
// Make newform 1050.3.p.i in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1050_p();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1050_p_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_1050_3_p_i();".
// 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_1050_3_p_i();".
// 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 := [101626561, 294123256, 1087013404, -630430976, 595763862, -91953512, 64376800, -6808704, 4836403, -298200, 197216, -7728, 5846, -112, 92, 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], [1202728311133779179554909109, -9955887688266404429946106, -35874681437029057265441973, 4685579583089306764764583, -10591214719874231223793348, -3497231496721962844392765, -1467974756910134628134465, -306572240997688184788012, -76261143494901216401580, -16545566601055515429559, -2793391292439344623383, -455975368108340251668, -54078590776251682475, -8239103585403158159, -676562680999836294, -57754008957283349], [-364575409398831078887728279110394681911412, 908337323214930810545762959208351226961414, -391063312801232933505474632734192188104159, 125553897923934483082749534593575562768264, -43017604942605706493389121177310876552377, 10045248662052121950280368262485018501526, -3078254928738380300885234729392369408894, 518393018039589118943098682457300057850, -120485837746082046446534418358041078995, 17357216638024717520736861099164996862, -3333370131956257781242715612074616370, 394619982775641320842481938865321022, -49249559559979064058398506987306073, 5113403739807495199248389450038520, -463929575136617004234823710411877, 46018860137918251422611848916178], [-1342637407867531028623664059931524385, 18026328457882421765670697323056283672, -13772890004567177944112685563754762092, 10378818141701407363254998583323267424, -1885642184797314535191161423590090530, 1118207658408946318903529367006504424, -141790839418260960053496277234405320, 84172908290927660560603113151996608, -6167850325881365022247857940572659, 3425112192815123728105742003362520, -164820722237809092990089468647608, 101575342244927525528895400573584, -2441936278614792467764753655926, 1592398885966382922411001797120, -5447157168753271526835174868, 17400352139076767691403754752], [-107999579002538473664858731240086213321125, 455299469949949001006808967750814029947180, -246911752503096306017796830784635952976919, 117570780319527025900418282659080593337321, -30051537552343231396274055237124441558824, 10483004614526267309457404262166594319923, -2572169999826035957556653521121901023631, 667012850104717150279246253304669351102, -111787268318451401044906829423789082260, 24663363768518711116676474220149700881, -3428346482911612093472398677927059241, 667779590038153606343241043238558634, -53934149533217133028668957518264761, 9535509196408409190209302920836991, -497615566110261825195943736500894, 102029534763959390276405841394999], [-9829450534674094451308149999632187248805, 131112916615924963752999305865652963348499, -100750239749972322740616761667196754934902, 76280258944814714871953810544519573613285, -14127347938893413955566753290761285684409, 8321014304598085656622695434060879347468, -1043154428996648611051999107178863913973, 