
// Make newform 1840.2.e.h in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1840_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1840_e_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_1840_2_e_h();".
// 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_1840_2_e_h();".
// 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 := [1024, -512, 128, 4336, 15457, 9568, 2464, -1594, 1572, 702, 210, -196, 100, 6, 2, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [953326853515704207873024, 556650317363414170826464, -304907534037070840065976, -2028057361772854751575673, 3918219876554594510372320, -84234101814344138888912, 169048474594930667975626, -537061025633670808973620, 523491302527906903267042, -135549389895600500067810, 22923942005700239322180, -5798966345385152663476, 5615701405841246795546, -1533252977882876294850, 241867657278618280682, -564796151969688089], [-4340717231201691735808, 69896317461757032781472, 493638074924085942890092, 267645466920360925996151, 73974887245017554157920, -56255878203811954909816, 49351663880097415453818, 18861957671154867372740, 6577290238793402720266, -6371259042835319510930, 3167278369797318418580, 101144014888243416932, 63600734450499465858, -64646323960557883050, 31617792777529048366, -669081183175634857], [-4340717231201691735808, 101771277845332840877952, 461763114540510134793612, 267645466920360925996151, 73974887245017554157920, -56255878203811954909816, 49351663880097415453818, 18861957671154867372740, 6577290238793402720266, -6371259042835319510930, 3167278369797318418580, 101144014888243416932, 63600734450499465858, -64646323960557883050, 31617792777529048366, -669081183175634857], [-93942894839672333928576, -3335136930759889767274816, -2575592145277412776789596, -1232478353488625770263843, 255183413268314405913440, -52052886724240319553832, -9697761543566197765634, -114373918633476906704980, 2054109647928134982382, 5800880410569448661690, 4424452632901492664220, -4150958833832665209716, 80508305370487005526, 98141784974047919250, 51817197715454225402, -36082429475973024899], [-140707114255972724633344, 1194224413274039153395616, 2635056706542329997920176, 4286478475719333680330703, 115121855831410995632480, -2054926237083438692768608, 952700810705768643736474, 458943318236379704058780, 52212436728277648932258, -316526821134642417546610, 167555889184928866487620, -23161774996655402727844, 726871944630147495194, -3285807383750098948290, 1841229998511317961538, -318969229217745937361], [-1299074755592200570567680, 1227514889489791068838304, -653589136267567895383144, -4552141809370651565356611, -3264034951805109084713312, -519643278737243550826480, 568205842884924476879182, -501938626773392038340220, -239291997595419620454186, -53565452553247896896982, 63267592666342922762540, -32676821886335924461052, -2037151664803358920002, -508649117447187324278, 