
// Make newform 1064.2.q.n in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1064_q();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1064_q_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_1064_2_q_n();".
// 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_1064_2_q_n();".
// 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 := [256, -128, 1280, 736, 4688, 1376, 4560, 8, 3204, -62, 839, -19, 159, -2, 15, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-233244093841280, -1066686719972864, -7592361290310144, -5626733575860240, -7490464984730624, -3589595465252720, -4897476918238464, -2623214172465900, -1240172855303516, -691104095414161, -234620536245287, -135681102976851, -22061790698884, -13171357172271, -1621149621626, -887991240725], [81247815120256, 182455911050240, 338317725714176, 243008252638736, 326444577847296, 132337661273488, 207318990934624, 58410208152940, 53431360026644, 14036480431633, 10037162516759, 2435537980379, 953328046732, 235055440711, 65538410882, 14068115277], [113662878812800, -970070517083136, 34971034077568, -3469399868568272, -731915319670720, -3134294518746512, 229822296226640, -2445186494156332, 110954881408500, -647614156114233, 26960277777057, -125746184909531, 2754752149212, -11918886590167, 74255719302, -810574810813], [-35169887843968, -951893337440768, -121800031476992, -1305166508577008, -38130599038080, -1562079233957104, 393069278434784, -839841355419604, 125141579564692, -210471150872935, 27603120276079, -36330971679053, 2678729722988, -3309757518145, 164538036370, -204905039787], [-770795160447616, -3711273657583104, -23404554005083648, -19365128459647408, -25301719258064000, -14880934278277936, -16426930474636256, -9028198334938436, -4164946945743468, -2324795793996291, -788348746957845, -432681058698385, -74457739040052, -41601999321077, -5441931926454, -2689825004519], [861362289784704, 530866034229248, 3580160077276672, 2655478264221136, 3662588781234544, 1698938827624368, 2308256804479040, 1252530362554664, 582112627729628, 330377236922557, 109195740583175, 64950712395523, 10247092847008, 6302277954239, 750640917806, 425431690537], [-127746035785088, 977494501291520, 176245078812416, 2573217916964144, 393946120082304, 2200066100140336, 40506159472960, 1283648427035908, -5736755156452, 324075334315243, -2408221942867, 58645372805081, -225714105596, 5457395599933, 33860631830, 351188739807], [-83894204762752, 