633905589310301125652383803769358202665, -45307871148339960168856971658866756539, 26032403279396509166363967498551796676, -1187602293972815521886016745964354547, 777466972868003211094199096781514407, -17364450778627773798376683355140498, 12275699116190510793754047669137227, -21147198422644956592033966560067, 134409898005376741703512285310102], [127583315037424866443615513010121944293123, -384924078941954250051628044382522005810305, 199585585169109011459715236576290498221034, -79852042573139128475593209350397361599772, 22763847665916121668171336262827172713853, -7504704529887628407915597917160544171127, 1751519196048072184977880770154539327173, -450429116516935856988558071283198281261, 72576108656938834733873180020190675023, -16515783030420790586451322867154864685, 2105230386131833286396313926379492627, -426153734346560417635925481170864283, 32186289069847532763699370799632672, -6058198328557099232364083716716484, 281955796478742744637109265247727, -61429677070949408200027856490357], [332620630527866859603644264420455887243781, -1960783729694641949917456365948039861600382, 1051248648034845819145134509371481920184423, -547359250072724672130276739710666189278081, 120793939383614343286456218705872694316670, -52370124407479989893736668802219718738453, 9338607444230212435261866262508110336291, -3475512386687862410882944576509713856712, 386598745086566790290265119671927189358, -133328159436232991050173996482621121435, 11125643556603443772271167433574439981, -3701661767541818481701476276118272548, 167152824309762061536221654529237215, -55159269492954233108528000900109671, 1234786430244768602649223970134258, -578228859242670890019630861418833], [-150047106815802304043838392869738305966163, -349277930052592314980757639714142037399485, 200201743004848695051712474644747888074184, -290099220935800787089817808530069128969265, 48411746950158510698117655770254497756179, -32630367349604644112815306182136062684042, 3547981588116210787777744986319141518619, -2593298217042442425835158171358377109819, 163064328623877625820325835387507160841, -108683734477779696037434128185832402446, 3832189527266138571340990611201666789, -3314551785894273503186827021273739961, 53115740924217261817200701198805364, -53447719211766743887590190142035487, -258508606171066072621522689902829, -590324686062022489650236255727896], [616255702328965055804953913386729088858320, 369313159463611314235124000569397039197921, 71303909434058108877767004607039584431382, 377906663597366344649799908074579775348711, -11051529960712246421494988328443122682738, 42519809444656034523121695752486923838640, -669218737813224075926497432520094802768, 3290125964118469409722113329853724673599, -44662939311649959397306110606241779294, 136532495622602826178264580026097756656, -465793120713505585635077717021177136, 4103393468622234627576197828635461129, -949095276335381753698053659019262, 65820550234839117861624840400322321, 871779212349417800468728537652262, 711986579795629656228430470516998], [823575393607085165297041934848114219274089, 1201687191208601644859962899545577262093720, -386509635782297256474317492285413606810557, 921088407323036584077684769629324604072937, -78934700970238174358503967333927967003272, 