639944811506344667566, -328086612481608009763], [-107593881030981543276544, -3939967276533955427877024, 1549991709567331298025256, 10856809983417508211138923, 9891141608439985398920800, 1350815537979474984914032, -1382131543294845713179486, 1037048304461043182970780, 854491089510550619162458, 76879314495638174174630, -152213113407290106228940, 81160127013367967422236, 7835326530712091726514, 614001498766608317510, -1535375885944134270142, 841925339302843628619], [-23061364425557027753728, -733530723301828313458784, -606086541272980490786928, -2205874733214161441488273, -1555862406023457302487712, -773356052025354301260832, 359794110659287610999514, -239794915024620709995940, -105935593052591979557342, -75638304146095329018290, 50823127396663014338116, -20011902708376888146916, -858852135854606012262, -748886722375901601090, 531602863276948713442, -207962376041506841905], [4455281824367152369273856, -7551143469775208630023904, 4011928424236009986551736, 28059114061089377308872313, 25958054272212434904396320, 3423207863830156165574352, -3540845704199600616790986, 2808490629040326766600500, 2055560822793859187884318, 242799571396014564554210, -391518309605053145121860, 206585678634484606502836, 18550348604166644455654, 2132038220718324369090, -3953155420784877654762, 2118185236141746471449], [-20422553595681348505600, -591486967727086100610080, 421969477678591882074120, 2965824734178438435235855, 2423008249317927822153440, 398765757204922242534320, -363797928128516647631366, 284665898450793362180972, 202760223167881268421202, 26651039266139952258734, -40411825872447519151324, 21089846935305423863020, 1835948719271709389514, 230984654232606903502, -408368235739557435366, 216832476518453171567], [-120787935180359939517184, -2610188589858814773945824, -3590288385190530566279984, -15001687353545491172481717, -10003211668266940224284960, -4509793508171907734061728, 2235828650644175579974914, -1585255762916027177913780, -688987991591815620730262, -434567402565538232210490, 303961652938262561218260, -124348621205043592627444, -5638677744919485282366, -4285355384275934205450, 3164479609138396473898, -1282048923496956438741], [-3181384674636323243527168, 8597925159288225369863712, -4568188273249911843048328, -31880221946092013629388959, -29807609026592396802595040, -3641530117296416951306416, 4187584486409972800336038, -3075848072628037146480300, -2618657823981625025068594, -184409721653063663205070, 458174412566717236259100, -247870842045704795997228, -23832307778719830030122, -1300938577750539720430, 4615436144052771597446, -2588229413723518163967], [-448532962164852865847552, -4246082060596735800332192, 10409412107504918728814928, -18490637950705990608063631, -17183680768812435879037280, -12018735453826353009247904, 6950964135040340249863142, -2503967682918694073411740, -1117119434130211646290466, -1161038922935840085830670, 