80173165426176, -347332382905984, -272058697425520, -500568422777600, -214367384973280, -238837003395824, -229375750636916, -55741803266736, -63000513650151, -9357603859449, -12973171701001, -836729119824, -1257457268597, -65002779306, -88198852655], [-918889371704192, 2743897321585152, -8497918771647872, 6788149393737328, -9078567171603712, 10366389494085856, -9256849763956384, 6954537806399252, -2604517297831248, 1811850113799231, -515558496262335, 333364954141009, -49264868546880, 30779792913437, -3031790914950, 2030314903703], [1314601402099328, 5705608627616256, 5820835873481216, -10209227561493520, 2649634451897728, -17942216498989552, 7307910961291648, -15423304307469836, 2274150901209828, -4179379074795177, 472486928172657, -813699122229115, 45953130310332, -77164507985591, 2346883868598, -5244700706093], [472230018270848, -15019186394098176, -30714906966664960, -50812331390046544, -34295267259827840, -38559108962554576, -20035475427156416, -24612989387258588, -4897587114493092, -6329245059124317, -911991703609611, -1187823512763295, -85764307633980, -112716780288347, -6857065436634, -7415041100777], [518001185001600, 526633776519680, -1666632002964608, -15510066748381616, -8116566088565696, -15387289160498112, -1730392506840640, -13555293653234356, -183234561054040, -3661473258291431, -1844402747305, -726252196117697, 1023801855592, -69446380585885, -582473115970, -4771753121831], [-1287743730750336, 4197912374516480, -7853459524596416, 10976605479526576, -11685298902396128, 17971408524512784, -12492104630391544, 10511025807388268, -3533109813540112, 2682057305391013, -699548588005093, 470421547551451, -66543603564584, 42850402073615, -4121934292162, 2700796668517], [-1605614924870784, 1981015106403328, -4284953758961152, -8663083660596592, -12839819484107008, -11128677529052688, -2209924722238976, -12051988426735124, -198112420450724, -3343191330072847, 35042677420231, -680749049937037, 5707585000580, -65505508351409, -60070110278, -4572753596683]];
    Rf_basisdens := [1, 1, 1593234061512192, 23090348717568, 796617030756096, 144839460137472, 1593234061512192, 199154257689024, 144839460137472, 34635523076352, 796617030756096, 1593234061512192, 1593234061512192, 796617030756096, 398308515378048, 531078020504064];
    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_1064_q();"
function MakeCharacter_1064_q()
    N := 1064;
    order := 3;
    char_gens := [799, 533, 913, 1009];
    v := [3, 3, 2, 3];
    // 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_1064_q_Hecke();"
function MakeCharacter_1064_q_Hecke(Kf)
    N := 1064;
    order := 3;
    char_gens := [799, 533, 913, 1009];