105175386632805186771827763933309569582763, -4934523094765596199389693350950648914187, 8124889475512256874501227693939632642810, -223525280693826435916728566384972140716, 338414406325381687458027282043473872617, -3970280009377581384619015772999411037, 10162960799976213744515356011572115906, -39202978391233464444306360048122959, 163391456903595146744732956966072127, 1852285003049929087052570331696528, 1764975938393729366963681385799343], [6844632397334610746953902438407673207116, -16324539167775629589823491710647695828176, 12524101853154380970686107173503865531343, -9388816670061924410780420177258727817240, 1697419715174601685899290126118881569318, -991040252297617005748512087372373597736, 123584635058899329799419588508184463458, -73949487471406031390488864572123656636, 5333734875396681470304930008869104748, -2985451639876647182371085823671089968, 140068532503481168323378183787351526, -88383851722041552421306118227957608, 2070620591675343319876280413062238, -1383484706094836359184994704240272, 3852719496783125874239699105885, -15146962332167086975911046675596], [-10024458526124237254005202888221833667740, -16316108629673610537042574548471360931136, 12561613093130083522431064835280464968224, -9493281883173746776756996790696940683128, 1780534733659258888547308382234455799002, -1039620097324413530414056993030116382704, 136011162532639634621553271253032164995, -77756476527255158388334910202906987716, 5913591831251674034883305831660820984, -3161847919541951827639329091269140816, 159427282385337336181335626996264889, -92909929232580146755250982584235360, 2353772139163922525796446278024976, -1450749857931551628175441224995896, 6742312326680500100812962302115, -15695346419457846209281835223520], [-23926228959773806185858731136579338464835, -38842532014981225602471254399462455798901, 30811849435233691208527130104227031144716, -22935035325509782054919587376143953816331, 4520471450461910640553969711249964502511, -2453773174632007402144600068324174455250, 352528661124042910024133953243239757835, -184749851970209278666049080923998853833, 15359759593270696106552771791237598865, -7494710080670555291162470938804763982, 421418216815192265268055091790458409, -221957303143100369653138224121882179, 6311670521194816310556410853682532, -3460138607408100129199227288797647, 23931632759137214177457474370647, -37843762023245203184992623239608], [16376788416025990146487916910940766026640, -39102822762957642047137623259186055795171, 29862159504958191221748980176572973620103, -22679191757799153968692465101503969996980, 4121245281097393216646345375497892732779, -2460174618963658787599447356617919097961, 289136703397288624104556325096432039318, -186056032092382098363420621791607929693, 12515860647284234638283153739172269637, -7591692863182153553006588244409170495, 320351452082265573667292205927002610, -225774485227290092779433605920617583, 4708240138935360355625740218514339, -3553062423310914581399872258706620, 2421288208827724994970270679457, -38896735311986822333587883641137], [-640801390875761684646160678410369543684936, -5161020851537811333074591816408832173877376, 2265970777726001005839696776040395953940179, -2317267719056477742937543846810017003765821, 