 57], [1105, -398, -116, -135, 576, -115, -27, 163, 56, -75, -52, -23], [470, 296, 165, -3, -200, 32, 47, 52, -170, 36, 48, -14], [-261, -429, -29, -89, 312, -111, 58, -88, 162, -75, -156, 67], [617, 448, 26, -39, -308, -99, -153, 117, -140, 113, -80, -103], [-3368, -84, -35, -173, 168, 88, 303, -258, 194, -196, -244, -42], [-424, 192, 3, -113, -368, -53, -146, 148, -90, -65, 156, 5], [-1537, 851, -55, -135, -236, -13, 256, 28, -158, -93, 160, 53], [-4109, -640, -66, -158, -100, -100, -239, -32, 291, 104, -226, -47], [141, -654, -193, -213, -337, -243, 78, 159, 35, -59, 27, 22], [200, 474, 142, 364, -526, -16, 280, -138, -42, 62, 94, 102], [-582, -1394, -41, 0, 86, -8, 220, -192, 262, -386, -143, 159], [-2158, -666, 97, -45, 96, 51, 194, 296, -366, 7, 26, -53], [-2801, 1512, 183, 346, -534, 89, -71, -105, -123, 137, 23, -8], [6248, 781, -79, 138, -288, 26, -271, 195, 98, -114, 52, -144], [-110, 458, -43, -300, -106, 56, 44, 6, 172, -34, -215, 41], [-2324, -76, -78, -200, 358, -62, 298, 140, -66, -308, 16, 14], [-3254, -372, -30, -37, 490, -65, -96, -141, 267, -7, -178, 83], [6833, 40, -155, -172, -370, -164, -4, 153, -74, 2, 72, -102], [-4605, 946, 133, 89, 193, 113, 372, -15, 79, -43, 319, 120], [-3540, -268, 41, 177, -720, -70, -214, 42, 136, 102, -16, -144], [4558, 204, 94, 339, 144, -265, -312, 123, -151, 85, -18, -81], [1115, 943, 29, 59, -356, 55, 466, -52, -96, -129, 68, 121], [1082, 436, -135, -78, 19, -71, 225, 21, 137, 13, 31, 42], [-3868, -1394, -25, -216, -338, -56, 402, -18, -174, -70, 39, 69], [4018, 123, 67, 462, -124, -6, 177, -269, -68, -134, 220, -40], [1669, -290, -190, -101, 346, 125, 20, -157, 370, 139, -194, 11], [-4416, 1058, 129, -81, -92, 167, -120, 109, -219, 103, 261, 122], [-2876, 58, -39, -245, -82, -75, -538, -86, 272, 499, -266, -7], [-10131, -972, 26, 5, 308, -157, 181, -183, 232, 31, -12, 19], [-3033, -40, 145, 199, -428, -40, -547, -282, 198, 218, 65, 241], [6369, 125, 242, 56, -124, -86, 102, -180, -190, -10, -104, -110], [5883, 888, -169, -266, -222, -287, -111, 79, 53, -423, -121, -68], [-6846, -72, 225, 265, 56, 158, 26, -242, -63, 92, -9, -132], [1178, -460, -6, 78, -486, 130, -86, 38, -20, -30, 178, -298], [4852, 1042, 94, -198, -54, 98, 184, 130, -10, 256, 292, 66], [387, -100, -156, 11, -54, -221, -436, -133, 176, -303, -266, 1], [-3294, 940, 171, 46, -752, 187, 110, -39, 106, -193, -21, 201], [-1029, -1288, -44, -180, -182, -36, 103, 336, -233, 180, 576, 47], [-8283, -1130, -166, -111, -1060, 153, -62, 151, -100, 109, 86, -73], [-2816, -791, 36, 259, 159, 155, -108, -89, 175, -56, 209, 98], [-5516, 976, 166, -336, 258, -194, -402, 202, -192, -110, -86, -152], [27, -2611, 0, -190, -52, 2, 188, -114, 336, -2, 64, 274], [-6434, 847, 80, 85, -763, 99, -380, -89, -81, 130, -65, 24], [4990, -805, -110, 15, 441, 57, -650, -149, 267, 300, 155, -2], [2087, 488, 139, -47, -610, -288, 423, -122, -122, -290, 55, 175], [3941, 1992, 157, 142, -6, -301, -687, 283, -311, 339, 247, -186], [2934, 1664, -258, 73, -148, -157, -196, 475, -41, -15, 