
// Make newform 6724.2.a.k in Magma, downloaded from the LMFDB on 02 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6724_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_6724_2_a_k();".
// 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_6724_2_a_k();".
// 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 := [-419, -2396, 90, 12793, 5697, -25826, -11767, 27346, 8920, -16810, -2731, 6021, 139, -1213, 97, 125, -19, -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, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [3116004854587, 15809958092496, -10685259727101, -78769453872953, 9297430533512, 138054852077165, -5130891802539, -114439381711673, 5925586176453, 49734475422503, -3971444565708, -11788214166918, 1182197159602, 1492723212643, -155505124518, -91033227445, 7257188382, 1991517643], [-10083193492405, -47461994701591, 42125222800165, 243449932168627, -75336500922212, -440353094234389, 107494348078290, 367051997634590, -102114105595312, -152542982400265, 50303498028408, 31817896739628, -12482996651044, -2921985164762, 1472930210424, 44758789498, -64254528752, 5117334346], [1444927175198, 6202735728448, -10763194799416, -34134027938943, 33044467694233, 63343137430515, -55778225919714, -50611916210578, 49059148113724, 17400821302788, -22073321217837, -1710057538425, 5121834540362, -371054096832, -576585972870, 92196221641, 24356871092, -5173972350], [5600600764393, 26840671393652, -27115629095933, -139664906978463, 63599285039357, 254326049844237, -103306772316510, -209959303318169, 95947657753223, 83369073851682, -45404717907189, -15275604128697, 10923266588875, 815983493973, -1260964950102, 83886922991, 54189470506, -7921568979], [2750677696801, 12515504000850, -13635196433008, -64428752588925, 30073379654729, 115195251418598, -44239028078093, -93958575058698, 38778131464912, 37509476855445, -17896834799664, -7168664501697, 4257842347081, 490814930554, -488949149590, 19321281480, 20960946416, -2655782727], [-11166522772634, -52981715336441, 52699851349205, 270647198758838, -111663471815599, -484673771971403, 165622034849829, 397599561133138, -148998460682897, -160457570758472, 70045212979986, 31408624167846, -16857072206324, -2354081778793, 1949876969196, -49120926157, -83975285857, 9827313929], [-2648485890131, -9672363656964, 22817189135175, 50705520484711, -78545831139745, -92406150778999, 131078825772882, 70495805190064, -110621710941825, -20201425584490, 48162957923889, -482505727770, -10922929596545, 1328598195322, 