
// Make newform 5520.2.be.a in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5520_be();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_5520_be_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_5520_2_be_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_5520_2_be_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 := [64, -896, 6272, -1664, 2160, -18048, 62624, -45408, 16460, -168, 1088, -920, 373, 28, 8, -4, 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], [94236309393562936299712, -1332221403888018014991392, 1286677423391995969163200, -3755942242288309796882992, 2003438432384731993177024, -1621157387025078645325152, 825981882874189932557056, -449503757829354878585096, 38959469703544326596092, -27782783728011205760238, 15856188225716940095240, -9091844016900447538055, -526628042123880972776, -175071232472117903174, 79205853823130329892, -22393307326694017831], [-311438671188616313007904, 4173357676598058424304, -99682082229361135065120, 1007413334526259350866064, -2744439062106117518346848, 3217485504535708805590224, -1712413047279119646456252, 435067467376470129669992, -42773987978022886175024, 61329501638043960405891, -35865907280266927386845, 8635195911545456863880, 430477237088200538532, 345190768387595378903, -135876466122824388149, 25569479030280566412], [-72370372478847279136032, -8823571164277865133088, 226667030484296414298880, -2355692976644302317089648, 6833395910542852553834256, -7511743716854025606700288, 3940945426884564607889904, -965514734333909107594024, 115423269614068896266898, -141781898740870143152142, 82758888276838958876400, -19198824197009526898095, -889272987826610412594, -797609807237825424606, 310762723733366637388, -57299681650542813799], [3743822249151046185922688, -76921847512402003069408, 1654837676509585409533440, -15979835659401989048171248, 43448370476958442988560896, -51191725696677861835307008, 27883629363643201451158784, -7432691074169667193750664, 700416946484716590460688, -964687428832831710042042, 581239161047249858179860, -151253320213586326551555, -7995018565696568059124, -5561764669179045278226, 2241122015124087814768, -439455554879435383139], [17380390028458691226688, -8800959934029242894816, 117111471819228528626944, -803716592544779228231536, 1293457952188598334348928, -2643931979850558069220992, 1734559757843010960652096, -625533344564557583278984, 16736683687098622304212, -47875325984313322571514, 34947175687175212714580, -13835641266715917580139, -974908431881953254128, -321061485849858930146, 151526112378019231664, -37104260317816956443], [844306681429309503644608, 46133565624937104022592, -309337175284090395834640, -107340536214601554404528, 7777687897607258688480656, 313489945615288993111112, -2953484078021429244894616, 2321667818194567952846876, 148038962812064696378968, -16471505765686500866422, -50079493872898502763745, 58315309974111227789410, 5332779684253886100096, 350397790926717630654, -352863803987486548657, 139701703083084843466], [163776380549154061343424, 10748186561286905391968, -80106538518255115634544, 