    char_values := [[1, 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, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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 := 2;
    raw_aps := [[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, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [-2, 0, 2, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, -1, 1, -1], [-1, 0, 0, 1, 0, 1, 0, -1, 0, -1, 0, 0, 0, 0, 0, -1], [0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -4, 0, -1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0], [2, 0, 2, 0, 0, -2, 0, -1, 0, -1, 1, -3, -1, 1, -2, 2], [0, 0, 1, 0, 0, 0, 0, -1, 1, -2, 0, -1, 0, 1, 0, 1], [0, 0, -1, 0, 2, -1, 0, 0, 0, 0, 2, -1, -1, 0, -2, 0], [-2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, -1, -1, 0, -1, 0, 0, 0, -2, 0, 0, 2, 0, 0, 1, -1], [0, 0, 5, 0, 1, -1, 1, 0, 1, 0, -1, -3, 0, 2, 0, 0], [-2, -3, 0, 0, 0, 0, 0, 3, 0, 2, 0, 2, 0, 0, 0, -2], [-2, -2, 1, 2, 0, 1, 0, 1, 1, -1, 0, 0, 0, 1, 0, -1], [0, 0, 2, 0, 1, 0, 1, 0, 1, 0, -3, 0, 1, 0, 2, 0], [0, -1, -1, 0, 0, 0, 0, -1, -1, 0, 0, 1, 0, -1, 0, -1], [0, -2, 1, 2, -2, 2, 2, -1, 0, -2, 0, -1, -1, 1, 1, -1], [1, 3, 1, -1, 0, -3, 0, 0, -1, 0, 0, 1, 0, -1, 0, 2], [0, 0, -2, 0, -4, 0, 0, 0, -1, 0, 2, 0, -1, 0, -1, 0], [-6, 2, 0, 0, 2, 1, 0, 1, 1, 2, -2, 1, 1, 1, 0, 0], [0, 0, 0, 0, 3, -1, 0, 0, 1, 0, 1, -2, 0, 1, -3, 0], [-1, 1, 2, 1, 1, 0, 1, -3, -1, -2, 1, -3, -3, 1, -1, 1], [-3, 1, 1, -1, 0, 1, 0, -1, -1, 0, 0, 1, 0, -1, 0, -2], [0, 0, 0, 0, 3, -1, 1, 0, 0, 0, 1, 1, -1, -2, -1, 0], [0, 0, -4, 0, 1, -1, 0, 0, 0, 0, 1, -2, -1, 1, -1, 0], [1, 0, 2, -1, 0, -4, 0, -1, -3, 1, 0, 1, 0, -3, 0, 3], [3, 0, 3, 3, 0, 0, 3, -1, 1, -3, 0, -5, -1, 1, 0, 0], [1, -3, 1, 1, -3, 2, 1, 0, 3, -2, 1, -2, 0, 0, 0, 0], [0, 0, 0, 0, 1, 2, -1, 0, 1, 0, 0, 2, 3, 0, 2, 0], [3, -2, -4, 1, 0, 3, 0, -4, 2, -2, 0, -2, 0, 2, 0, -1], [-6, 0, 0, 2, 0, 5, 2, 3, 3, 0, -2, 3, 3, 3, 2, -2], [0, 0, -2, 0, -2, 2, 1, 0, -2, 0, -3, 1, 0, 1, 1, 0], [-5, -1, 2, 3, 0, 5, 0, -3, 1, -2, 0, -2, 0, 1, 0, -3], [-4, 3, 3, 2, 0, 1, 0, -1, -2, 3, 0, 0, 0, -2, 0, -1], [0, 0, -4, 0, -2, -1, -2, 0, 2, 0, -2, 1, 1, -2, 2, 0], [4, 2, 1, 0, 2, 0, 0, -1, 0, 2, -2, -1, -1, 1, -1, 1], [0, 0, -3, 0, 4, 1, 1, 0, -1, 0, 0, 2, 0, -1, 2, 0], [-8, 2, 6, 1, 0, 0, 0, -2, -1, -3, 0, 2, 0, -1, 0, -2], [3, 2, 0, -2, 2, -3, -2, -1, 1, -2, 4, -3, -1, -3, -2, 2], [0, 0, -10, 0, -4, 3, -1, 0, 0, 0, -2, 3, 3, 0, -1, 0], [1, -2, -3, -1, 0, -2, 0, 1, -4, 3, 0, 4, 0, -4, 0, -2], [9, -1, 3, 3, -1, 1, 3, -3, 0, -2, -1, -5, -3, 1, 