
// Make newform 1216.3.d.e in Magma, downloaded from the LMFDB on 13 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1216_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1216_d_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_1216_3_d_e();".
// 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_1216_3_d_e();".
// 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 := [19859428, 10041872, 6110408, 4221760, 619800, -1201304, -419708, 65416, 66793, 12048, -7342, -2372, 707, 116, -30, -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], [-61668090459014287736763536, 70048023111716646028814284, 21297483765283067963991964, -50054041505096501281468800, -10570322133592234108937268, 5010011209263472651188052, 1529065906675012464553656, 481596207599784996842056, -383209455960866841156876, -81987393760171937983845, 46282270471383206746691, 2551110228466579848476, -2171743799076503593269, -33716435683692955912, 71942636170106415259, -4517671988737445075], [153928541468852783618173468826536329006932, 59482415269426300405617102445345205796344, -285519813728759851710933336813445577678524, -94358433840036760028099622197520731663004, 34664396046078663171287700709152070415112, 7733919977099961145701724298664473792036, 3353618392162143285789304067639813136120, -943728373346818966593552370205321870992, -1081972621748826151630019851168683406347, 291889414903599346781758935616456602978, 52285983975789826093468681872144611509, -18058966436658322914814285878936074711, -1976350128924154480631899645288485927, 837079779939679872680268494802687889, -12165070779345105581860968180339856, -10332064623441439038332243194502461], [761745585734022290071340203330284135166648, 25448414776410639674918655188255489807668, 85502899571720972206626462537004753623484, 32068597058180761284569901600206985925416, -3386900758382301674023844758653144834204, -5977681402907385643675042990846389736172, -2409862270559545332741195564123776979384, 191387796600496646723773138059011623288, 708317949811196235665859237105175522878, 27765168353922418088689561211177744997, -98290824371121259805615557964667286741, 6404749752176814200184565769533499822, 4507205580227846128813513060890241587, -339243401835365955675785532076945534, -156818636350762964603372774813501759, 20977174827808192284620577906783061], [-407524617602704037890719638944586608048976, -373407530314207035866674834960089676567724, -121008774204415569574128781597847177401628, -29405908704615667433105662989512212455168, 44089629916875342810303745686385393653540, 18395545069733522697802366113428167409884, -2274438127543092990697230244951658622216, -2875298950739947623662262688117376855336, -577382941196569566488690794764731053092, 298273111134462954089564428236365511597, 121237877720894106326235262790525005541, -32238898190452001102257774445964317308, -5911132281367919830174428867783457683, 1384391829229516676563882092925433144, 213031101611376483736942668358145773, -50127807521442554765095977140729765], [-407524617602704037890719638944586608048976, 99910785873986151481189130343387179742772, -121008774204415569574128781597847177401628, -29405908704615667433105662989512212455168, 44089629916875342810303745686385393653540, 