1210501821138, -230371176244, -50603464797, 11803141429], [9047936110015, 42374489575258, -46511822439691, -220047682933342, 111477525775799, 398513814592351, -175114738828536, -326783730572012, 157297317857782, 128762207692726, -73050456522636, -23283325928445, 17392761930154, 1166759169824, -1995721301484, 140687773235, 85458250280, -12706462453], [-3319137402201, -13379478509291, 27553612601762, 74225002852414, -94535728505100, -140983155412297, 161112947668344, 113316452983224, -139298908025489, -37213916783767, 61735268970033, 2291048915370, -14174506040616, 1314403417348, 1584375684936, -270768827727, -66622887610, 14581625611], [-6681662979711, -28744642678583, 42088982381393, 151415229995020, -119399617063218, -276859681903246, 193084476817374, 224454182063040, -167696590794944, -83191318721869, 75366387176691, 12019532656149, -17524825558758, 365468328223, 1978326966189, -246434536365, -83790223759, 15573324721], [-13206886037810, -59376665730388, 72235224284866, 308555790769380, -182093616122701, -559656749084574, 286692846728613, 457237843066132, -253236716346664, -177282316486935, 116026956085509, 30420133624599, -27379666078688, -1000148999592, 3123471135903, -278095912429, -133248857093, 21496924875], [-9503983646651, -41084140120265, 62896266670892, 214909166736059, -185530400946604, -391976009745326, 301089085580478, 315888878583973, -259401854034584, -114823562535575, 115729076679657, 15309571898499, -26768818836839, 970288356284, 3010800581655, -404134485886, -127204175650, 24502450997], [24215425416438, 111696177343726, -119169371261137, -576214282899728, 263605588244454, 1039499940524957, -396748083152055, -854791903023753, 355309854803212, 342639566853530, -165999383243025, -65351603037402, 39762033539457, 4361662830109, -4584109108035, 199773094002, 196973552528, -25546997768], [11849230825906, 52028770979178, -73664313099501, -273182512495898, 210352965398084, 499476640716707, -342605240819139, -405139844414294, 298982211042972, 150389313742214, -134721427642401, -21855754875327, 31371968167408, -614205646850, -3544430197479, 439074870923, 150193646706, -27849538169], [-8358561843844, -32067113594142, 73244369064915, 171888190759013, -252532425913328, -317433173924702, 421127983641438, 246648387025376, -355899689429790, -74723933022020, 155287394364447, 979462178145, -35294318721076, 3927005792066, 3918845783796, -718183719098, -164067200772, 37415229194], [-41569411538363, -186934372131905, 240795696286250, 974621525875457, -645182932918291, -1771612930073546, 1031630403073839, 1441996856128720, -904075199033918, -549447748023506, 410285557618356, 88798896717471, -96105901350554, -1091367528379, 10905779624805, -1138731763930, -463557042919, 79656384008]];