0, 0], [-1, -2, -2, 1, 0, 3, 0, 0, -3, 1, 0, 0, 0, -3, 0, -3], [2, -1, 0, -2, -1, 2, -2, 2, 3, -3, 5, 3, 2, 3, -2, 2], [5, 3, 1, 1, 3, 0, 1, -4, 1, -4, 3, -6, -4, -4, 0, 0], [-2, -1, 5, 2, 0, 1, 0, -5, 1, -1, 0, -4, 0, 1, 0, 3], [0, 0, 8, 0, 5, -4, 0, 0, 1, 0, -2, -5, -3, 1, -5, 0], [0, 0, 3, 0, 2, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0], [13, -5, -3, -3, -5, -1, -3, 2, 1, -1, 4, 3, 2, -3, 0, 0], [-4, 3, 7, 2, 0, 3, 0, -2, 1, -1, 0, -6, 0, 1, 0, 3], [0, -1, 1, 0, -1, -2, 0, 0, -2, 0, 0, 0, 0, 2, -1, 1], [0, 0, 2, 0, -3, 5, -1, 0, -6, 0, 1, 5, -1, 0, 3, 0], [-4, 0, 9, -1, 0, 0, 0, 0, 5, -2, 0, -5, 0, 5, 0, 5], [-4, 0, 6, -2, 0, -4, 0, -2, -4, 2, 0, 2, 0, -4, 0, 2], [0, 0, 4, 0, 2, 1, -1, 0, 1, 0, 0, 2, 2, -1, -3, 0], [1, -5, -1, -3, 0, -1, 0, -1, -1, 0, 0, 1, 0, -1, 0, 0], [-5, -2, -3, 1, -2, 6, 1, 3, 1, 3, -4, 7, 3, 1, 4, -4], [1, -5, -3, -1, 0, -3, 0, 2, -4, 7, 0, 5, 0, -4, 0, -2], [-5, 1, 5, 3, 1, 2, 3, -4, 2, -3, 0, -7, -4, 3, -2, 2], [6, -2, -7, -3, -2, 1, -3, 4, 0, 3, 0, 8, 4, -6, 4, -4], [3, 0, -6, -3, 0, -1, 0, 0, -4, 5, 0, 4, 0, -4, 0, -3], [0, 0, 8, 0, -2, -6, 0, 0, 4, 0, -1, -3, -2, -3, -2, 0], [0, 0, 4, 0, 4, 0, 0, 0, -2, 0, -6, 0, -2, 0, 3, 0], [0, 0, -8, 0, -1, -3, 0, 0, 1, 0, 1, 0, -2, -3, 2, 0], [3, 8, 1, 1, 8, -2, 1, -3, -1, -1, 0, -5, -3, -3, 0, 0], [-6, 1, 9, -2, 0, -5, 0, 6, 1, 2, 0, 2, 0, 1, 0, 3], [2, -1, -1, 2, -1, -3, 2, -5, -7, -4, 2, -3, -5, -5, 3, -3], [1, 1, 1, -3, 0, 0, 0, -1, 0, 3, 0, -2, 0, 0, 0, 2], [0, 0, -8, 0, 0, 1, -1, 0, -5, 0, 0, 4, -4, -3, 1, 0], [10, -7, -12, -1, 0, 3, 0, -3, -1, -2, 0, -1, 0, -1, 0, -2], [0, 0, -6, 0, -4, 3, 4, 0, 0, 0, -1, 0, 3, 3, 5, 0], [2, 8, 0, 2, 8, 2, 2, 0, 1, 1, -3, -1, 0, -1, 2, -2], [0, 0, -4, 0, -4, 0, -2, 0, 3, 0, -4, 2, 3, -2, -1, 0], [-6, -1, 7, -2, 0, 1, 0, 3, 6, -4, 0, -2, 0, 6, 0, 1], [0, 0, 4, 0, -1, -2, -3, 0, 3, 0, 2, -2, 1, 0, -4, 0], [0, 0, 8, 0, 1, 3, -1, 0, -2, 0, 3, -3, 1, 6, -5, 0], [-4, -4, 6, -4, 0, -6, 0, 8, 2, 4, 0, 4, 0, 2, 0, 2], [-1, 2, 0, -1, 2, -3, -1, 7, 6, 3, -2, -1, 7, -1, -1, 1], [8, 0, 4, 2, 0, 0, 2, -1, 4, -1, -1, -7, -1, 1, -2, 2], [-5, 3, 1, -1, 0, -3, 0, 6, -4, -2, 0, 7, 0, -4, 0, -4], [6, -2, 0, -2, -2, -6, -2, -4, -8, -2, 4, 0, -4, 0, -2, 2], [0, 0, 0, 0, 2, 0, 0, 0, 8, 0, 0, -2, 8, 2, 0, 0], [0, 0, -2, 0, -3, -2, 1, 0, 3, 0, -2, -6, 1, 4, -2, 0], [-7, 7, 0, 3, 7, 4, 3, 5, 3, 6, -9, 3, 5, 3, 3, -3], [0, 0, -2, 0, 2, 0, 3, 0, 5, 0, -3, 0, 5, 0, 3, 0], [-10, 3, -3, -2, 3, 7, -2, 5, 3, 5, -3, 11, 5, 5, 1, -1], [5, 3, 1, -2, 3, -6, -2, 2, 4, -3, 5, -6, 2, -4, -3, 3], [0, 0, -10, 