18395545069733522697802366113428167409884, -2274438127543092990697230244951658622216, -2875298950739947623662262688117376855336, -577382941196569566488690794764731053092, 298273111134462954089564428236365511597, 121237877720894106326235262790525005541, -32238898190452001102257774445964317308, -5911132281367919830174428867783457683, 1384391829229516676563882092925433144, 213031101611376483736942668358145773, -50127807521442554765095977140729765], [-1161374216486244766926971296, -3939786290999348462011137826, -2294972628814137668352462136, 968241119704979781578518770, 95424450365899599046981152, -68054722076369893135207453, -10292445969983110996472934, -4177219579104308850834199, 5803606386111136804830984, 1305958543213426046790795, -1518470418757817768451074, 150146172344732490957751, 73981279508823200406696, -8315982046452121145597, -2980854830183604972586, 553761430169698066805], [-2198531006689395972187516425295058074510544, 12307718059904540901045094573977735902428, 81959654609083669748909025581272530205948, -1285470102280381985647152603443719595431248, -251588423427398192875303356941926800381972, 159042522881611293704968826312033490282124, 70995826992106204660044926461806682096200, -2233858553682330723498048751195172085728, -10938264740882338229983183874157865647036, -1405167731225219328745495646789555935653, 1277457418288155343904436982499155291259, 36191109288435020769574258099182638372, -56131190616266241374746957551414826653, -984054334098558918405451688137338160, 1938336531247524702188306704546041955, -108000940999534579859766264934585763], [-383379380067133651845490917683974220781668, 609150375511723627627874126379158851588132, -1365511985858784147275276841893614268324360, -476447943318372511672708530605880874466340, 309918193405296633439526665432818619418604, 127798742459935611347573209282097594591440, -13261631925851910476372148710110871844312, -20425892023523350084275990582248053199384, -2274229808276314807562105399193044866857, 2600996281468189788089676367103226923781, 247133769242596611811544778356388800282, -178816620989110424320705818455045115445, -11504725945930543807049904012464723130, 7194818140013404020157268061445934975, 232173663832819365129583176629057095, -160831251056868240723831507775280378], [11468596753205415970306653510239705152056700, 17091969714897752832349832025967013518363012, 6002468015676247136626093502887804503653992, -807250832784679368556264475761328316562548, -1909706710120677459576291966386250170452740, -246625646812523786511473765840634046494824, 161404864347640549520891190013831378723176, 71897745945936324152326946289885416518768, -6697644278410794576432991552379902574337, -5834006269072893383918470777198694533383, -1253578760717998095756113790642957294610, 576378196780432863297186103138827865163, 67618581907876732794228532505069482938, -21985999712245311000885223627842020017, -3364017925439331447563378765906335421, 828188597057137378456931952689271082], [-1940074962871204040460907880277576032454460, 247225483125712259467136027140559903378540, -160416329980541461668667720558323796242472, -4622349411464361406676721205466107835676, 32377311579775636606419972630043789367364, -46858069223573463902999432103543567780448, 