325720156884053455436956898910230887692869, -249429330587068245084062957173810405121123, 23660231471496971098367824036470558791800, -18414668516933299176037168718231046053374, 1031686452696127710883915229583974343955, -748086755713860568427618745384862604993, 26208089259582258949141613245951511568, -21996973972180950255733413299929515798, 372752220955071808996323249023281809, -345792902538290606979636753407083907, -477901293928498865022635782945573, -3726774205013522964103392462287743]];
    Rf_basisdens := [1, 852577600778545978990177524, 95852909214978668011338765982386839392644, 6206482078151946905681090778450326301, 31950969738326222670446255327462279797548, 31950969738326222670446255327462279797548, 13693272744996952573048395140340977056092, 95852909214978668011338765982386839392644, 95852909214978668011338765982386839392644, 95852909214978668011338765982386839392644, 95852909214978668011338765982386839392644, 726158403143777787964687621078688177217, 726158403143777787964687621078688177217, 1228883451474085487324855974133164607598, 1228883451474085487324855974133164607598, 95852909214978668011338765982386839392644];
    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_1050_p();"
function MakeCharacter_1050_p()
    N := 1050;
    order := 6;
    char_gens := [701, 127, 451];
    v := [6, 6, 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_1050_p_Hecke();"
function MakeCharacter_1050_p_Hecke(Kf)
    N := 1050;
    order := 6;
    char_gens := [701, 127, 451];
    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, 1, 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 := 3;
    raw_aps := [[0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 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, 0, 0, 0, 0], [1, 1, 1, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1], [-2, 0, 0, -4, 0, 0, 1, 0, 0, -1, -1, 0, 0, 0, 0, -1], [-1, 1, 0, 0, 0, 1, 0, 0, -1, 1, -1, 0, 0, 1, 0, 0], [-4, -2, -1, 2, 0, 0, 1, 0, 0, -1, 1, 1, 0, 0, 0, 0], [1, 0, 0, 1, -1, 0, 1, -1, -1, -1, -2, 0, 0, 0, 0, 0], [3, -7, -2, -3, -1, -4, 1, -1, 0, -3, 2, -1, 1, 0, 0, -1], [10, 1, -1, 5, 0, 1, -1, 1, -1, 1, -1, 0, 0, -1, 0, 0], [-5, -6, 2, -4, 1, -5, 1, 0, 0, 3, 1, 0, 0, -2, -1, -2], [-3, -5, 0, -7, 3, -12, 1, 1, 2, -3, 2, 0, 0, 0, 0, -1], [2, 5, 0, -1, -1, -2, 3, -1, 2, -1, 4, 0, 0, 0, 0, -1], [0, 0, -2, -4, 0, 0, 0, 2, 0, -2, -2, -1, 0, 0, 2, -2], [-2, 0, -2, 2, -2, -14, 2, 2, 0, -2, -2, 0, 0, 0, 0, 0], [8, 10, -4, 1, -2, -1, 2, -6, -2, -4, 1, 0, -3, -2, 0, -1], [2, -4, -5, -9, 3, 3, -1, 3, 0, -2, -4, 1, 0, 0, -2, -3], [3, 2, 5, -17, -1, 29, -1, -1, -2, 1, 2, 0, 0, 0, 0, 1], [-9, 4, 2, 4, 2, 5, 3, 1, 2, 6, 1, -1, 1, 0, 0, 0], [26, 23, -1, 12, 0, 8, 5, -5, 1, -1, 4, 0, 0, 1, 0, -3], [34, -2, -1, 34, -2, -2, 1, -2, -2, -1, -3, -1, -2, 4, 2, 0], [-12, 2, -1, -25, 4, 1, 4, 2, 3, -8, 0, 0, 0, 0, 0, -3], [-15, 12, -3, 6, -2, -17, 2, 2, -1, -7, 0, 0, 0, 0, -4, -3], [13, -27, 1, 26, -1, -55, -3, 1, -1, 3, 2, 0, 0, 3, 3, 4], [44, 15, 7, 27, 3, 15, 1, 5, 3, 6, 0, 0, -3, -2, 0, 0], [-20, -6, -13, -3, 2, -1, 4, 8, 1, -9, -4, -2, 0, 0, 2, -7], [19, 22, -6, 15, -1, 18, 1, -3, 3, -11, 0, 1, 2, 0, 