98, 209], [-3909, -2146, -26, -211, 412, 104, 259, -318, 184, 180, -320, 310], [5549, -818, -16, 37, -390, 69, 418, 187, -176, -141, 360, -335], [-4971, -234, -324, 483, -928, -88, -877, 274, -172, 116, -74, -246], [430, 1138, 65, 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904, -1], [-37488, -3064, 382, -906, -2128, 1564, 1708, -132, -2124, -1276, 1406, -1368], [-44466, 2408, -798, -1108, -270, -320, -150, -826, 506, -170, -1330, 1526], [3864, -878, 1651, 2214, -3122, -476, -3610, -3418, 2360, -506, -721, -541], [-5672, 15419, 363, 3406, -1876, -1136, -3397, 449, -2106, 672, 268, -2646], [-6560, 5478, -638, 300, 1172, 908, -4454, 1434, -484, 1536, 668, -1772], [16529, 3098, -104, -1591, 7120, -137, 499, -1297, 1828, 695, -1096, 559], [-24078, 3652, -454, -740, -5344, -1504, -304, 2272, -1028, -924, 614, -554], [-25557, -3478, 346, 409, -4596, -219, -3455, 381, -1872, 1265, -652, -51], [29789, -7192, -124, 868, -4847, 756, 1362, -2528, 2050, -2164, -1672, -30], [-36144, 2603, -1401, -2374, -3837, -520, -855, 2413, 259, -1667, 293, 157], [59224, -735, 276, 361, 4556, 1475, 2097, -945, 804, -1921, 268, 839], [50612, -7378, -359, -5, -598, -7, -720, 275, 1231, 585, 1261, 410], [751, -2114, 787, -1641, 2967, 603, 786, -577, 1291, -1657, 725, 1246], [32617, 5106, 861, 2599, 2043, 2635, -1718, -2399, 287, 299, -567, 496], [-14760, 3060, -1422, -364, -1004, 336, 2854, 1320, 1294, -1500, -2274, 252], [17099, 5478, -247, 541, -3431, -541, -2932, 883, 863, -2709, -1677, 92], [6869, 2648, 348, 907, -564, 1968, 855, 2559, -2074, -1532, 1260, 296], [-17406, 976, -690, -283, 1582, -1665, -3174, 3435, -1943, 1021, 1102, -2045], [25637, -3276, -409, -3106, -112, -2870, -3996, 1073, 1180, 2126, -934, 1450], [48198, -9550, 1598, 32, 4872, 922, -2515, -2956, 2552, 1338, -2930, -328], [-4298, 1765, -1660, 1273, -473, 105, -716, 1353, -1173, -514, -507, -412], [-48860, -2900, -611, 856, -4280, -477, 3406, 1879, -776, -1125, -661, 501], [-26431, 4179, 278, 730, -2644, -816, -182, -1402, 2850, -2404, -576, 2608], [-62174, 517, -662, 885, -2469, -1869, -5782, 45, 537, 1172, -655, -670], [6264, -7224, -60, -2454, 686, 158, 3302, 486, -594, 2230, 728, -1856], [-11217, -656, -126, -2378, 4042, 100, 1469, -858, 719, -1568, 212, 221], [231, -3739, 1958, 1454, -8008, -984, 1712, -2334, -1586, -1488, -180, 820], [15741, 4202, -312, -3479, 1048, -1452, -1426, 2837, -286, 3844, 992, -576], [85395, -5551, 867, -393, 2960, 1979, 5892, 378, -1456, 879, 916, 225], [24855, 2078, -2323, -749, 37, -39, 2448, 837, 1237, -1619, -685, 2310], [62164, -10850, 784, -1740, 1748, -2106, -878, 1298, 424, 690, 858, 332], [-60444, -1186, 13, -1616, -654, 510, -5080, -912, 1022, 1276, -2055, -1963], [-72455, 5194, 1352, 2789, 2274, -323, -1406, 835, -2100, 315, 484, 125], [-76, -3514, -849, -1998, -3758, 678, 3070, 778, -1274, -568, 1085, 615], [48842, -7628, -346, -1802, 2576, -1558, -1316, 1046, -1882, 570, -602, -1596], [-15260, 6704, 538, 1908, -4546, 334, 2140, -1174, 632, -3346, -1478, 1966], [41580, -3913, 