2475240271357420903646813329202155758360, 2596169935444242947428824310968194725304, -58901348607844231642027628595913044855, 494415260748863130331723450405632173919, -422390956406519217315365728634226029510, 40993027522101478081161190175011690837, 21256769154854400209011223120049232398, -1992058956191665478190246207550702223, -1042110912366538905346738179074183011, 181896281271558998798736093744441926], [139272324017443017729226846064381345848204, -115589335694446601062273966332821675298660, 231357086121196488969190262494601100612512, 68551883093975535122567636556955921947500, -50663584954576931856793528639779625822076, -14754439659966968087627921151672348686104, 425303616235697409143713733248870449352, 2562790706026363890943596537685402188536, 801373183621815663625065367431493926067, -400260702422576292111011049493817251757, -67325724977106295213789998022870338764, 27962340534759432219652441224415949255, 3028540871915935166903215013892067920, -1179623306320652846281103039513322765, -54120996794139392221499314225035811, 24288198463469000106663625804055772], [39409922390338991258051917960952147795564, 30501425168660207523493257942928471322612, 12075512990682947766525478409386584523592, 5086568971919733736756217110440270421788, -3004991985851950051783296242998419735348, -1618241779122634160904129055113752018104, 127646836218595463586804840563979017640, 131266980693825893317149894085812385568, 56933337497047066088915601206493748971, -8675258735616300628900224954524169267, -10130437432071565841977550799273877034, 1349939152452257152197048919245562903, 494254858521988219596430799747018418, -62663089532086111538848914375208085, -17335286914719278070493511036825425, 2756543058008694826893547136053298], [-662044750581298933737639241280582514538388, 206641125812902719056094425045082279690712, 110950186715897487255509962719549509757532, 96861676535805403152314277401550051631500, 15760730135507674234294457247589388081232, -35751597212155950318420954399387839409572, -1413457810288916063712832039096536231912, 2442656764316551373729825614115050651072, 415526268046439478966736344647047501455, 235065278802847403826200316694193985426, -210243071316962236371597911831399167621, 8965171241520615347084194660232427467, 10276054194658006444486640234131823991, -375309444335745180659073918433835821, -475992940727439106725043977272894972, 61566134550827893060058085489960973], [3751355121264329269121831363635011803277620, -1842388186789813075119660373795440780597772, -1591957322595264092644595279794479475355392, 2179390897479378760863096143028145861332708, -16831721548313853780272759673088360044180, -384855127862444530485078680786151768213136, -61944808139244257523646144272343890910776, 18340375034340444208394666134801088842832, 21980931887622115328152031383813370275557, 72433911946074498430672888785544961733, -3466433790062169003736453957279025226380, 350919281533459576393716175762420938697, 163684327226957522819114883378782427672, -17580349184118744621218835015026669123, -5829269410994488757451198253287296029, 980716079614608876889364338450550948], [-3010612148223900070042047944770458433207660, -2155690049879677581156942916883881645272916, -786384134065121822927949033134988427245976, -216405623861665958710527775588300964856876, 