89009254303847462446576, 1425441230980083855222048, 430569310029225112211832, -850100044846283359262720, 566002065946927646456116, 28027087138017880071620, 3150971691525804233672, -15179392427921230632309, 13968426198284995440156, 1236916786086591809948, 116684157278614951436, -93876249017534482861, 34009524100498954924], [55233526678960890115296, -738358944973925573408256, -1858437542654510627821200, 1801469580584110661948304, 2128792998735165940548672, -13129113754973767703626776, 9878598436726013996313568, -3638831396467228504089708, 623219353097996059986, -225167864663606741739234, 200933097563041183630095, -83457016995930477068460, -6445092595785024774798, -1717983330107770032102, 880505055132253233071, -223356151832978209908], [-1932955519447723140567872, 391637599271987046216352, -4166312496714838271935200, 20939510598395836955100592, -11673278022187112926811424, 71186232994978237024199152, -56117189935335908355064496, 24499444218239838062427296, -91948359191815807646372, 1190786430340385736616338, -1098043412608702503400770, 569272045850750951194965, 44025670310952507825176, 9692947659699732558114, -5233002171242895530902, 1467508247969570962901], [817166959957868757296, -11216975671400262226896, 507745037561589969520, -970114662909385438936, 24908556001061761521512, -110192195306696806738916, 81918401702948551124388, -30518839909115709877418, 161278877668106868691, -1899083034361956529764, 1653992395133468254305, -696404637902714743190, -53067119951003531693, -14306448104402867992, 7283657711690606901, -1856365266029439028], [1876980550567798417556032, -25987382272721391873487392, -1211545795163318363457600, -23214628329792689250941552, 52211043531645188945835584, -189225266087830922589685312, 135911502779325343997747216, -52314610125801861039516216, 486411393437322269222212, -3308875375372790427186838, 2713239457625169804239320, -1195903546080911591781535, -89828524897078368355296, -24185853183290763727854, 12185535405329615195492, -3161330175133885254831], [-117283563291563124756928, 1617327804848636083628576, -749426359272182604170864, 838043210186307474026992, -3408341800529148310923648, 13689244198922517790399968, -10142129746831373977569608, 3791267880997429381455816, -27015465087264750298364, 235921791902117618025326, -204495788443437004457355, 86281763985010554218963, 6545072874457425453552, 1773054069533715792634, -901020790322433139847, 229762909361918816175], [-711791183647280820656192, 9764350823638301822300192, -3441505923093088420226080, 263816343263634857645072, -21834891657963530120968224, 97753879874209701251464832, -73166458764427204113160696, 27082925975940619456063336, -137215035263440896446332, 1679834085428337891531678, -1479479863038053949806330, 617241953105613212778205, 47105813531921546323536, 12726226693636926236734, -6489058375714485668502, 1647786134581718440981], [-4930276316315155162179328, 67813411026733915122914208, -14771338934975639530906080, 18778730745673106956448528, -147200343497065264843608896, 