    Rf_basisdens := [1, 1, 60941428749, 60941428749, 60941428749, 60941428749, 20313809583, 60941428749, 60941428749, 60941428749, 60941428749, 60941428749, 60941428749, 60941428749, 60941428749, 60941428749, 60941428749, 60941428749];
    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_6724_a();"
function MakeCharacter_6724_a()
    N := 6724;
    order := 1;
    char_gens := [3363, 5049];
    v := [1, 1];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

function MakeCharacter_6724_a_Hecke(Kf)
    return MakeCharacter_6724_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, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1], [-1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [-2, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, -1, 1], [-1, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, -1, 0, 1, -1, 1, 1, 0], [-3, 0, 0, 0, -2, 1, 1, -1, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1], [-2, 1, -1, 1, -2, 2, 1, -1, 0, -3, 1, -1, -1, 0, 1, 0, -1, 0], [2, 0, 0, 1, 1, 1, 0, 0, 0, 1, -1, -1, 0, -1, 0, 0, 0, 0], [-3, 0, -2, 1, -1, 0, 0, 1, 1, -2, 0, 1, -1, -1, 1, -1, 0, 1], [1, 1, 0, 1, 1, 0, 0, 2, 2, -2, 1, 0, 1, 0, 0, 0, 1, -1], [0, 2, -1, -1, 1, -1, 1, 1, 0, -2, 1, 1, 1, 1, 1, -1, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 0, 0, 2, 1, -1, -1, 2, -1, -1, 0, 2, 0, 0, 1, 0, 0, -2], [2, 0, 0, -2, 0, 0, 1, -3, 0, 4, 0, -3, 0, 0, -2, 1, 1, 2], [-1, -2, 1, -1, 1, -1, 1, -2, 0, 1, -1, -3, 3, -1, -1, 2, -1, 1], [0, 1, 1, 0, 0, -2, 1, 0, -1, 1, 1, -1, 1, 1, 1, 2, -1, 2], [0, -1, 1, 1, 0, 0, -1, 0, -1, -1, 1, 0, -1, 1, 1, 1, 0, -2], [-1, 0, -1, -2, 2, -2, 1, 0, 3, -1, -1, 4, 0, 0, -1, -3, 0, 1], [1, 0, 3, 0, 1, 1, -2, 0, -1, 3, -2, 1, 0, -1, 1, -1, -1, 1], [2, 2, 0, 3, -1, 4, -1, -1, -1, -1, 0, -2, -1, 1, 2, -1, -1, -1], [2, -1, 0, -3, 1, -3, 0, 2, 0, 5, -2, -3, 1, 1, -1, 2, -1, 0], [-6, 2, 0, -2, 0, -2, 0, 1, -2, 2, 0, -2, 2, 4, -1, 4, 0, 0], [-4, 0, -1, 2, 0, 4, 0, 0, 3, -5, -1, 2, -2, -1, 0, -1, 0, 0], [-3, -3, 1, -2, 0, -1, -2, -4, -1, 2, 0, 3, -2, -1, -1, -1, 0, 3], [-6, 0, -1, 2, -2, 2, -2, -2, -3, -4, 1, -5, 0, 4, 0, 2, -1, 0], [-4, 3, 0, 1, -2, 2, 0, 1, 3, -4, 0, 4, -2, 2, -2, -1, 1, 0], [-1, -1, 3, 1, 3, -4, -3, 2, 4, 2, 2, 7, 1, -3, 3, 0, 0, 0], [-5, 0, 0, 1, -2, 2, 2, -2, -1, -5, 