762176261837051384990140, -250389486494265538272092, -8707977648300977704858, -11457146342780235799230, 7849910266256606593854, -2642888945664910864623], [-150605332666768364582752, 50175033137633848711128, 1058982307105449261342133, 3248825928763831075053889, 1984719931099756217426160, 478381537503435627193246, -323581731844098202246038, 318883474884996061259370, 139489491216171912857464, 40301528362365775227230, -38258112235964529343520, 19224867526049022000338, 1201904523584497153952, 382730328923720987360, -390372336275624352541, 189621980412060243137], [-495178165772403862599424, -5841096957539534050870624, 4428663247793606577665776, -25100415896991282501011497, -21446083076286990646032800, -13689050286567661049021408, 6701984866603385326334154, -3059708196081617915726020, -1409871110534423896856302, -1367315172251677890979810, 860187450071408174746660, -303190775204524701949444, -11093928427081377174326, -13659830033743327363890, 8944116696865206225138, -3214385228459778752201]];
    Rf_basisdens := [1, 2039997464548851718174720, 127499841534303232385920, 31874960383575808096480, 509999366137212929543680, 2039997464548851718174720, 407999492909770343634944, 1019998732274425859087360, 203999746454885171817472, 2039997464548851718174720, 203999746454885171817472, 1019998732274425859087360, 2039997464548851718174720, 2039997464548851718174720, 127499841534303232385920, 2039997464548851718174720];
    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_1840_e();"
function MakeCharacter_1840_e()
    N := 1840;
    order := 2;
    char_gens := [1151, 1381, 737, 1201];
    v := [2, 2, 1, 2];
    // 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_1840_e_Hecke();"
function MakeCharacter_1840_e_Hecke(Kf)
    N := 1840;
    order := 2;
    char_gens := [1151, 1381, 737, 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]];
    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, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [-1, 0, 2, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, -1, -1, 1], [0, 0, 1, -1, 0, 1, 0, -1, -1, 0, 1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 1, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, -1, 0, 0, 0, 0, 1, 0, -1, 0, 0, 1, 0, 0, 0, 0], [-2, -1, 0, 0, 0, 0, 0, 0, -1, 1, 0, 1, -1, 0, 0, 0], [-1, 0, 0, 1, 1, -1, 0, -1, 0, 0, 1, -1, 0, 0, -1, 1], [1, -1, 0, 0, -1, 0, 2, 1, 0, 0, 1, 0, 0, 0, -1, 0], [0, 0, 2, -1, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1], [-1, 0, 1, 1, 1, 0, 0, 1, 0, 0, -1, -1, 0, 0, -1, 0], [1, 0, 0, 0, -1, 2, 0, 0, 2, 0, 0, 2, 0, 0, 1, -2], [4, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, -1, 0, 0, 0, 0], [2, 0, 0, 0, -1, 0, 0, 1, 0, 0, 1, 0, 1, 0, -1, 0], [1, 0, 0, 1, -1, 1, 0, 1, 2, 0, -1, 1, 0, 0, 1, -1], [-3, -3, 0, 0, -1, 0, 0, 2, 1, 0, 2, -1, 2, 0, -1, 