625505202864271357582465648, -464019093201352332432477584, 173235171883211922179021424, -1044812632511988236449528, 10782076755025973164713922, -9362653473555985745334190, 3949213686776762072093155, 300354967909233596033724, 81105100589352480044706, -41256382875221750700458, 10521267573052040758179], [861240134394725537790048, -11810244640222325675946208, 3458967843824083491365280, 80965926607595525060032, 26514840899483907674201376, -119495653065311981325644808, 89399980582648206502360744, -33105615204951311090397084, 163825187962440470325338, -2054011490558481066866392, 1807616767329908505829800, -754745724599480418409495, -57610372955257229156194, -15554783592512782860496, 7931243684916089679668, -2014702384092608461819]];
    Rf_basisdens := [1, 346931609914134839972480, 86732902478533709993120, 173465804957067419986240, 1040794829742404519917440, 69386321982826967994496, 173465804957067419986240, 34693160991413483997248, 173465804957067419986240, 1040794829742404519917440, 810587873631156168160, 1040794829742404519917440, 34693160991413483997248, 173465804957067419986240, 1040794829742404519917440, 173465804957067419986240];
    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_5520_be();"
function MakeCharacter_5520_be()
    N := 5520;
    order := 2;
    char_gens := [4831, 1381, 1841, 4417, 1201];
    v := [1, 2, 2, 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_5520_be_Hecke();"
function MakeCharacter_5520_be_Hecke(Kf)
    N := 5520;
    order := 2;
    char_gens := [4831, 1381, 1841, 4417, 1201];
    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], [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], [-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, 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, 1, 0, 0, 0, 0, 0], [-1, 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], [1, 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, 1, 0], [0, 0, 0, -1, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1, -1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0, -1], [0, 0, 0, 0, 0, -1, 1, -1, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 1, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, 1, 0, 1, -1], [0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, -1, -1], [0, 1, 0, 0, 0, 0, 0, 0, 1, 0, -2, 0, 1, 0, 1, 0], [-1, 0, 0, 1, 2, -2, 2, -1, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, -1, 0, 2, 0, 2], [3, 0, 0, 1, 0, 0, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0], [2, 0, 1, 1, -1, 1, -3, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, -2, -2, -2], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, -1, -1, -2], [0, 0, 0, -2, -2, -1, -1, -1, 0, -1, 0, 0, 0, 0, 0, 0], [4, 0, -2, 1, 1, -2, 2, -1, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 1, -1, -1, 1, -2, 3, 0, 1, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -2, -1, -2], [0, -2, 0, 0, 0, 0, 0, 0, 1, 0, 4, -1, -1, 0, 1, -3], [0, 2, 0, 0, 0, 0, 0, 0, -1, 0, -2, 1, 0, 1, 1, -1], [0, -1, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 