0, -8, -3, -1, 0, 2, 0, 4, 0, -1, -3, -3, 0], [2, 4, 4, 4, 0, 0, 0, -1, 5, -2, 0, -6, 0, 5, 0, 6], [-3, -3, 4, 1, 0, -1, 0, -3, -5, 4, 0, 0, 0, -5, 0, 1], [1, 4, 0, 5, 4, 5, 5, -1, -4, -7, 2, 3, -1, 3, 5, -5], [0, 0, -6, 0, -3, -1, -4, 0, -5, 0, 6, 0, -6, -1, -3, 0], [6, -5, -7, 0, -5, 1, 0, -2, -9, 0, 0, 8, -2, -6, 7, -7], [0, 0, -1, 0, -1, 0, 0, 0, 3, 0, -1, -7, 3, 7, -1, 0], [3, -6, -5, -1, 0, 4, 0, -5, 2, -7, 0, -2, 0, 2, 0, -2], [0, 0, 12, 0, 1, -2, 1, 0, -2, 0, 2, 3, -4, -5, 2, 0], [0, 0, -2, 0, -2, -4, -5, 0, 4, 0, 1, -1, 0, -3, -3, 0], [10, 0, 0, -1, 0, -1, -1, 5, 2, 9, -8, 3, 5, 3, -1, 1], [5, 6, 0, 3, 0, -2, 0, -2, 5, -6, 0, -3, 0, 5, 0, 5], [-4, -2, 7, -4, 0, -8, 0, 7, 3, 0, 0, 5, 0, 3, 0, 3], [0, 0, 22, 0, 2, -4, 4, 0, 0, 0, -4, -6, -4, 2, 0, 0], [5, 10, -8, -3, 0, 2, 0, -1, 0, 5, 0, 1, 0, 0, 0, -3], [-10, -12, 8, 2, 0, -1, 0, 4, -1, -1, 0, 3, 0, -1, 0, -2], [0, 11, 9, 4, 11, -5, 4, -7, -1, 2, -6, -15, -7, 3, -5, 5], [0, 0, 0, 0, -5, 2, -1, 0, -1, 0, 1, -4, 1, 6, 0, 0], [-5, 7, 3, 1, 0, 1, 0, -8, -6, 2, 0, 1, 0, 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-6], [-5, 3, 7, 7, 3, -1, 7, 2, 0, -5, -2, -6, 2, 8, 0, 0], [29, -9, -5, -3, -9, -5, -3, 9, -2, 0, 3, 9, 9, -1, 2, -2], [0, 0, 6, 0, 2, 2, -6, 0, -2, 0, -2, 7, 0, -5, 2, 0], [-10, -5, 5, 2, 0, 6, 0, -5, 5, -2, 0, -1, 0, 5, 0, -5], [14, 3, -5, 0, 3, 6, 0, 2, -2, -9, 9, 10, 2, 0, 5, -5], [0, 1, -11, 2, 0, 3, 0, 5, -2, 9, 0, 8, 0, -2, 0, -11], [0, 0, 6, 0, -4, -5, -5, 0, 6, 0, 2, 6, 1, -11, 1, 0], [19, 1, -23, -5, 0, 3, 0, -5, -1, -2, 0, 1, 0, -1, 0, -4], [18, 6, -18, 5, 0, 8, 0, 4, 5, -5, 0, -8, 0, 5, 0, 0], [1, 10, -3, -3, 10, -2, -3, 7, 1, -1, 4, 7, 7, 1, 0, 0], [0, 0, 7, 0, -15, 6, -7, 0, -5, 0, 7, 4, 1, 2, -2, 0], [-5, 7, 3, -3, 0, -11, 0, 10, -4, 14, 0, 13, 0, -4, 0, -2], [4, 2, -4, -7, 2, -5, -7, 3, 4, 1, 6, 1, 3, -7, -3, 3], [-2, -3, -1, -2, 0, -7, 0, 9, 2, 8, 0, 10, 0, 2, 0, -3], [-4, 1, 16, 2, 1, -10, 2, 5, 9, -2, 0, -16, 5, 16, -14, 14], [-10, 3, 8, 1, 0, 7, 0, -5, 9, -1, 0, -5, 0, 9, 0, -2], [0, 0, -12, 0, 4, -2, -7, 0, 7, 0, 10, -3, 5, 1, -11, 0], [17, -8, -14, -5, -8, 0, -5, 11, -10, 9, -4, 26, 11, -2, 9, -9], [14, 2, -2, 2, 2, 7, 2, -4, -1, -5, 3, 2, -4, -2, 4, -4], [-5, 11, 7, 1, 11, -8, 1, 2, 6, 3, -4, -13, 2, 1, -6, 6], [7, 4, 0, -1, 0, -2, 0, 4, 5, 12, 0, -5, 0, 5, 0, 7], [0, 0, 10, 0, -5, 4, 7, 0, -8, 0, -12, -1, -4, 5, 0, 0], [-3, -16, 0, -1, 0, 8, 0, -3, 1, 1, 0, -5, 0, 1, 0, -3], [17, 9, -11, -7, 0, -4, 0, 6, 6, 0, 0, -2, 0, 6, 0, 6], [0, 0, -6, 0, -1, -7, -7, 0, 13, 0, -2, -6, 6, -1, -10, 0], [17, 