1, 3, -1, -2, 2, 1, 0, 2], [0, 0, 1, -3, -1, 1, 2, 0, 0, 1, -3, 2, -1, -1, -1, -2, -1, 3], [4, 1, 0, 3, -2, 3, 1, 1, -3, -2, 1, -4, -2, 0, 2, 3, 0, -1], [-3, 1, -2, 0, 0, 1, 1, -3, 0, 0, -1, -2, 0, 2, 0, -1, 3, -1], [-3, 1, 3, -1, 3, -1, 0, 2, 2, 1, 0, -1, 1, -1, -2, 2, 0, 0], [-4, 3, -2, 4, -2, -2, -3, 0, 5, -7, 2, 4, -2, -1, -1, -4, 4, -3], [-8, 1, 1, -2, -2, -2, -1, -1, 1, -5, 0, 5, -1, 2, -1, -1, -1, 0], [2, 0, 1, -1, 2, -2, -2, -1, -3, 7, 0, 2, 0, -1, -3, -3, 1, 1], [-6, 2, 1, -6, 0, -1, 2, -1, -7, 4, 0, 0, 0, 3, -2, 1, -3, 3], [-3, -1, 1, 2, 2, 0, -3, 7, 0, -1, -1, 3, 0, 1, 2, 2, 2, -2], [0, -1, 1, 0, 2, -1, 0, -4, 1, 5, 0, -8, 0, 2, -2, 2, 0, -2], [-3, -2, 2, -2, 0, 0, -1, -3, 1, 0, -2, -3, -3, -1, 1, -1, -2, 3], [3, 1, 0, 2, 4, 1, 1, 5, -4, -2, 3, 0, 4, 0, 5, 0, 0, -5], [0, 0, 1, 1, 1, -8, -4, 3, 3, 5, 2, 6, 1, 0, 2, -2, 3, -1], [3, 2, 1, -3, 1, -1, -2, -5, -6, 5, -3, -7, 1, 1, -3, -4, 2, -1], [2, 6, -1, -2, -2, -2, 1, 1, -4, 5, -2, -7, -2, 0, -1, -4, 1, 0], [8, 2, -4, 4, 0, 2, 0, 5, 3, -1, 2, -4, -1, -3, 3, -2, 0, -3], [3, 1, 1, 1, 0, -3, -2, 4, -1, 4, 0, 5, -2, 0, -4, -4, 2, -1], [7, 0, 0, 7, 0, 5, -4, 5, 1, -6, 5, -1, 3, -2, 6, 3, 0, -6], [12, 5, -1, -3, 1, -4, 2, 3, -5, 12, -1, -6, 0, -4, -3, -5, 2, -3], [-4, -1, 5, -4, 6, 1, 0, -5, -1, 2, -2, 1, 0, -3, -3, -1, -3, 1], [-2, -3, 0, 1, -2, 3, 1, 2, 0, -2, -2, -2, -1, 3, 4, 0, -2, 2], [1, -1, -1, 2, 5, 1, 0, 1, 4, 1, 1, 6, 2, -2, 1, 0, 1, -2], [0, 1, -2, 4, 2, 3, -4, 4, -2, -3, 0, 7, -2, -3, 2, -5, 3, -4], [-3, -4, 0, 1, 0, 3, 0, -2, 0, 2, -1, -4, 3, 3, -3, 5, 0, 1], [-1, 2, -1, 0, 3, 4, 1, 0, 2, -4, -3, 5, -2, 1, -5, -2, 1, -1], [-9, 0, 3, 0, 1, -3, -3, 4, -1, 3, 0, 3, 3, 2, 1, 1, -3, 1], [4, 0, 0, 4, 3, -4, -5, 9, 3, 3, 1, 14, 0, -4, 4, -5, 1, -3], [1, -1, 4, -2, -2, 3, 2, -8, -1, 1, 0, -5, -2, -1, 1, 2, -3, 1], [2, -1, 1, 0, 0, -2, 4, -4, 1, 8, -2, -3, -4, -1, -5, -2, 5, 0], [-1, -2, -3, 3, -3, -6, -6, 7, 5, -1, 0, 6, -4, 1, 1, -5, 1, 0], [1, 1, 0, 0, 2, -5, 4, -2, -2, 0, -1, -3, 2, -1, 2, 0, 2, -3], [1, -1, -3, 0, -2, 2, 2, -7, -2, -8, 1, 0, 0, -3, 2, 1, -2, 1], [-10, -4, 0, 0, 2, 6, -1, 0, 4, -5, -3, 5, 0, 1, 0, 4, -2, 0], [3, 1, 0, -2, 0, -1, 2, -9, -5, 2, -2, -13, 3, 1, 0, 1, -1, 1], [6, -5, 3, -5, 2, -1, 2, 6, -4, 8, 0, -8, 6, 0, 1, 4, 1, 0], [-4, -4, -1, 8, 1, 5, -1, 0, 2, -15, 2, 11, -2, -5, 3, 1, -2, -1], [-1, 4, 2, -4, 0, 2, 0, 1, -4, 12, -1, 3, -2, 1, -3, 0, 1, -2], [8, -6, 2, 1, 2, -3, -3, -4, -3, 15, -1, -6, -1, -5, 2, 0, -1, 0], [2, 0, -3, 0, -3, 2, 4, -3, 2, -1, -3, -10, -4, 