0, 2], [-1, 1, 0, -46, 0, 0, 2, 2, -1, -5, 0, -6, -3, 0, 0, -1], [24, -12, -6, -4, 2, -4, 2, 0, -2, 0, 0, 2, -2, 0, 0, -2], [-25, 12, 7, 7, 0, 3, -4, -4, -3, 2, -4, 0, 0, 2, -2, 5], [18, -26, 2, -12, 0, 32, 0, 0, 2, 6, 0, 1, 0, 0, -4, 2], [-7, -5, -12, -44, 2, 15, 2, 2, 5, -1, -1, 6, 3, 1, 2, 2], [-9, -8, -2, -18, 2, -19, -4, 3, 2, -1, 6, 5, 5, -6, -6, 7], [90, 46, -6, 86, 0, 42, 0, -2, 4, -10, 2, 3, 6, 0, 0, 2], [3, 1, 4, 55, 2, 4, -2, -2, -1, 1, 8, -6, -3, -1, -2, 5], [-32, -12, 12, -8, 2, 6, -4, 8, 0, 6, -8, 0, 0, 2, 0, 6], [76, -52, 8, 76, 4, -52, 14, -6, 4, -4, 0, 2, 4, -4, -2, -10], [-40, -40, -9, -77, 0, -75, -2, -2, 3, -2, 2, -5, -5, -4, -4, 3], [-60, -4, 4, 66, 0, -2, 0, -6, -2, 0, 6, -1, 0, 0, 4, 4], [-2, -2, -4, -62, -8, -8, 0, 0, 2, 10, -6, 0, 0, -2, -4, 4], [28, -39, 5, 56, -5, -83, -7, 5, -5, 7, 2, 5, 5, -2, -2, 12], [-86, -14, 12, -82, 0, -10, 8, 0, -4, 4, -8, 2, 4, 0, 0, -4], [-2, 3, 1, -62, 0, -68, 5, 5, -3, -13, -6, 12, 6, 1, 2, -9], [104, 47, 11, 8, -3, 2, 8, -6, 7, 3, 11, -3, 3, -3, 3, 2], [41, 43, 10, 29, 4, 38, -8, 16, 1, 11, -8, 0, 1, -1, 0, 5], [29, -17, -7, -6, 1, -15, 3, -9, -7, -5, 2, 5, -5, -1, 1, 0], [2, 20, 4, 4, -4, 32, -2, 8, -4, -2, -2, -10, -10, 2, 2, 10], [-107, -88, -13, -68, -5, -63, -1, -9, 6, -21, 12, 0, 5, 2, 0, -1], [-35, -32, 8, 59, 8, 40, -8, -8, 0, 16, 0, -4, 0, 0, -2, 8], [30, -40, 6, 32, 8, -38, -4, 8, 6, 10, 12, -5, -10, 0, 0, -2], [74, -46, 2, 2, 0, 8, -10, 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147, 22, -160, 58, -14, 43, 23, 28, 0, 31, 7, 0, -7], [-607, 156, -59, -605, -22, 158, -94, 33, -24, 45, 17, -8, -16, 4, 2, 53], [395, 161, -40, 805, -5, 340, 43, -3, 10, -65, -48, -1, -1, -12, -12, -31], [-994, 76, -40, -32, 8, -10, 0, 22, -4, -6, -8, 26, -26, 2, -2, -18], [-252, -68, 62, 348, -26, 36, 26, -64, -2, 32, 90, -7, 0, 0, 4, 62], [508, -194, -80, 225, -36, -95, -134, 62, -94, -14, -121, 0, 0, 32, 0, 63], [-129, -68, -121, -58, 53, -57, 39, 31, 2, -64, 8, -35, 0, 0, -32, -29], [-60, -44, -104, -552, 40, 86, 16, 16, 44, -28, 50, 52, 26, -4, -8, 54], [-234, 8, 162, -12, 34, 130, 26, 42, 20, 82, -48, 0, -12, 40, 0, 34], [-630, -88, -40, -656, 2, -114, -34, 20, 28, -62, 18, -21, -42, -56, -28, 44], [268, 6, -82, -78, 4, -10, 22, 68, 30, -18, 18, 28, -28, 12, -12, -56], [-627, 506, 5, -640, 27, 493, -13, 31, 40, -30, 36, 43, 86, -4, -2, 17], [203, 248, 24, 420, -66, 525, 36, -15, -52, 17, -102, 13, 13, 24, 24, -37], [-371, 570, 51, -348, 0, 593, 56, -28, -23, 41, -28, 5, 10, 24, 12, -51], [308, -76, 112, 360, -24, -24, -10, 70, -76, 72, -108, 22, 44, 44, 22, 42], [-442, -170, 66, -894, -36, -330, 2, -20, -46, 74, -38, -53, -53, 40, 40, -32], [-69, -121, -45, -82, -37, -13, -115, 69, -21, -107, -78, 1, -1, -4, 4, -12], [-86, -56, -16, 58, 28, -16, -14, -22, -24, -36, 8, 46, 0, 0, -40, -2], [-100, -36, -136, -360, 36, -242, 64, 64, 36, -128, -88, -30, -15, 22, 44, -88], [60, -42, -23, -79, 42, 63, -54, 20, -15, -11, -74, -54, 0, 0, 34, -35], [-467, -316, -26, -463, -54, -312, -37, 7, -58, -12, -78, -20, -40, 32, 16, 57], [14, -11, 3, 936, 28, 444, -25, -25, 11, 33, 58, 8, 4, -9, -18, 41], [676, 516, 128, 416, 54, 346, 88, 20, 80, 82, 60, 0, -8, 12, 0, -34], [611, 