316955730474516403706180174143369807261860, 145341928914662756630530403330665465563392, -24518671402714165932463989599596507014648, -17943980431348865630571931801099984969384, -5008154933678296318236777891484759163979, 2315182753997864612605431599459366081319, 900660782381509751330601305388198883750, -247535092684288847682024586379232185519, -44931805849114698078515408646195813654, 11147597189852018301627111324745902301, 1491138167810198467306323505900757693, -381372472033414906493473476831413446]];
    Rf_basisdens := [1, 72977246294245082107079760, 118329579047048296836965991325869214077624, 118329579047048296836965991325869214077624, 236659158094096593673931982651738428155248, 236659158094096593673931982651738428155248, 2352078227971109756747226420, 394431930156827656123219971086230713592080, 473318316188193187347863965303476856310496, 2366591580940965936739319826517384281552480, 473318316188193187347863965303476856310496, 52590924020910354149762662811497428478944, 5024610575246212179913630204920136478880, 78886386031365531224643994217246142718416, 1183295790470482968369659913258692140776240, 394431930156827656123219971086230713592080];
    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_1216_d();"
function MakeCharacter_1216_d()
    N := 1216;
    order := 2;
    char_gens := [191, 837, 705];
    v := [1, 2, 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_1216_d_Hecke();"
function MakeCharacter_1216_d_Hecke( : Kf := false)
    N := 1216;
    order := 2;
    char_gens := [191, 837, 705];
    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]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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]];
    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, 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], [1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1], [-2, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0], [-3, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, -1, 0, 1, 0], [-9, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0], [0, -1, 0, 0, -3, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0, 1], [5, 0, 1, -2, 0, 1, 0, 0, 2, 0, 0, 0, 0, -1, 0, 0], [-7, 0, 0, 2, 0, -2, 0, 0, 2, 0, 0, -2, 0, 1, 0, 0], [0, -2, 0, 0, -4, 0, -3, 1, 0, 0, 0, 0, 0, 0, -2, 2], [0, -2, 0, 0, 7, 0, 5, -1, 0, 1, 0, 0, -1, 0, 2, 2], [22, 0, 0, -2, 0, -2, 0, 0, 1, 0, -1, 3, 0, -1, 0, 0], [0, -2, 0, 0, 0, 0, -6, 3, 0, 2, 0, 0, 0, 0, 2, -2], [-6, 0, -4, 0, 0, 4, 0, 0, -1, 0, -1, 2, 0, 2, 0, 0], [0, 1, 0, 0, 5, 0, -4, -3, 0, 0, 0, 0, 0, 0, 2, 1], [0, 7, 0, 0, 10, 0, 2, 2, 0, 1, 0, 0, 1, 0, -3, -1], [-3, 0, -2, -8, 0, 0, 0, 0, 2, 0, -8, 2, 0, 4, 0, 0], [0, 2, 0, 0, 3, 0, 8, 0, 0, -5, 0, 0, 5, 0, 1, -2], [0, 6, 0, 0, 6, 0, -8, 1, 0, 0, 0, 0, -2, 0, -2, 1], [-13, 0, 2, 2, 0, 8, 0, 0, 4, 0, 0, -2, 0, 1, 0, 0], [-25, 0, -4, 2, 0, 5, 0, 0, -3, 0, -5, 4, 0, 4, 0, 0], [-19, 0, 1, -6, 0, -4, 0, 0, 1, 0, -8, -3, 0, 3, 0, 0], [0, 5, 0, 0, -10, 0, -4, 0, 0, -5, 0, 0, 5, 0, 1, -3], [0, -14, 0, 0, -4, 0, 2, 3, 0, 0, 0, 0, -6, 0, 0, -2], [7, 0, 1, -2, 0, 0, 0, 0, 4, 0, -6, -1, 0, 3, 0, 0], [-21, 0, 0, 12, 0, -4, 0, 0, 0, 0, 6, 10, 0, 1, 0, 0], [0, 5, 0, 0, -21, 