2, -1, -3, 3, 0], [-2, -1, 0, -6, 1, 1, 5, 0, -1, -5, -3, -4, 4, 2, 0, 1, 4, -1], [4, -4, -1, 0, -1, 2, 3, 4, 0, 3, 4, 1, 3, -2, 1, 6, 2, -3], [-4, -7, -5, 7, -4, 2, -1, 1, 8, -8, 4, 6, -1, -1, 4, 3, 0, 0], [-6, 3, 1, -1, 3, 4, -3, 3, 2, -9, 2, 2, 6, 4, 2, 1, -4, -4], [-6, -1, -2, -4, -7, 2, 8, -8, 1, 1, -6, -3, -5, 5, -1, -3, -3, 5], [-7, 2, -4, -2, -2, -1, 3, 4, -4, -1, -3, -2, -2, 4, 0, -4, 1, 0], [-10, -1, 0, -1, -2, -9, 0, -3, 5, -4, -1, 6, 2, 6, 3, -1, 0, 2], [-10, -2, -3, 4, -4, 2, 2, -2, 6, -12, 3, 4, -4, 0, 2, -1, -1, 2], [-8, -2, -4, -2, -3, 5, 4, -7, 1, 0, 0, -4, 0, -2, -5, 2, 1, 3], [-10, 1, 3, 1, -2, 5, -5, 2, 4, -6, 2, 10, -3, 1, 1, 3, 0, 0], [-3, 0, 1, -6, -1, 3, 4, -9, -8, 14, -7, -11, -1, 2, -8, 1, -3, 4], [5, 4, 1, -1, 0, -7, -6, -2, -4, 11, 1, 3, 0, -4, -5, -8, -1, 2], [-6, -7, 2, 6, -1, 1, -2, 2, -3, -5, 2, 3, -1, -4, 3, 1, 1, 0], [-12, 4, 2, -7, 0, -4, 3, -4, 1, 5, -3, -1, 0, 2, -8, 1, 2, 4], [2, -11, 4, -4, 5, -6, -1, -6, 9, 0, -3, 6, 2, -2, -1, 0, 1, 5], [-4, -1, -3, 2, 5, -2, -1, 3, 8, -4, 4, 3, -1, 2, -1, 1, 4, -4], [-8, -4, 1, -4, -1, -5, 4, -6, -1, 0, -2, -3, 1, 7, 1, 7, -1, 5], [0, -4, -2, 0, 1, 1, 0, -1, 4, -9, -1, 1, -2, 3, 4, 0, 0, 1], [9, 3, 2, 2, 3, -9, -1, 1, -5, 14, -4, -6, -2, -1, -3, -4, 4, -4], [0, 3, 0, -4, -3, 0, 4, -8, -3, 4, -7, -5, -8, 4, -2, -6, 1, -1], [7, -4, -4, 2, 0, 1, 3, -4, 3, 4, 7, -6, -3, -1, 2, 1, -3, 1], [4, 4, -3, 3, -3, -1, -2, 1, 4, -8, 1, 11, -7, -2, 0, -7, -1, 1], [1, -2, -1, 3, 2, -3, 2, -7, 1, -1, 5, -1, -1, -8, 1, 0, 3, -1], [1, -1, -2, 2, -3, 0, 2, 1, -6, 3, 0, 1, 2, -2, 1, 1, -1, 0], [-3, -7, 5, 2, 1, 0, -3, 3, 0, -4, 3, 17, 5, -4, 9, 5, -3, 0], [-5, 2, -1, -4, -1, -1, 4, 0, -3, 5, -7, -12, -1, 6, -2, -1, 1, -1], [8, -4, 3, -4, 0, 2, 4, -2, 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-2, -6, 3, 20, 9, -7, 14, 6, -9, -5], [8, 8, 6, 5, 2, 9, -11, 11, -8, -3, 4, 7, 3, -6, 6, -7, 1, -10], [12, 1, 4, 0, 4, 5, 0, 9, -12, 13, -1, -12, 6, -1, -2, -2, -3, -4], [-25, -5, 3, 3, 6, 1, -12, 5, 27, -16, -5, 15, 6, 4, 2, -8, 2, -4], [-3, 1, -4, -13, 2, -4, 10, 11, -8, 19, -11, -9, 10, 2, -11, 0, 6, -1], [-16, -2, 0, -23, 8, -17, 7, -4, -2, 18, -1, -25, 20, 13, -13, 8, -6, 2], [-2, -6, 4, -6, -3, 21, 9, -6, -10, 20, -15, -16, -1, 6, 0, 8, -13, -1], [5, 7, -5, 5, 5, 5, 11, 8, -8, -4, 6, -25, 22, 10, 11, 8, -3, -13], [5, -8, 11, 6, 5, -14, -9, 14, -6, 24, 2, 5, 6, -10, 3, 12, 0, -3], [-19, -1, 6, -3, -5, -15, 9, -10, -3, -9, -6, -4, 4, 6, 0, 2, -9, 8], [-12, 8, -10, -10, -7, 8, 22, -20, -8, 