2482, 543, -1655, 103, -712, 533, -231, 1850, 899, -1616], [7490, -2208, -279, -3345, 3556, 20, 1592, 3530, -3191, 2406, 561, 252], [-2582, 8788, -1173, -2300, 634, -239, 2154, 1245, -2204, -3247, 205, 1903], [-62742, 2822, -254, 356, 218, 1678, 1883, 1460, 590, -108, 540, -898], [-14880, 7896, -1115, 1192, 5465, -2031, -3327, -699, 1279, 429, -1861, 3116], [-18822, 13686, 1267, 1415, 868, 931, -3560, -1571, -453, 2843, -1389, 852], [31911, -1466, 330, -509, 1516, 1066, -4022, -673, -2012, 782, -536, -762], [-45798, -1182, -798, -298, 2540, -2604, 644, -644, 1052, -1232, -2224, 2664], [20859, 2732, -247, 1844, 582, -1985, 1907, -737, 801, 1663, -1029, -254], [-48345, -14484, -352, -96, -1214, 1114, 401, -2788, 3693, 422, 190, 1725], [-45967, 4290, -763, -898, 7892, -760, 1766, -965, 2116, -1880, -1286, 2444], [-3738, 962, -411, 751, -1320, 389, -2892, 708, -2182, 249, -650, -815], [-13274, 4944, -2240, -2333, -4832, 59, -764, 1059, -595, -803, 56, 629], [32646, 10054, -743, -3410, 336, -1948, -1262, 1332, 1710, 118, -1417, 693], [-8944, 3457, 1756, 3065, 1128, -33, 6199, -641, -1316, 543, 2404, 659], [-20552, -1194, 720, -1734, -2316, 938, -244, 722, 1716, 2678, 944, 1674], [-63096, -2320, 136, -72, 88, -80, -566, 800, 2350, -688, -1344, -374], [-10887, 4974, 700, -25, -8328, 2390, -3443, 1101, 544, 1262, 608, -442], [-57164, -816, 1750, -2578, 4240, 908, 292, -246, -1374, -792, -204, -2224], [-51710, -3614, -617, -240, -3584, -590, -2630, 1122, 96, 1804, -2031, 1161], [70850, 18179, 2208, 361, 6916, 1437, -2181, -2157, 2000, 393, -1576, 1837], [39967, 7174, -998, 243, 236, -916, -80, -925, 1700, 848, 1708, 260], [6009, -2708, -1274, 769, -1118, -687, 27, 877, 1700, -2505, 410, -1788], [3122, -4268, -735, -524, -1165, -1545, -4849, 373, 2085, 911, -1891, -666], [-27537, 10510, -101, -516, -4936, -364, 2138, 3593, -1368, -2632, 1588, 394], [25266, -3596, -1060, 1104, -4788, -526, -1744, -166, -1920, -2074, 1140, -1680], [9864, -3946, 2211, 273, 720, -813, -76, 1154, -2778, 3527, 1932, -2835], [34639, -11239, -853, 1883, 3372, -1191, 3364, -222, -832, 697, -2144, 1811]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1856_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1856_4_a_bl();".
// 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_1856_4_a_bl(:prec:=12)
    chi := MakeCharacter_1856_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_1856_4_a_bl();".
// 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_1856_4_a_bl( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1856_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![-11605016, 34164648, 11333891, -18457158, -3164363, 2483536, 113150, -122236, 2402, 2360, -129, -14, 1]>,<5,R![2158837575300, 31718775060, -161375255195, -3397758554, 4379523877, 112129888, -54452022, -1372684, 331222, 6652, -943, -10, 1]>,<7,R![-51510514450432, 31912018231296, 12388290966528, 520863366144, -169954399488, -15344836096, 376753216, 69463360, 566608, -100512, -1844, 44, 1]>],Snew);
    return Vf;
end function;