6, 7, 1, 6, -4, 1, -13, -7, -3, 2, -11, -13, 3, -6, 6], [0, 0, 4, 0, -3, -4, -6, 0, -7, 0, 16, 0, -11, -4, -13, 0], [0, 0, 10, 0, 8, 3, 0, 0, 0, 0, -6, 9, 3, -6, 14, 0], [-12, -2, 17, -2, 0, -3, 0, 1, 5, 1, 0, -2, 0, 5, 0, 5], [0, 0, -2, 0, 2, 10, -6, 0, -6, 0, -2, 14, 4, -4, 6, 0], [0, 0, -4, 0, 15, -3, 6, 0, 5, 0, -12, -4, 2, 1, 1, 0], [-20, -12, 18, -6, 0, 2, 0, 0, 2, -2, 0, 0, 0, 2, 0, -2], [1, 17, 5, 5, 0, -4, 0, 0, 2, -1, 0, -2, 0, 2, 0, 6], [-13, 3, -17, -7, 3, 1, -7, 5, -5, 3, 4, 18, 5, -16, 10, -10], [27, 4, -5, -1, 4, 2, -1, 2, -3, 12, -11, 8, 2, -2, 4, -4], [0, 0, -16, 0, -11, 12, 3, 0, -5, 0, -6, 12, 7, 0, 10, 0], [0, 0, -18, 0, -7, 1, -6, 0, 1, 0, 10, -2, 2, 3, -11, 0], [5, -5, 7, 1, 0, -5, 0, -8, 3, -11, 0, -7, 0, 3, 0, 12], [32, 0, 7, -1, 0, -13, -1, 0, -2, -1, 2, -10, 0, 4, -8, 8], [21, -3, -15, -5, 0, -3, 0, 9, 12, 3, 0, -3, 0, 12, 0, 6], [-13, -9, -4, -1, -9, 4, -1, 7, -1, -2, 3, 13, 7, 5, 3, -3], [10, 9, -21, -4, 0, -3, 0, 5, -8, 9, 0, 14, 0, -8, 0, -11], [17, -3, -13, -3, 0, -3, 0, -2, -4, 9, 0, -1, 0, -4, 0, 4], [0, 0, 4, 0, 13, 7, 1, 0, 2, 0, -13, -3, 9, 10, -1, 0], [2, 0, 2, -2, 0, -8, -2, -6, -2, 0, 2, -10, -6, -6, -4, 4], [1, 25, 7, 2, 25, -5, 2, -7, 1, -3, 1, -15, -7, -1, -5, 5], [-27, 2, 12, 7, 0, 20, 0, -4, 3, -5, 0, -5, 0, 3, 0, -15], [0, 0, 16, 0, 8, -10, 4, 0, -5, 0, 0, -12, -15, 2, 5, 0], [4, 5, -1, -6, 5, -3, -6, 3, 7, 12, -6, -1, 3, -3, -5, 5], [0, 0, -28, 0, -7, 0, -3, 0, -6, 0, -2, 9, -6, -9, 6, 0], [17, 12, 8, 7, 12, -9, 7, -15, -14, -1, -6, -17, -15, -1, -1, 1], [0, 0, 13, 0, 13, -6, 0, 0, 3, 0, -4, 5, -3, -11, 11, 0], [0, 0, 6, 0, -8, -9, 2, 0, 4, 0, 9, -6, -5, -3, -1, 0], [-12, 15, 5, 8, 0, 13, 0, -3, 3, -12, 0, -6, 0, 3, 0, -7]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1064_q_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1064_2_q_n();".
// 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_1064_2_q_n(:prec:=16)
    chi := MakeCharacter_1064_q();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_1064_2_q_n();".
// 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_1064_2_q_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1064_q();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![256, -128, 1280, 736, 4688, 1376, 4560, 8, 3204, -62, 839, -19, 159, -2, 15, 0, 1]>,<5,R![9216, -18432, 81408, 79104, 203968, 61984, 102256, 24984, 36576, 5078, 5199, 253, 516, 9, 28, -1, 1]>,<11,R![64, 96, 2352, -6816, 68948, -72372, 203949, 255093, 280827, 134394, 53829, 11535, 2717, 414, 87, 9, 1]>],Snew);
    return Vf;
end function;