7, -11, -26, -4, 1, -4, -1, -4, 9], [-19, -3, 5, -7, -7, 10, -5, -9, -6, 20, -5, 4, -15, -10, -14, -4, 11, 12], [12, -4, 1, 20, 5, -4, -14, 28, 16, 4, 16, 9, 5, -4, 8, 7, 10, -12], [-5, 9, -8, 19, -12, 8, -9, 0, 0, -9, 5, 7, -23, -4, 8, -15, -7, -3], [-10, -5, 2, 0, 17, -8, -7, 9, 1, -12, 9, 13, 14, -6, 10, -5, -5, -4], [-25, -1, 5, 8, -14, 6, 6, -9, 7, -48, 4, 13, -3, 8, 8, 13, -8, 1], [-10, 3, -10, 10, 6, -1, 2, 12, -3, -13, 3, 5, 15, 1, 6, 0, 0, -19], [9, -4, -3, -2, -2, 1, 7, -24, -6, -6, -7, 8, -2, -8, 0, -2, 0, 15], [5, -6, 6, -14, 0, 1, 15, -28, -27, 16, -10, -28, 10, -4, -3, 20, -12, 12], [16, 1, -4, 5, 7, 0, 7, 8, -5, -5, -1, -15, 16, 7, -4, 5, 10, -12], [-10, 3, -10, 4, -5, 2, 8, -1, 14, -11, 6, -10, 3, -9, 8, -3, 2, -8], [-25, 2, -9, 3, -1, 5, -2, 12, 2, 9, 1, -2, -3, 14, 1, 12, -1, -2], [-5, 8, -5, 12, -1, -11, 6, 1, -9, 10, -7, -9, -1, -1, 2, -1, 4, -10], [2, -12, 16, -8, 16, -14, -13, 0, 2, 17, 4, -4, 14, -7, 4, -2, -6, 4], [-31, -3, -1, -10, 0, 0, 14, -10, 8, -15, 1, 7, 12, 12, 5, 21, -9, 13], [8, 10, -1, 3, 1, 1, 7, 30, -1, -9, 4, -12, 10, 13, 4, 8, -2, -14], [-7, 9, -4, -4, 20, -3, -3, 19, 2, 7, 0, -2, 20, 10, 1, 0, 0, -14], [12, 10, -8, -1, -3, -18, 10, -3, -14, 22, -5, -31, -4, 3, -11, -13, -1, 2], [10, -10, 10, 9, 11, 5, -7, -4, -16, 19, 4, -12, 4, -16, -3, 11, -6, 0], [13, 14, -1, 12, 8, -16, -14, 15, 16, 1, 14, 16, 2, -10, 6, -18, 5, -14], [-1, 6, -7, 24, 1, -5, -10, 14, 13, -12, 10, -5, 15, 10, 8, 6, 14, -14], [-27, 14, -2, 12, -13, 11, -8, 4, 11, -22, 6, 18, -6, -12, 5, -13, 7, -8], [12, 6, -2, -1, 4, -10, -6, 14, -1, 13, 6, -2, -3, 4, 0, 5, 0, 1], [19, -11, 16, 12, 6, 1, -12, -15, 0, 13, 4, 11, -8, -8, 3, 5, 1, 6], [-37, 10, 2, 2, -10, -2, -12, -1, -1, 0, 10, -11, 4, 12, 6, 8, -5, 7], [-18, -6, 8, 8, 0, -11, -6, 20, 4, -12, 3, -2, 9, 1, 14, 19, 1, -1], [21, 11, -8, -3, -3, 19, 10, 12, -6, 0, -16, -2, -10, 2, -1, -5, -8, -4], [7, 0, 4, -5, 0, 1, -2, -16, 8, 11, -10, 8, -15, 4, -23, -17, 9, 6], [11, 7, 6, 4, 16, -21, -15, 15, 9, 0, 3, 32, 13, -10, 7, -19, 10, -14], [-3, -15, 2, 5, 5, 0, -14, 9, -9, 8, 8, 16, 1, -12, 13, 2, 4, -2], [-13, -13, 21, -10, 4, 12, 6, 6, -24, 8, -5, 10, 3, -4, 1, 10, -14, 10], [-43, -5, -1, 22, -2, -4, -9, 1, 6, -23, 7, 26, -12, -2, 6, 12, -3, 5], [2, 7, -9, 9, -8, 11, 11, 1, -5, -13, 1, 3, -4, 8, 6, 9, -6, -2], [6, -2, -8, 1, 10, 8, 8, 2, 26, -1, 5, 1, -2, -3, -4, 6, -3, -9], [-13, -5, 7, 0, 7, -9, 4, -1, -11, -8, 4, -8, 14, 0, -4, 7, -13, 7], [-15, -4, -1, -12, -12, -2, 6, -6, -2, 25, -10, 2, -7, 8, -7, 9, -8, 