296, -54, -8, 46, 119, -60, 127, -4, 63, -106, 5, -5, -4, 4, -25], [-56, -41, -97, -985, 65, 407, 15, 15, 41, -54, -52, 6, 3, 20, 40, -76], [22, 137, -118, 89, -17, 388, -53, -69, 28, 49, 36, -48, -48, -54, -54, 29], [-367, -458, -55, -238, 13, -311, 81, -55, 80, -41, 148, 0, 15, -36, 0, -81], [47, -288, 32, 3, -12, 298, -2, -20, -2, 16, 18, -26, 0, 0, 24, 18], [-171, 396, 173, -144, 77, 423, 134, 12, 50, 12, 8, -6, -12, -14, -7, -92], [-106, -294, -84, -64, 20, -6, -20, 58, -16, -10, -40, -33, 33, -10, 10, -34], [885, -292, 48, 1764, -30, -573, -99, -17, -36, 158, 69, -35, -35, -14, -14, 76], [3, 16, 19, -1394, 9, -193, 13, 13, -16, -51, 88, 22, 11, -20, -40, 63], [400, -126, 53, -298, -6, 140, -1, -44, 2, 51, 43, 13, 0, 0, 52, 46], [819, 80, -45, 383, 1, -13, 55, -53, 10, -7, 60, 0, -6, -18, 0, -51], [480, -750, 43, 477, 15, -753, 101, -55, 18, -40, -16, 21, 42, 12, 6, -67], [392, 136, -24, 770, 66, 258, -18, 0, 52, -20, 84, -77, -77, 2, 2, 4], [-126, 140, -114, -56, 48, -278, 42, 22, -2, -70, 20, 51, 0, 0, -4, -24], [24, 119, 62, 39, -35, 160, 29, 69, -44, -45, -64, -56, -56, -22, -22, 31], [-564, 60, -74, -347, -10, 3, -96, 76, 36, -66, 25, 0, -19, 10, 0, 21], [37, 13, 50, 31, -42, 176, -24, -24, -13, 77, 84, -18, -9, 15, 30, 113], [-1022, -36, -184, -150, 34, -124, 28, 26, -48, -48, -6, -17, 17, -6, 6, -68], [-4, 78, -50, -40, 10, -34, -43, 6, -48, -29, -53, -12, 12, -2, 2, -1], [-482, -80, -40, 442, 42, 16, 8, -12, -2, -2, 20, -37, 0, 0, -12, 10], [24, 48, 72, -470, -76, -260, 24, 24, -48, -20, -36, -72, -36, 2, 4, -8], [170, 365, -23, 352, -13, 695, -57, 47, -1, -1, 44, 13, 13, 2, 2, 116], [-852, -1166, 26, -424, -6, -580, -14, 2, 10, -28, -10, 0, 52, -50, 0, 26], [-145, 776, 64, -137, 10, 784, 5, 51, 2, -22, -36, 5, 10, -4, -2, 33], [27, -3, 24, -22, 86, -60, -30, -30, 3, -23, 62, -14, -7, -20, -40, -21], [1011, 54, -27, 503, -1, 93, -139, 137, -24, 21, -80, 0, -8, 0, 0, 57], [332, -65, 41, 622, 85, -116, 34, -56, 43, 21, 51, -4, -4, 13, 13, -132], [-4, -196, 96, 114, -52, 178, -10, -64, -30, -16, 54, 16, 0, 0, -40, 34], [-34, -5, -39, -676, 57, 411, 29, 29, 5, -110, -96, 76, 38, -28, -56, -148], [-256, 222, -78, -80, -2, -148, 24, -68, -54, -70, 26, -24, 24, 26, -26, -12], [-276, 470, -140, -225, -32, 144, -56, -8, -30, -66, 12, 0, -12, -8, 0, -10], [-100, 90, -80, -154, -58, 224, 59, 2, -12, -95, -117, 70, 70, -4, -4, -11], [476, 106, -74, 190, -28, 10, -66, 10, -34, -50, -30, 0, 62, -16, 0, 24], [-18, -180, 38, 29, -55, 177, 14, -7, -10, -37, 21, -22, 0, 0, -35, -3], [465, -41, -56, -51, 5, -104, 65, -53, -14, -25, 60, 23, -23, 2, -2, -21], [52, -50, 42, 84, 18, -192, 76, 72, -2, -126, -58, 91, 91, -24, -24, -24], [-5, 737, 97, 52, -45, -165, 59, -217, -43, -63, 104, 10, -10, -18, 18, 70], [-81, -49, -130, 520, 46, 633, 32, 32, 49, -61, -115, 46, 23, -33, -66, -112], [-87, -48, 79, 188, -71, 139, 11, -5, 14, 36, 16, 7, 0, 0, 36, 19], [23, 5, 28, 629, 22, -120, -18, -18, -5, 9, 60, 104, 52, 22, 44, 33], [-820, 715, -11, -426, -36, 330, -49, -23, -51, -57, -70, 0, -12, 19, 0, 55], [-270, -686, 80, -221, -25, -637, -14, 45, -74, 91, -81, -18, -36, 30, 15, 21], [200, -26, 48, 354, 90, -102, -28, 4, 44, 