1, 1, 1, 1], [0, -1, 0, 0, 0, 0, 0, 0, -1, 0, -2, -1, -1, -2, -2, -3], [0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 3, 2, 1, 1, 4, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 3, 1, -1, 2, 2, -1], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, 1, 0, 2, -3, 1], [0, 3, 0, 0, 0, 0, 0, 0, -1, 0, 1, 1, 2, 4, 2, 2], [0, -2, 0, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, -2, -4, -2], [0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -1, -1, -1, -2], [2, 0, 1, 1, 3, -1, 1, 1, 0, -2, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 3, 1, -1, 1, 4, -2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, -2, -2], [-8, 0, 0, 2, 2, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -2, -2, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0], [8, 0, -1, -1, 3, -3, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0], [2, 0, -1, -1, 1, -1, 1, -3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 3, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 3, 1, 2, 2], [0, 2, 0, 0, 0, 0, 0, 0, 4, 0, -2, 0, 0, 0, 4, -2], [-4, 0, -2, 4, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -6, -2, 0, 2, -2, 0], [0, 0, 1, -3, -3, 1, -1, 5, 0, -2, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 0, -1, 0, 5, -1, 2, 0, 4, 0], [0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 4, 0, -1, -4, -1, -4], [-4, 0, 0, 2, -2, 4, -4, 4, 0, 0, 0, 0, 0, 0, 0, 0], [3, 0, 1, 1, 4, -5, 1, -2, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 3, 1, -1, 1, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0], [6, 0, -1, 0, 2, 2, -5, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 5, 0, 0, 0, 0, 0, 0, -3, 0, -7, -2, 1, 3, 2, 2], [0, 0, 2, -2, 4, -2, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0], [0, 3, 0, 0, 0, 0, 0, 0, -4, 0, -9, 1, 3, 4, 0, 1], [-5, 0, -2, 2, 1, -2, 0, -2, 0, 2, 0, 0, 0, 0, 0, 0], [0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 2, -2, 2, 0, -1, -2], [0, 2, 0, 0, 0, 0, 0, 0, 1, 0, -6, -3, 0, -1, -1, -1], [0, 1, 0, 0, 0, 0, 0, 0, 2, 0, -4, 0, -1, -1, -1, 1], [0, -6, 0, 0, 0, 0, 0, 0, -2, 0, -1, 0, -1, -2, -2, -3], [-7, 0, -1, 3, -4, 3, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 4, 3, -3, -1, 1, 1], [0, 1, 0, 0, 0, 0, 0, 0, -2, 0, -8, -2, 1, 1, -1, -3], [0, -4, 0, 0, 0, 0, 0, 0, -2, 0, -4, 2, 0, 2, -2, -2], [0, 0, -3, 5, 1, -2, 4, -5, 0, 3, 0, 0, 0, 0, 0, 0], [-5, 0, -1, -1, -2, -5, 3, 0, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, -2, 2, -6, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, -2, 1, 2, -4, 0, -3, 0, -1, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 0, 0, 5, 0, 0, -3, -2, 1, 1, 0], [6, 0, 1, -1, 1, 1, 1, -1, 0, 2, 0, 0, 0, 0, 0, 0], [-4, 0, -1, -4, -2, -1, 4, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 8, 2, 0, -2, 0, 0], [0, 0, -2, 0, -2, 2, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -8, -4, -2, -4, -4, -4], [4, 0, 3, -1, -1, 0, 2, 3, 0, -1, 0, 0, 0, 0, 0, 0], [12, 0, 0, 2, 2, -4, 1, 2, 0, -2, 0, 0, 0, 0, 0, 0], [0, 5, 0, 0, 0, 0, 0, 0, -3, 0, 2, -2, -1, 