0, -4, -6, 0, 4, 0, 0, -4, 0, -8, 3], [0, -1, 0, 0, -4, 0, -1, 4, 0, -1, 0, 0, 17, 0, 3, 0], [85, 0, -2, -4, 0, -8, 0, 0, 0, 0, 6, -2, 0, 2, 0, 0], [0, -10, 0, 0, -14, 0, 7, 4, 0, 4, 0, 0, -20, 0, 2, 1], [21, 0, 8, 0, 0, -5, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0], [0, 4, 0, 0, 2, 0, -18, -2, 0, -14, 0, 0, 6, 0, 0, -2], [0, 0, 3, 8, 0, 9, 0, 0, -2, 0, -1, 15, 0, 7, 0, 0], [0, 7, 0, 0, 42, 0, 10, -5, 0, 5, 0, 0, -3, 0, -5, 6], [0, 14, 0, 0, 5, 0, -4, -8, 0, -7, 0, 0, 13, 0, -5, 8], [-3, 0, 5, 0, 0, -8, 0, 0, 7, 0, 3, 0, 0, -2, 0, 0], [0, 19, 0, 0, 15, 0, -6, 4, 0, -18, 0, 0, 4, 0, 4, -1], [-27, 0, -5, -16, 0, 10, 0, 0, 11, 0, -9, 8, 0, 10, 0, 0], [0, 18, 0, 0, -11, 0, -4, 7, 0, -13, 0, 0, 20, 0, 7, -2], [102, 0, 4, 4, 0, 7, 0, 0, 1, 0, 13, 0, 0, -9, 0, 0], [11, 0, 4, 6, 0, -3, 0, 0, -7, 0, 1, -10, 0, 0, 0, 0], [0, 0, 0, 0, 39, 0, -14, 1, 0, -10, 0, 0, 9, 0, -10, 1], [0, 3, 0, 0, 0, 0, 20, 10, 0, -14, 0, 0, 20, 0, 8, -1], [0, 6, 0, 0, 25, 0, -24, 0, 0, 7, 0, 0, -9, 0, 5, 4], [0, 28, 0, 0, -20, 0, -16, -7, 0, 2, 0, 0, 8, 0, -10, 2], [41, 0, 11, 2, 0, 10, 0, 0, 10, 0, -11, -5, 0, -3, 0, 0], [-66, 0, 13, -5, 0, 12, 0, 0, -5, 0, 0, 11, 0, -1, 0, 0], [0, -4, 0, 0, -37, 0, 22, 3, 0, -6, 0, 0, -11, 0, -2, 3], [-1, 0, -6, -18, 0, 18, 0, 0, -12, 0, -10, -22, 0, 1, 0, 0], [0, 19, 0, 0, 44, 0, 13, -4, 0, 17, 0, 0, -15, 0, -7, 6], [13, 0, 4, 6, 0, -19, 0, 0, -9, 0, -17, -16, 0, -2, 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0, -42, 0, -31, -74, 0, 69, 0, 0, -79, 0, -85, -18], [0, -112, 0, 0, 113, 0, 58, -11, 0, 49, 0, 0, -12, 0, 65, 10], [41, 0, 40, -75, 0, 38, 0, 0, 14, 0, -24, 106, 0, 48, 0, 0], [204, 0, 2, 22, 0, 228, 0, 0, -27, 0, -39, -32, 0, 22, 0, 0], [0, -44, 0, 0, 214, 0, 36, 37, 0, 76, 0, 0, -142, 0, 14, -3], [0, -265, 0, 0, -77, 0, -12, -38, 0, -52, 0, 0, -6, 0, -12, -71], [0, 30, 0, 0, 222, 0, 82, -70, 0, 66, 0, 0, -78, 0, -58, 20], [0, -79, 0, 0, -242, 0, -173, 41, 0, -95, 0, 0, -51, 0, -89, 19], [0, -67, 0, 0, -78, 0, 197, 95, 0, -28, 0, 0, 105, 0, 49, -79], [-461, 0, -9, -68, 0, 25, 0, 0, 17, 0, -95, -59, 0, 40, 0, 0], [-254, 0, 64, 6, 0, 11, 0, 0, 23, 0, 27, -82, 0, -37, 0, 0], [0, 106, 0, 0, -31, 0, -127, 55, 0, -53, 0, 0, -139, 0, 8, -38], [0, -85, 0, 0, -357, 0, -159, -16, 0, -82, 0, 0, 106, 0, -91, -26], [0, 3, 0, 0, 35, 0, 76, 26, 0, -44, 0, 0, 6, 0, 46, -63], [0, -124, 0, 0, -76, 0, -180, 4, 0, 2, 0, 0, 78, 0, -14, 6], [0, 84, 0, 0, 225, 0, 110, -45, 0, 59, 0, 0, -44, 0, 63, 20], [4, 0, -11, -34, 0, -71, 0, 0, -26, 0, -149, -47, 0, -53, 0, 0], [-709, 0, -46, 88, 0, 18, 0, 0, -56, 0, -25, -17, 0, -65, 0, 0], [0, -114, 0, 0, -36, 0, 151, 52, 0, 15, 0, 0, 128, 0, 34, -63], [-197, 0, 2, -44, 0, 72, 0, 0, 40, 0, -66, -12, 0, 83, 0, 0], [-392, 0, 32, 108, 0, -70, 0, 0, -53, 0, 37, 70, 0, 54, 0, 0], [0, 12, 0, 0, 104, 0, 188, 23, 0, 82, 0, 0, -4, 0, 12, -85], [238, 0, 21, -68, 0, -185, 0, 0, 3, 0, -88, -101, 0, -21, 0, 0], [0, 69, 0, 0, -207, 0, -194, 18, 0, 44, 0, 0, 50, 0, 54, 25], [-519, 0, -29, -14, 0, 55, 0, 0, 2, 0, 20, 20, 0, -41, 0, 0], [445, 0, 135, 182, 0, 22, 0, 0, 97, 0, 134, 103, 0, -63, 0, 0], [0, -126, 0, 0, 195, 0, 38, 14, 0, -81, 0, 0, 221, 0, -37, -32], [0, 102, 0, 0, -62, 0, 138, 42, 0, 8, 0, 0, 56, 0, 12, 34], [-81, 