0], [1, 0, 1, 1, -1, 2, 0, 2, 1, 0, -2, 0, 0, 0, 1, 0], [1, 4, 0, 0, -1, 0, -1, -1, 0, 2, -1, 0, 1, 0, -1, 0], [0, 0, 4, 2, 0, -2, 0, 1, 2, 0, -1, 0, 0, 0, 0, 0], [-5, 0, 0, 0, -1, 0, 1, 1, 2, 0, 1, -2, 1, 0, -1, 0], [-1, 0, 1, 0, 1, 0, 0, 1, -1, 0, -1, -1, 0, 0, -1, 1], [2, 0, 0, 0, 0, 0, 0, -2, 0, 2, -2, 0, -2, 0, 0, 0], [0, 0, 1, -3, 0, 1, 0, -1, -1, 0, 1, 2, 0, 0, 0, 0], [-2, 0, -4, 1, 2, -5, 0, 0, 0, 0, 0, -1, 0, -2, -2, 3], [-3, 0, 0, 0, 2, 0, -2, 0, -1, -1, 0, 1, 0, 0, 2, 0], [-2, 0, 6, -1, 2, 1, 0, -1, -4, 0, 1, -3, 0, 2, -2, 3], [-1, 0, -3, 1, 1, -4, 0, -4, 1, 0, 4, 0, 0, -2, -1, -2], [-5, 1, 0, 0, -2, 0, 1, 2, -1, 0, 2, 1, 2, 0, -2, 0], [0, 0, -1, -1, 0, 1, 0, -1, 0, 0, 1, 1, 0, -1, 0, -3], [1, -3, 0, 0, 0, 0, -1, -2, 1, 0, -2, -1, 0, 0, 0, 0], [0, 0, 0, 0, 3, 0, -1, -1, -1, -3, -1, 1, -1, 0, 3, 0], [-3, -1, 0, 0, -1, 0, -2, -1, -2, 0, -1, 2, 2, 0, -1, 0], [0, 0, -2, -3, 0, 3, 0, 2, 0, 0, -2, 3, 0, -2, 0, -1], [1, 0, 0, 2, -1, 1, 0, -3, 1, 0, 3, -1, 0, 3, 1, -1], [0, 0, 6, -2, 0, 0, 0, -1, -1, 0, 1, 1, 0, 0, 0, -2], [-1, 0, 5, 1, 1, -5, 0, -1, 4, 0, 1, 3, 0, -2, -1, -2], [9, 5, 0, 0, -2, 0, 1, 1, 0, 2, 1, 0, 2, 0, -2, 0], [0, 0, 0, 0, -1, 0, 0, 1, -1, 1, 1, 1, 2, 0, -1, 0], [-7, 0, 0, 0, 1, 0, -4, -3, 2, 2, -3, -2, -2, 0, 1, 0], [1, 0, -1, -1, -1, 2, 0, 2, -3, 0, -2, -2, 0, 2, 1, 0], [2, 0, -2, -2, -2, 3, 0, 2, 0, 0, -2, 2, 0, 0, 2, -2], [4, 6, 0, 0, -2, 0, -1, 0, -2, 0, 0, 2, 3, 0, -2, 0], [-2, 2, 0, 0, -2, 0, 0, -2, -2, 2, -2, 2, 2, 0, -2, 0], [-2, 0, 6, 4, 2, 2, 0, 0, -2, 0, 0, -6, 0, 2, -2, 0], [-1, 0, 2, 1, 1, -5, 0, -3, 2, 0, 3, 1, 0, -2, -1, -1], [-10, -6, 0, 0, 0, 0, 1, 1, -4, 0, 1, 4, -3, 0, 0, 0], [-1, 0, -1, 1, 1, 4, 0, 0, -3, 0, 0, -4, 0, 4, -1, 4], [9, -1, 0, 0, -2, 0, 3, 1, -2, 2, 1, 2, 2, 0, -2, 0], [5, -6, 0, 0, 1, 0, 0, 3, 0, 0, 3, 0, 2, 0, 1, 0], [-7, -2, 0, 0, 2, 0, 1, 0, 1, -3, 0, -1, 1, 0, 2, 0], [-1, 0, 3, 3, 1, -2, 0, -3, -2, 0, 3, -5, 0, 0, -1, 0], [-2, 0, 7, 0, 2, 2, 0, -2, -5, 0, 2, -5, 0, -2, -2, 1], [1, 3, 0, 0, -2, 0, 3, 3, 0, -2, 3, 0, 2, 0, -2, 0], [-1, -6, 0, 0, 2, 0, 3, 1, 1, -1, 1, -1, -3, 0, 2, 0], [1, 0, -15, 1, -1, -2, 0, -1, 4, 0, 1, 3, 0, 2, 1, -2], [0, 2, 0, 0, -2, 0, -3, 2, 2, 2, 2, -2, 1, 0, -2, 0], [0, 0, 4, 2, 0, 4, 0, 1, 0, 0, -1, -2, 0, 2, 0, 2], [-1, 0, 8, 0, 1, -4, 0, 2, -6, 0, -2, -6, 0, 2, -1, 4], [-1, 0, 1, 3, 1, -7, 0, 1, 4, 0, -1, 1, 0, 2, -1, -2], [-5, 1, 0, 0, 2, 0, -1, -5, 2, 6, -5, -2, -4, 0, 2, 0], [2, 0, -1, 1, -2, -1, 0, 1, -2, 0, -1, -3, 0, 3, 2, -1], [-2, 0, 3, -1, 2, 3, 0, 0, -3, 0, 0, -2, 0, -3, -2, -1], [5, 5, 0, 0, 0, 0, -3, -2, 3, -2, -2, -3, -4, 0, 0, 0], [1, 0, 5, -2, -1, 6, 0, 3, -6, 0, -3, -4, 0, 3, 1, 2], [1, 0, -7, 1, -1, 1, 0, 0, -3, 0, 0, -4, 0, 4, 1, -2], [-7, 1, 0, 0, -2, 0, 1, -3, 0, 4, -3, 0, -2, 0, -2, 0], [5, 0, -3, -1, -5, 2, 0, 0, -1, 0, 0, 0, 0, 4, 5, -4], [1, 4, 0, 0, 1, 0, -7, -1, 4, -2, -1, -4, 1, 