2, 3, 2], [12, 0, -2, -2, 2, 2, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0], [0, -5, 0, 0, 0, 0, 0, 0, -1, 0, 13, -1, 0, -2, 0, -6], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, 1, 0, -2, -3, 1], [0, -1, 0, 0, 0, 0, 0, 0, 2, 0, 12, 2, 1, 5, 1, 4], [6, 0, -1, 0, 10, -6, 3, -2, 0, 4, 0, 0, 0, 0, 0, 0], [0, 6, 0, 0, 0, 0, 0, 0, 6, 0, 7, 2, 1, 4, 8, 1], [0, 0, -2, 4, 6, -2, 4, -4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 6, -3, 0, -3, -5, -3], [-10, 0, 3, -3, -5, 3, 0, 5, 0, -1, 0, 0, 0, 0, 0, 0], [-8, 0, 2, -4, 0, -2, -1, 0, 0, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 12, 3, -4, -5, 3, -5], [0, -4, 0, 0, 0, 0, 0, 0, 3, 0, 9, 3, 0, -6, 0, -2], [0, 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0, 0, 0, -8, 0, 0, 2, 2, 2, -2, -6], [-2, 0, 3, 0, 0, 5, 0, -5, 0, 1, 0, 0, 0, 0, 0, 0], [-2, 0, -1, 9, 11, -7, -3, 7, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, -6, 0, -16, 0, 3, 3, -5, 1], [0, 2, 0, 0, 0, 0, 0, 0, -3, 0, 4, -5, -4, 1, -13, 1], [0, 0, -2, 0, 16, -6, 5, -2, 0, 6, 0, 0, 0, 0, 0, 0], [-38, 0, 3, 3, 1, 3, 5, -7, 0, 4, 0, 0, 0, 0, 0, 0], [-14, 0, -3, 6, 6, 1, 2, 0, 0, -2, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 0, 9, 0, 4, 3, -2, -3, 19, 2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, -2, -6, -2, 2, -4], [-16, 0, 1, -8, 2, -3, 2, -2, 0, -4, 0, 0, 0, 0, 0, 0], [0, -6, 0, 0, 0, 0, 0, 0, 2, 0, -24, 2, -4, 2, -8, -2], [1, 0, -3, 9, 2, 3, 3, 2, 0, -1, 0, 0, 0, 0, 0, 0], [0, -8, 0, 0, 0, 0, 0, 0, -2, 0, 20, 6, 2, -6, -5, -5], [0, 13, 0, 0, 0, 0, 0, 0, -1, 0, 10, 2, 3, 0, 11, 0], [-2, 0, 0, 8, 0, 6, -12, 8, 0, 0, 0, 0, 0, 0, 0, 0], [-19, 0, -5, 5, 2, -5, 7, -4, 0, -1, 0, 0, 0, 0, 0, 0], [-2, 0, -4, 0, 0, 0, 6, -10, 0, 4, 0, 0, 0, 0, 0, 0], [-15, 0, 0, -1, 6, -10, 2, 1, 0, 3, 0, 0, 0, 0, 0, 0], [16, 0, -3, 3, 1, 3, 0, 3, 0, 1, 0, 0, 0, 0, 0, 0], [-8, 0, -2, -1, -3, 3, -2, -2, 0, 4, 0, 0, 0, 0, 0, 0], [0, 3, 0, 0, 0, 0, 0, 0, 1, 0, -29, -1, 0, 6, 8, 4], [-2, 0, 3, -7, 3, -1, -1, -3, 0, 2, 0, 0, 0, 0, 0, 0], [-28, 0, -2, 3, -5, 6, -6, -2, 0, 1, 0, 0, 0, 0, 0, 0], [0, 3, 0, 0, 0, 0, 0, 0, 4, 0, -19, -5, 1, 8, 6, 3], [0, 8, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 6, 8, 4, 4], [1, 0, 2, -6, -1, -2, -6, 2, 0, 4, 0, 0, 0, 0, 0, 0], [-2, 0, -3, 3, 17, -7, 5, -1, 0, 2, 0, 0, 0, 0, 0, 0], [-8, 0, 2, 3, -7, 9, -4, 0, 0, 2, 0, 0, 0, 0, 0, 0], [-13, 0, 2, 3, 4, 0, -12, 1, 0, 1, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, 0, 0, 0, 0, 2, 0, -5, 2, 1, 8, 6, 1], [12, 0, -5, 3, 3, 7, 1, -3, 0, 0, 0, 0, 0, 0, 0, 0], [3, 0, 2, -5, 2, -8, 10, -8, 0, -6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -5, 0, -12, -5, -4, 1, 5, -9], [0, 3, 0, 0, 0, 0, 0, 0, 2, 0, -6, 6, 5, 3, 5, 8], [8, 0, -2, -1, -5, 1, -10, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 3, 0, 0, 0, 0, 0, 0, -7, 0, -4, -4, 1, -10, -5, 0], [0, 0, 2, 8, 8, 2, -1, 0, 0, 4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 14, -2, -4, -8, -10, -6], [30, 0, -8, 3, -3, 0, -2, 4, 0, -1, 0, 0, 0, 0, 0, 0], [-32, 0, -6, 6, -6, 