0, 16, 110, 0, 89, 0, 0, -44, 0, 182, 144, 0, 12, 0, 0], [848, 0, -18, -88, 0, 43, 0, 0, 20, 0, 10, 13, 0, 62, 0, 0], [-260, 0, 77, 79, 0, -90, 0, 0, 9, 0, 14, 55, 0, 15, 0, 0], [0, 244, 0, 0, -89, 0, -86, -78, 0, 44, 0, 0, 44, 0, 12, 46], [-316, 0, 74, -62, 0, 112, 0, 0, 15, 0, -105, -89, 0, -23, 0, 0], [763, 0, 2, -28, 0, -252, 0, 0, -30, 0, -8, -26, 0, -68, 0, 0], [0, 154, 0, 0, -130, 0, -94, 49, 0, -60, 0, 0, 106, 0, -40, -90], [815, 0, -66, -14, 0, -11, 0, 0, 48, 0, 66, -40, 0, -48, 0, 0], [442, 0, 68, -6, 0, -82, 0, 0, 49, 0, 27, 23, 0, -77, 0, 0], [0, -98, 0, 0, -117, 0, -12, -70, 0, 98, 0, 0, -106, 0, 66, -16], [-701, 0, -74, -4, 0, 32, 0, 0, 46, 0, -47, 27, 0, 67, 0, 0], [0, -38, 0, 0, 249, 0, -204, 9, 0, 70, 0, 0, 143, 0, 18, 31], [-109, 0, -46, -70, 0, -188, 0, 0, -18, 0, -112, -58, 0, 17, 0, 0], [0, 312, 0, 0, -78, 0, 163, -30, 0, -14, 0, 0, 59, 0, -7, 16], [0, -12, 0, 0, -368, 0, -97, -28, 0, -14, 0, 0, -146, 0, -24, 95], [582, 0, -104, 86, 0, -180, 0, 0, -21, 0, 51, -51, 0, -69, 0, 0], [-29, 0, -28, -52, 0, 33, 0, 0, -89, 0, -57, 12, 0, 46, 0, 0], [0, 270, 0, 0, 108, 0, 95, -6, 0, 90, 0, 0, 58, 0, -56, -33], [0, 77, 0, 0, 32, 0, 38, 24, 0, -11, 0, 0, 213, 0, 13, -9], [0, -78, 0, 0, 84, 0, 337, 10, 0, -130, 0, 0, 153, 0, -53, -22], [0, 76, 0, 0, -1, 0, -174, -11, 0, -5, 0, 0, -72, 0, 15, -8], [-351, 0, 117, 100, 0, 167, 0, 0, 41, 0, 67, 143, 0, 56, 0, 0], [0, -106, 0, 0, -122, 0, -241, 24, 0, -72, 0, 0, 248, 0, 0, 53], [0, -31, 0, 0, 207, 0, 239, -72, 0, 20, 0, 0, 100, 0, -47, 26], [-1047, 0, 60, 52, 0, 70, 0, 0, 72, 0, 13, -31, 0, 35, 0, 0], [25, 0, 45, 80, 0, 76, 0, 0, 49, 0, 213, -92, 0, -82, 0, 0], [0, 204, 0, 0, -221, 0, 148, 49, 0, 78, 0, 0, 47, 0, 42, 53], [-21, 0, -40, -84, 0, 25, 0, 0, -5, 0, -149, -116, 0, 92, 0, 0], [768, 0, -41, -32, 0, 115, 0, 0, 36, 0, -19, 63, 0, -3, 0, 0], [0, -115, 0, 0, 150, 0, -403, -33, 0, -1, 0, 0, -61, 0, 29, 67], [1154, 0, 70, 56, 0, -118, 0, 0, 22, 0, 152, -20, 0, -34, 0, 0], [0, 68, 0, 0, -96, 0, 202, -32, 0, -46, 0, 0, 18, 0, -50, -60], [337, 0, 29, 24, 0, 224, 0, 0, 47, 0, 91, 36, 0, -2, 0, 0], [-274, 0, 12, -108, 0, 74, 0, 0, 44, 0, -42, 36, 0, 54, 0, 0], [0, -19, 0, 0, -90, 0, -152, -14, 0, 95, 0, 0, -105, 0, 21, 3], [-283, 0, -82, 42, 0, -217, 0, 0, 66, 0, -92, 62, 0, 56, 0, 0], [0, 108, 0, 0, 219, 0, -70, -14, 0, -38, 0, 0, 58, 0, -126, 22], [0, -216, 0, 0, -244, 0, 89, 63, 0, 48, 0, 0, 46, 0, -30, -102], [184, 0, 115, -53, 0, -48, 0, 0, 143, 0, 122, 1, 0, 5, 0, 0], [605, 0, -27, -136, 0, -81, 0, 0, -17, 0, -217, 43, 0, 20, 0, 0], [0, 34, 0, 0, -7, 0, 238, 29, 0, -42, 0, 0, -7, 0, 90, 83]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1216_d_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1216_3_d_e();".
// 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_1216_3_d_e(:prec:=16)
    chi := MakeCharacter_1216_d();
    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_1216_3_d_e();".
// 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_1216_3_d_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1216_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![640000, 0, 8243200, 0, 8992704, 0, 3478224, 0, 644369, 0, 63436, 0, 3382, 0, 92, 0, 1]>],Snew);
    return Vf;
end function;