0, 1, 0], [5, 0, -4, -2, -5, 4, 0, -1, 0, 0, 1, 2, 0, 2, 5, -2], [-2, 0, 12, -4, 2, -4, 0, -1, 0, 0, 1, 4, 0, -4, -2, 0], [-5, -2, 0, 0, -2, 0, -1, -2, 0, 8, -2, 0, -2, 0, -2, 0], [1, 0, 10, -1, -1, 5, 0, 4, -4, 0, -4, -3, 0, 4, 1, 5], [-3, 4, 0, 0, -2, 0, -1, -4, 0, 8, -4, 0, 2, 0, -2, 0], [-2, 0, -10, -2, 2, -1, 0, -3, -1, 0, 3, 1, 0, -2, -2, 4], [5, 6, 0, 0, 1, 0, -2, -7, 0, 6, -7, 0, -2, 0, 1, 0], [6, -9, 0, 0, 2, 0, 2, 2, -3, 3, 2, 3, -3, 0, 2, 0], [4, -12, 0, 0, 2, 0, 1, 3, 4, -4, 3, -4, -5, 0, 2, 0], [0, 2, 0, 0, 1, 0, 0, -3, 1, 1, -3, -1, -8, 0, 1, 0], [-7, 2, 0, 0, 1, 0, -1, 3, -2, 4, 3, 2, 1, 0, 1, 0], [3, 0, 3, -1, -3, 9, 0, 0, -2, 0, 0, -1, 0, 3, 3, -1], [2, 9, 0, 0, -2, 0, -2, -1, 4, 3, -1, -4, -1, 0, -2, 0], [1, 0, 0, 0, -1, 5, 0, 0, -6, 0, 0, -6, 0, 3, 1, 1], [-5, -2, 0, 0, -2, 0, 3, 4, -2, 1, 4, 2, -1, 0, -2, 0], [1, 0, -8, 0, -1, 4, 0, -5, -6, 0, 5, -6, 0, 2, 1, -2], [7, 3, 0, 0, 0, 0, 3, -4, 1, 0, -4, -1, -4, 0, 0, 0], [6, 0, 0, -2, -6, 10, 0, -4, 0, 0, 4, 2, 0, 4, 6, -8], [3, 0, -2, -2, -3, 6, 0, 2, 0, 0, -2, 2, 0, 2, 3, -6], [3, -2, 0, 0, 1, 0, 4, 1, 6, -2, 1, 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[0, 0, -22, 1, 0, -11, 0, 3, -2, 0, -3, -3, 0, -2, 0, -9], [22, 14, 0, 0, -4, 0, 0, -2, 2, 10, -2, -2, 6, 0, -4, 0], [1, 0, -6, 3, -1, -3, 0, 3, -2, 0, -3, -5, 0, -10, 1, -1], [25, 9, 0, 0, 0, 0, 3, -3, 0, -8, -3, 0, 4, 0, 0, 0], [-3, 1, 0, 0, 1, 0, 2, 4, 3, 0, 4, -3, -6, 0, 1, 0], [5, 0, -4, -4, -5, 6, 0, 6, 2, 0, -6, 6, 0, 10, 5, -2], [-5, 1, 0, 0, -8, 0, 1, 3, 0, 10, 3, 0, 6, 0, -8, 0], [1, 0, -3, 1, -1, 0, 0, 5, 0, 0, -5, -1, 0, -6, 1, -4], [-21, -21, 0, 0, 6, 0, 1, 1, -2, -6, 1, 2, 2, 0, 6, 0], [2, 0, -20, -6, -2, 0, 0, -3, 1, 0, 3, 7, 0, 0, 2, 6], [-16, -11, 0, 0, 4, 0, 2, 6, -9, -3, 6, 9, -1, 0, 4, 0], [3, 0, -7, -5, -3, 6, 0, -1, -4, 0, 1, 1, 0, 0, 3, -6], [4, -2, 0, 0, 3, 0, 3, -3, -5, 7, -3, 5, -9, 0, 3, 0], [3, 12, 0, 0, 0, 0, 7, -6, 0, 0, -6, 0, -8, 0, 0, 0], [-2, 0, -15, -1, 2, 1, 0, -7, -7, 0, 7, -6, 0, 0, -2, 2], [-4, 0, 22, 12, 4, 2, 0, 5, 2, 0, -5, -10, 0, -2, -4, 4], [31, -6, 0, 0, -3, 0, 0, 3, 6, -2, 3, -6, -2, 0, -3, 0], [0, 0, 21, -5, 0, 9, 0, -7, -8, 0, 7, -3, 0, 1, 0, 3], [-12, -6, 0, 0, -4, 0, 5, 0, 2, 4, 0, -2, 5, 0, -4, 0], [4, 0, -14, -2, -4, -2, 0, 7, 5, 0, -7, 7, 0, -2, 4, -4], [3, -5, 0, 0, 2, 0, -5, -6, -7, 4, -6, 7, 0, 0, 2, 0], [-21, -22, 0, 0, -4, 0, 8, 12, 0, 0, 12, 0, 9, 0, -4, 0], [2, 0, -6, -2, -2, -4, 0, 5, 4, 0, -5, 6, 0, -2, 2, -4], [-5, 0, 24, 2, 5, 5, 0, 0, -12, 0, 0, -14, 0, 5, -5, 3], [-13, 0, 3, 5, 13, -10, 0, -5, -4, 0, 5, -9, 0, -8, -13, 8], [-1, 0, 23, 7, 1, 9, 0, 6, 1, 0, -6, -6, 0, 2, -1, -8], [-8, 0, 0, -3, 8, -11, 0, -6, -4, 0, 6, -1, 0, -6, -8, 11], [-4, -2, 0, 0, -2, 0, 0, 3, 7, 0, 3, -7, 0, 0, -2, 0], [-1, 0, 0, -5, 1, -5, 0, -3, 2, 0, 3, 7, 0, -8, -1, -5], [2, 0, -2, 2, -2, -8, 0, -2, 4, 0, 2, 2, 0, 0, 2, 4], [-2, 