6, -2, -6, 0, 6, 0, 0, 0, 0, 0, 0], [0, 4, 0, 0, 0, 0, 0, 0, -8, 0, -12, -8, -2, -2, 4, -8], [0, 6, 0, 0, 0, 0, 0, 0, 1, 0, -4, -3, -2, -3, 5, 5], [0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 16, -6, 2, 2, -7, 1], [0, 3, 0, 0, 0, 0, 0, 0, 1, 0, 17, 3, 4, 0, 12, -4], [0, -8, 0, 0, 0, 0, 0, 0, 4, 0, 8, -2, 4, -10, -14, -4], [0, -4, 0, 0, 0, 0, 0, 0, 3, 0, 7, 5, 4, 2, -3, 9], [10, 0, 0, 0, -4, -4, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, 0, -18, 2, -2, 10, -2, 0], [-8, 0, 1, -2, 8, -7, 2, -4, 0, 2, 0, 0, 0, 0, 0, 0], [12, 0, -1, -1, -1, 7, -9, 9, 0, -2, 0, 0, 0, 0, 0, 0], [-12, 0, 0, -2, 2, -4, 5, 0, 0, -6, 0, 0, 0, 0, 0, 0], [7, 0, -6, -1, -2, -4, -2, -4, 0, -2, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 0, 4, 0, -10, 0, -4, 4, -10, 4], [4, 0, -2, 3, 11, -10, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0], [13, 0, -1, -3, -6, -7, -7, 2, 0, -3, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 32, -8, 0, -4, -5, -3], [0, -6, 0, 0, 0, 0, 0, 0, 2, 0, 4, 6, 2, -4, 0, -2], [0, -9, 0, 0, 0, 0, 0, 0, -3, 0, -7, 3, 0, 4, 0, -4], [0, -11, 0, 0, 0, 0, 0, 0, -4, 0, -12, -2, 0, -4, -17, -4], [4, 0, 5, -3, -7, 17, -13, 5, 0, -4, 0, 0, 0, 0, 0, 0], [0, 3, 0, 0, 0, 0, 0, 0, -8, 0, -28, 0, 1, 11, -5, 12], [32, 0, 5, -3, 5, -7, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0], [19, 0, -4, 3, 6, -12, 8, -6, 0, -2, 0, 0, 0, 0, 0, 0], [0, -3, 0, 0, 0, 0, 0, 0, -4, 0, 10, 4, 3, -9, -3, 4], [0, 4, 0, 0, 0, 0, 0, 0, 4, 0, 14, 6, 8, 2, 14, 0], [-24, 0, 0, 6, 14, -4, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 4, -3, -14, 8, -6, 4, 0, 2, 0, 0, 0, 0, 0, 0], [0, 6, 0, 0, 0, 0, 0, 0, 14, 0, -8, -2, -2, 8, 10, 4], [0, 1, 0, 0, 0, 0, 0, 0, -8, 0, -17, -7, -1, 2, -10, -9], [0, -7, 0, 0, 0, 0, 0, 0, -6, 0, 9, 1, -1, -4, -2, -7], [0, 6, 0, 0, 0, 0, 0, 0, -6, 0, -12, -8, -2, 4, -8, 0], [6, 0, 7, 0, -10, 5, -4, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 7, 0, 0, 0, 0, 0, 0, 7, 0, 50, 6, -1, 4, 17, 6], [0, 8, 0, 0, 0, 0, 0, 0, -4, 0, -22, 6, -2, 10, 7, 3], [0, -6, 0, 0, 0, 0, 0, 0, -7, 0, -13, -1, -2, -4, -10, -10], [0, -5, 0, 0, 0, 0, 0, 0, 5, 0, 19, 1, -4, 2, 2, -10], [2, 0, 1, -2, -2, -5, 6, 5, 0, -1, 0, 0, 0, 0, 0, 0], [-26, 0, -2, 5, -1, -2, 2, 6, 0, -5, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 6, 0, -15, -2, -1, 0, 6, 3], [2, 0, -2, -6, -2, -6, 6, -2, 0, 4, 0, 0, 0, 0, 0, 0], [0, 6, 0, 0, 0, 0, 0, 0, -6, 0, -26, 2, 8, 2, 2, 0], [0, -6, 0, 0, 0, 0, 0, 0, 7, 0, 10, -9, -5, -4, 1, -7], [0, -1, 0, 0, 0, 0, 0, 0, 3, 0, -26, -2, -5, -4, 5, -6], [-12, 0, 0, 5, -7, 4, 6, -10, 0, 1, 0, 0, 0, 0, 0, 0], [0, -6, 0, 0, 0, 0, 0, 0, 1, 0, 12, -5, 6, 1, -11, -1], [-8, 0, 2, 10, 10, -6, 8, -6, 0, -2, 0, 0, 0, 0, 0, 0], [0, -3, 0, 0, 0, 0, 0, 0, -4, 0, -26, 2, -2, 4, -1, -2], [-2, 0, -1, -5, -1, 0, -2, -5, 0, -5, 0, 0, 0, 0, 0, 0], [-7, 0, 5, -3, -2, -3, 7, 2, 0, 1, 0, 0, 0, 0, 0, 0], [0, -13, 0, 0, 0, 0, 0, 0, 4, 0, -4, 3, -1, -13, 1, 5], [0, -3, 0, 0, 0, 0, 0, 0, -4, 0, 16, -8, -1, 1, -5, 6], [-32, 0, -1, -3, -3, -5, -3, -5, 0, 6, 0, 0, 0, 0, 0, 