12], [16, 4, -2, 17, 3, 5, -8, -10, -5, 9, -1, -15, -4, -10, 6, -2, 9, -9], [17, 12, -8, -15, 10, 7, 16, -6, -10, 30, -8, -30, 8, 15, -6, 9, -8, -2], [16, -3, -2, -9, 6, -5, -2, 1, -7, 8, -6, -24, 14, 14, 8, 9, -6, -3], [-20, 8, -6, 2, 0, -14, -5, 3, 7, -11, 11, -7, 10, 10, 8, -2, 11, -10], [18, -9, -1, -5, 7, 18, 13, 8, -8, 6, 5, -33, 19, 1, 6, 13, 0, -4], [2, 15, -5, 3, 10, -15, -10, 8, 7, 12, 4, 22, -8, -10, -5, -24, 19, -13], [19, 2, 2, -13, 1, 1, -1, -12, -16, 19, -14, -15, -9, 1, -20, -16, 7, 9], [24, 8, -4, -17, 7, -4, 13, -24, -21, 25, -8, -18, 2, 0, -9, 8, -5, -1], [1, 0, 7, -10, 9, 4, 10, 3, -1, 5, -9, -11, 1, 16, 2, 5, -3, -5], [-27, -4, 1, -2, -3, 1, -9, -21, -1, -17, 2, -22, -1, 4, -11, 18, -2, 5], [11, 3, 4, 2, 4, 13, -1, -10, -13, 5, -14, 1, -4, -5, -12, -8, 0, -3], [-11, 2, -4, 7, -21, 12, 3, 2, -2, -6, -2, -17, -8, 3, 17, 9, -7, 2], [-34, 10, -2, -12, 2, 6, 11, -11, 7, -24, 7, -16, 13, 8, 4, 11, -7, 4], [-20, -5, -4, -3, 10, -15, -2, 29, 16, 4, 6, 26, 18, 3, -4, -10, 4, -6], [-4, -11, 15, -21, 7, -1, 19, -10, -26, 17, -15, -19, 8, 2, -4, 17, -7, 5], [13, 0, 4, 1, -5, 21, 10, 2, 2, -23, 0, -34, 8, 3, 1, 7, -7, 2], [-21, -2, 8, -9, -3, -17, 6, -1, -12, 26, 6, 6, -4, -6, -3, 6, -12, 17], [42, -8, 9, -6, 14, -11, -11, -15, -13, 27, -4, -11, 5, -14, 7, -1, -1, 2], [-10, -2, -4, 11, -2, -26, -15, -10, 9, 0, -2, 14, -18, 4, 1, -10, 7, -2], [-17, 1, 0, 14, -3, -2, -9, 8, 12, 2, 1, 27, -11, -20, 2, -16, 9, 8], [-5, 12, 4, 10, -2, -14, 5, -3, -12, 17, 6, 18, -9, -5, -3, 1, -2, 2], [10, 10, -4, -7, 18, -12, -4, 9, 1, 10, 2, -9, 16, 4, -2, -1, 7, -5], [-7, -6, -2, 5, 2, 0, 1, 24, -18, 2, -3, 5, 5, -4, 3, 3, -1, 2], [-20, -13, 1, -1, 12, -19, -11, 2, 1, -2, 6, 24, 12, -6, 1, -1, -3, -6], [-10, 3, 1, 18, 5, -11, -12, -9, -1, -8, 11, -17, 1, 8, 11, 7, -4, -8], [33, -7, 17, 0, 9, 1, -8, -3, -20, 20, 9, -13, 16, -5, 4, 7, 6, -10]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6724_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6724_2_a_k();".
// 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_6724_2_a_k(:prec:=18)
    chi := MakeCharacter_6724_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_6724_2_a_k();".
// 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_6724_2_a_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6724_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-419, 2396, 90, -12793, 5697, 25826, -11767, -27346, 8920, 16810, -2731, -6021, 139, 1213, 97, -125, -19, 5, 1]>],Snew);
    return Vf;
end function;