2, 0, 0, -1, 0, -4, -7, -6, 12, -7, 6, -9, 0, -1, 0], [-5, -2, 0, 0, 0, 0, 6, 14, 8, 0, 14, -8, -1, 0, 0, 0], [11, 4, 0, 0, -5, 0, -3, 3, 0, 0, 3, 0, 5, 0, -5, 0], [6, 0, -8, -3, -6, 23, 0, 4, -4, 0, -4, -1, 0, 8, 6, -7], [-1, 0, 16, -2, 1, 17, 0, 4, -4, 0, -4, -2, 0, -7, -1, -1], [3, 7, 0, 0, 2, 0, -3, -7, -10, 4, -7, 10, -8, 0, 2, 0], [-1, 0, 22, -2, 1, 18, 0, -4, -6, 0, 4, -4, 0, 4, -1, 8], [-1, 0, -27, -8, 1, 4, 0, -3, -12, 0, 3, -4, 0, 3, -1, 8], [0, -12, 0, 0, 6, 0, 7, 4, -4, 4, 4, 4, -13, 0, 6, 0], [5, 0, -27, -5, -5, -4, 0, -6, 3, 0, 6, 8, 0, -2, 5, -6], [3, 0, 7, -3, -3, -8, 0, 3, 10, 0, -3, 13, 0, -10, 3, -12], [3, 0, 33, -1, -3, 5, 0, -2, -3, 0, 2, -2, 0, 8, 3, 0], [-6, -8, 0, 0, -2, 0, -4, 10, 0, -2, 10, 0, 12, 0, -2, 0], [-3, 14, 0, 0, 1, 0, -6, -5, 4, 2, -5, -4, 6, 0, 1, 0], [-20, -8, 0, 0, 1, 0, 1, 7, -7, -5, 7, 7, 3, 0, 1, 0], [-5, 0, 21, 1, 5, -6, 0, -5, -2, 0, 5, -3, 0, -4, -5, 2], [17, 4, 0, 0, 5, 0, 2, 1, 8, -2, 1, -8, -8, 0, 5, 0], [3, 0, -1, 0, -3, 0, 0, 1, 9, 0, -1, 9, 0, 8, 3, -3], [-17, 2, 0, 0, -7, 0, -2, -4, -2, 12, -4, 2, 8, 0, -7, 0], [2, 0, -21, 1, -2, 9, 0, -1, 4, 0, 1, 3, 0, -1, 2, -7], [18, 6, 0, 0, 3, 0, -9, -8, 14, -4, -8, -14, -4, 0, 3, 0], [2, 0, -22, 8, -2, -8, 0, -2, 8, 0, 2, 0, 0, 8, 2, -4], [0, 0, 12, 1, 0, 1, 0, 0, -6, 0, 0, -7, 0, 6, 0, -3], [40, -2, 0, 0, -1, 0, 3, 4, -2, -6, 4, 2, 2, 0, -1, 0], [20, -6, 0, 0, 9, 0, -1, 2, -2, -12, 2, 2, -2, 0, 9, 0], [11, 2, 0, 0, 5, 0, -10, 2, 0, -12, 2, 0, 4, 0, 5, 0], [-4, 0, 4, -8, 4, -12, 0, -5, 0, 0, 5, 8, 0, -8, -4, 6], [2, 0, -18, 4, -2, 10, 0, -2, 0, 0, 2, -4, 0, 12, 2, -4], [-13, -8, 0, 0, 2, 0, 0, 2, -3, 1, 2, 3, 4, 0, 2, 0], [-4, 0, 8, 8, 4, -9, 0, -2, 2, 0, 2, -6, 0, -4, -4, 2], [-1, 2, 0, 0, 0, 0, -9, 2, 10, -4, 2, -10, -6, 0, 0, 0], [-9, -2, 0, 0, 1, 0, 8, 7, -2, -2, 7, 2, -4, 0, 1, 0], [4, 0, 0, 2, -4, -7, 0, 3, -3, 0, -3, -5, 0, 2, 4, -10], [13, 11, 0, 0, -4, 0, 1, -8, -3, 16, -8, 3, -2, 0, -4, 0], [-2, 0, 0, 6, 2, 3, 0, 9, -3, 0, -9, -9, 0, -2, -2, -4], [-6, 0, -7, 0, 6, -2, 0, -4, -4, 0, 4, -4, 0, -11, -6, 0], [4, 0, 4, -7, -4, -7, 0, 6, -2, 0, -6, 5, 0, -6, 4, 1], [15, -10, 0, 0, -3, 0, 0, -3, -4, 6, -3, 4, 8, 0, -3, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1840_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1840_2_e_h();".
// 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_1840_2_e_h(:prec:=16)
    chi := MakeCharacter_1840_e();
    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_1840_2_e_h();".
// 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_1840_2_e_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1840_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![64, 0, 1264, 0, 7441, 0, 16123, 0, 10815, 0, 3218, 0, 479, 0, 35, 0, 1]>,<7,R![541696, 0, 3052288, 0, 5335056, 0, 3332320, 0, 703096, 0, 69892, 0, 3621, 0, 95, 0, 1]>],Snew);
    return Vf;
end function;