26, 118, -32, -32, -6, -6, -14], [136, -224, -30, -244, -50, 232, 24, 58, 2, -76, -34, -34, 0, 0, -36, -56], [-1, -24, -30, 17, -47, -6, -1, -47, -65, 55, -46, 24, 48, 24, 12, -18], [10, 42, 52, 419, -34, -198, 32, 32, -42, -72, -104, 36, 18, 36, 72, -112], [-575, 70, -95, -572, -17, 73, -97, -1, -20, 90, 64, 45, 90, 0, 0, 13], [-96, 216, 176, 364, -54, -120, -66, -86, -30, 62, 20, -47, 0, 0, 34, 56], [474, -972, -22, 220, -2, -554, 98, -102, 8, -14, 68, 0, 44, 6, 0, -58], [3, -2, 1, 50, 20, 223, -5, -5, 2, -8, 77, -110, -55, 6, 12, 59], [328, -404, -92, 690, -10, -688, -44, -86, 24, 72, 34, -17, -17, -34, -34, -8], [-156, -11, -43, -264, -101, -11, 19, 37, -53, -51, -120, -37, -37, 74, 74, 66], [-50, -412, -44, -30, 16, 288, 42, -22, -8, -44, 64, -14, 0, 0, -44, 14], [-97, -366, -24, -109, 4, -378, -7, -5, 16, -20, 20, 78, 156, -24, -12, 3], [-78, 29, -49, 356, -76, 488, 107, 107, -29, -167, -46, 60, 30, 35, 70, 1], [-138, -35, -55, -74, -19, -106, 30, -32, -7, -69, 49, 60, -60, -27, 27, -24], [383, 1194, 11, 184, -14, 629, -107, 79, -2, -40, -61, 0, 15, 34, 0, 73], [387, 309, 60, 771, -51, 654, -3, -39, -54, 99, -48, 59, 59, 12, 12, -39], [56, 46, 102, -64, -86, 496, -10, -10, -46, 60, 62, -64, -32, -24, -48, 102], [184, 622, -190, 30, -40, 248, -78, -2, -146, 26, -90, 0, -23, -44, 0, -16], [153, 804, -99, 126, 33, 777, 30, -102, 60, 24, 138, -9, -18, 30, 15, -108], [231, -20, -122, 498, 14, 113, -21, -117, 50, 52, 35, 66, 66, -26, -26, -60], [287, 527, -65, -86, -21, -55, 36, 31, 37, -56, 57, 24, -24, 17, -17, -51], [-938, -442, -32, -478, 8, -254, 64, -48, 2, 22, 50, 0, -72, 32, 0, -56], [380, 250, -9, -376, 52, -286, -53, -20, -30, -17, -33, 55, 0, 0, -46, -10], [-82, -201, 71, -172, -47, -548, 42, 138, -55, -117, -89, -22, -22, -5, -5, 88], [-55, 288, -14, -17, -56, -274, 39, 43, 12, -46, -4, -42, 0, 0, 12, -31], [334, 299, -66, 675, 45, 612, -69, -7, 52, 17, 114, -82, -82, -8, -8, 69], [102, -308, 190, 164, -94, 302, 20, -136, -20, 56, 156, -58, 0, 0, -6, 116], [-908, -208, 22, -916, 54, -216, 26, 4, 62, 0, 78, 12, 24, 20, 10, -42], [-71, -35, -106, -877, 30, 204, 36, 36, 35, -67, 40, -50, -25, -45, -90, 45]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1050_p_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1050_3_p_i();".
// 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_1050_3_p_i(:prec:=16)
    chi := MakeCharacter_1050_p();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    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_1050_3_p_i();".
// 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_1050_3_p_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1050_p();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<11,R![9682651996416, -10143730662912, 26588869767936, 15929692886016, 26420738609664, -11416664448, 516442695552, 676772544, 6945139504, -7989152, 49322848, 129616, 252064, 352, 604, 4, 1]>,<17,R![13699042635136594176, -7639155812489338368, 988073795283754752, 240843456926443008, -5010295164885504, -2387574728931456, 24615413755776, 18306805272768, 641540452528, -29790743520, -1531993376, 40497360, 2594784, -23280, -1892, 12, 1]>],Snew);
    return Vf;
end function;