0], [0, -8, 0, 0, 0, 0, 0, 0, 4, 0, 2, 0, -8, -6, -6, -6], [0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 10, 0, -6, -10, 0, -4], [0, -2, 0, 0, 0, 0, 0, 0, -4, 0, 4, 2, 2, 0, 0, -2], [4, 0, -3, 2, 10, -11, 4, -7, 0, 3, 0, 0, 0, 0, 0, 0], [43, 0, -2, -1, -4, 0, 8, -3, 0, 1, 0, 0, 0, 0, 0, 0], [-2, 0, -2, -6, 2, -8, 14, 0, 0, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 22, -2, 4, 6, -2, 4], [0, 0, -2, 0, 14, -14, 8, -4, 0, 0, 0, 0, 0, 0, 0, 0], [10, 0, 4, -9, 1, 4, -2, 14, 0, 3, 0, 0, 0, 0, 0, 0], [0, 4, 0, 0, 0, 0, 0, 0, -3, 0, -22, -1, 5, 0, -9, 7], [0, -12, 0, 0, 0, 0, 0, 0, -3, 0, 15, -11, -8, -14, -8, -14], [-18, 0, -4, 9, 5, -5, -8, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, 0, 0, 0, 0, -1, 0, 20, -1, 0, -3, 9, 7], [12, 0, -10, -2, 6, -6, 2, -6, 0, -2, 0, 0, 0, 0, 0, 0], [0, 5, 0, 0, 0, 0, 0, 0, 6, 0, 26, -4, 4, 8, -3, 0], [0, -7, 0, 0, 0, 0, 0, 0, 2, 0, -2, 3, 1, -15, -15, -11], [-5, 0, 4, -3, -8, 0, 4, -4, 0, -8, 0, 0, 0, 0, 0, 0], [-10, 0, -6, 14, 4, -7, 3, -7, 0, 7, 0, 0, 0, 0, 0, 0], [31, 0, 2, 3, 4, -8, 6, 3, 0, 3, 0, 0, 0, 0, 0, 0], [0, 11, 0, 0, 0, 0, 0, 0, 1, 0, -8, 4, -3, 2, 11, -8], [10, 0, 9, 3, 1, -1, -5, 9, 0, 4, 0, 0, 0, 0, 0, 0], [-24, 0, 2, -4, -4, -2, 3, -6, 0, -2, 0, 0, 0, 0, 0, 0], [-16, 0, -4, 2, -14, 4, 4, -4, 0, 4, 0, 0, 0, 0, 0, 0], [13, 0, -9, 7, 16, -5, 1, -4, 0, 1, 0, 0, 0, 0, 0, 0], [0, 10, 0, 0, 0, 0, 0, 0, -8, 0, 10, 4, 2, 4, -4, 0], [0, -8, 0, 0, 0, 0, 0, 0, 6, 0, -15, 0, -5, -4, 0, -3], [0, 0, -2, -2, -6, -3, -5, -7, 0, 5, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, 0, 0, 0, 0, 2, 0, -2, 2, -8, -12, 4, -6], [0, 0, 4, -4, 4, 0, -2, 6, 0, 6, 0, 0, 0, 0, 0, 0], [0, 12, 0, 0, 0, 0, 0, 0, 10, 0, -8, 0, 2, 6, 16, 6], [-2, 0, -4, -8, -8, 4, -4, 4, 0, 0, 0, 0, 0, 0, 0, 0], [3, 0, -7, 5, 4, -15, 9, 2, 0, 3, 0, 0, 0, 0, 0, 0], [-25, 0, 0, 5, -2, -2, 4, -3, 0, 3, 0, 0, 0, 0, 0, 0], [0, -6, 0, 0, 0, 0, 0, 0, -8, 0, -24, 2, 2, -6, -8, 0], [0, -10, 0, 0, 0, 0, 0, 0, -2, 0, -20, -2, 0, -12, -13, -3], [0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 10, 3, -1, -7, 5, 9], [0, 0, 0, 0, 0, 0, 0, 0, -14, 0, -2, -2, 4, -4, 4, 2], [0, 10, 0, 0, 0, 0, 0, 0, 0, 0, -18, -4, 4, 4, 6, -4], [12, 0, 9, -13, -1, 5, -7, 9, 0, -10, 0, 0, 0, 0, 0, 0], [-1, 0, -8, 11, -2, 6, 2, 6, 0, 4, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, 0, 0, 0, 0, 8, 0, 10, -6, -2, -10, 11, -11], [-32, 0, -6, 6, 6, -6, 2, -6, 0, 6, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 5, 0, 3, 1, 6, -4, -8, -4], [0, -6, 0, 0, 0, 0, 0, 0, 0, 0, -8, 0, 6, 12, -10, 4], [10, 0, 2, 4, 8, -2, -4, -2, 0, -2, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, 0, 0, 0, 0, 12, 0, 22, 2, -4, -10, -2, -4], [0, 6, 0, 0, 0, 0, 0, 0, 2, 0, 14, 2, 6, 4, 16, 2], [0, -2, 0, 0, 0, 0, 0, 0, -6, 0, -8, -6, -2, 0, -16, -2], [-29, 0, -1, 3, 2, -9, -5, -4, 0, 7, 0, 0, 0, 0, 0, 0], [34, 0, -1, 3, -7, 11, -5, 5, 0, 0, 0, 0, 0, 0, 0, 0], [0, 16, 0, 0, 0, 0, 0, 0, 8, 0, 30, 4, 4, 6, 16, 8], [0, -7, 0, 0, 0, 0, 0, 0, -9, 0, -11, -4, 5, 5, -12, 4], [31, 0, -8, -1, 2, -12, 10, -4, 0, -2, 0, 0, 0, 0, 0, 0], [-27, 0, -2, 12, 13, -14, 6, -6, 0, 2, 0, 0, 0, 0, 0, 0], [-18, 0, 3, -4, 0, 7, 0, -6, 0, 0, 0, 0, 0, 0, 0, 0], [0, -5, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 5, -12, -11, -4], [48, 0, -3, 1, 5, -5, 5, -1, 0, 8, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, 0, 0, 0, 0, -9, 0, -31, -11, 0, -6, -12, -12], [0, -8, 0, 0, 0, 0, 0, 0, -1, 0, -4, -3, -4, -3, -3, -7], [-12, 0, 2, -14, -2, -2, -3, -6, 0, 0, 0, 0, 0, 0, 0, 0], [0, 3, 0, 0, 0, 0, 0, 0, -9, 0, -37, 1, 4, 0, -6, 4], [-12, 0, -3, -1, 7, 5, -3, -5, 0, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -12, 0, -18, 0, 0, -4, 4, 0], [-4, 0, 3, -1, -1, -9, -3, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-4, 0, 2, 0, -8, -3, -3, 1, 0, -5, 0, 0, 0, 0, 0, 0], [-8, 0, 3, 1, -9, 6, -8, 1, 0, -3, 0, 0, 0, 0, 0, 0], [-16, 0, -3, 11, -5, -3, -1, -5, 0, 0, 0, 0, 0, 0, 0, 0], [-33, 0, -2, 2, 5, 6, -2, 12, 0, 4, 0, 0, 0, 0, 0, 0], [0, -5, 0, 0, 0, 0, 0, 0, 4, 0, 10, -4, -6, -12, 3, -6], [10, 0, -2, 8, -2, -9, -1, -9, 0, 3, 0, 0, 0, 0, 0, 0], [0, 13, 0, 0, 0, 0, 0, 0, 6, 0, 16, 3, 5, 7, 17, 5], [0, -4, 0, 0, 0, 0, 0, 0, 0, 0, -17, 0, -1, -4, -16, -5], [-20, 0, 4, 8, 18, 0, -4, 10, 0, 2, 0, 0, 0, 0, 0, 0], [-44, 0, 2, -3, 1, -2, 2, 1, 0, 1, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, -2, 0, 8, 8, 5, 9, 5, 1], [29, 0, 4, 3, 8, -6, -2, -4, 0, 6, 0, 0, 0, 0, 0, 0], [17, 0, 9, -9, 12, 3, -3, 10, 0, -7, 0, 0, 0, 0, 0, 0], [38, 0, 4, -8, -8, 10, -2, 4, 0, -8, 0, 0, 0, 0, 0, 0], [-14, 0, 3, -3, 3, -1, -1, 9, 0, 2, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, -3, 0, -1, 3, 0, -10, -10, -4], [14, 0, 6, 0, 12, -2, 4, 6, 0, -6, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 18, -2, -5, -1, -11, -5], [0, -2, 0, 0, 0, 0, 0, 0, 5, 0, 18, 3, 0, 11, 15, 1], [0, 1, 0, 0, 0, 0, 0, 0, -4, 0, 4, -2, 0, 0, 11, -8], [-8, 0, 2, 6, 10, -6, 6, -10, 0, 10, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, -4, 0, 15, 9, -1, 4, 8, -3], [0, 2, 0, 0, 0, 0, 0, 0, -5, 0, 12, -5, -2, 7, -5, 0], [0, 5, 0, 0, 0, 0, 0, 0, -10, 0, -32, -2, 3, 3, -5, 2], [0, 3, 0, 0, 0, 0, 0, 0, 8, 0, -14, 2, -3, 5, -1, -5], [-13, 0, 6, -3, 0, -4, 0, 0, 0, -8, 0, 0, 0, 0, 0, 0], [46, 0, -3, 1, -5, 5, -5, 11, 0, -4, 0, 0, 0, 0, 0, 0], [6, 0, 12, -12, 0, 6, -12, 12, 0, -8, 0, 0, 0, 0, 0, 0], [-4, 0, 6, 8, 12, 11, 3, 9, 0, 1, 0, 0, 0, 0, 0, 0], [0, -6, 0, 0, 0, 0, 0, 0, 3, 0, 3, -3, -10, -8, -5, -9], [0, 22, 0, 0, 0, 0, 0, 0, 0, 0, -2, 8, 4, 8, 3, 9]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5520_be_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5520_2_be_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_5520_2_be_a(:prec:=16)
    chi := MakeCharacter_5520_be();
    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_5520_2_be_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_5520_2_be_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5520_be();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![36, -164, 37, 316, 39, -108, -25, 4, 1]>],Snew);
    return Vf;
end function;