
// Make newform 1680.3.n.c in Magma, downloaded from the LMFDB on 12 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1680_n();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1680_n_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_1680_3_n_c();".
// 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_1680_3_n_c();".
// 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 := [92164, -191418, 155185, -81896, 87351, -121640, 98500, -49514, 23723, -10997, 2453, 605, -535, 115, 6, -7, 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], [-3733556287735812568972, 4061422187595623919846, -1574984849740851348531, 227291479495220812064, -2081777642093942931102, 2132809003228625893332, -925421268323338367342, 468567568782019144854, -269345535648590571303, 69759297553716596239, 13032706252959764862, -13111368657365316102, 2742331780242735387, 181175436945772383, -166629305861773791, 20164875834556192], [-960792939111490724844, -905004298314197108714, 512035600332855575605, -219515096247192064736, 389791842556079290354, -545112784774331855884, 340451203286101107746, -123549104184020541754, 65612820364527807265, -28509207278162011385, 1846138178823774190, 3545289848794874698, -1347743712860600077, 79143204792591959, 59073746351088553, -13315832128803808], [-58763604128606214919009061815068, 131670842043932010093829929246094, -46203664582101451968299121982279, -997146869759303676291781327824, -59991302776270061945409663913478, 64987502795795310503445777258308, -27279004757497234835721254928918, 12859012329887969351220669033406, -7557931495378116458522850003147, 2047079366247964120042697573891, 351500279045276145087660834998, -376636950071278654339502297518, 80363911153991564520095169583, 5062240346976378465338941107, -4837795869840362887322373939, 582366636854799619277038288], [-13094510897902093659282518643064, -2123284095330733218023153318248, -2332235659508133320431484132252, 7277399878649258614896011736553, -8316080536691764063780097332504, 4173261952334018417225284446534, -1373523274579063075077336490604, 800798629550237246394499874218, -151111300778750901285336057416, -134319870208491950394222572437, 73834859004659402732742386279, -5668881192238634124444656554, -5064124786041075996290780736, 1337623935499866783763930141, 48410555193735353235158113, -56047855803089679899249191], [66329118670334543223990843406252, -121930406334169990493494597303686, 49284056043705051458968734753411, -9990891041239390842983902051204, 64667881361699864845135100833822, -65525031293538861698177571836812, 29023689290482498109339464485822, -14891326630481649206094730660574, 8503938559849503377979882470663, -2204197160777032400557374664059, -402896536170218454400510024922, 411611358134911033669080881422, -87105072614453014787413678427, -5385508743974265099941034963, 5244670371690778643109741691, -653440116454697544763613412], [-492657714220523891453, 573474421636053123784, -247894433283840714764, 61493275908093068816, -289805693373966048238, 310385991217264058438, -149532072502703865648, 73528198040302600146, -42065974347296857402, 12145458709178910626, 1445319702582612078, -2057313093604515718, 498297775623043408, 13877382408238632, -27614563320597354, 4045270610244428], [289965463667049565204, -388400377630040507062, 206618222870149974907, -104048739539033487963, 194104553896890261954, -248404394072046332874, 137863064513700905734, -59867045232295275528, 33404637660227258311, -11503770502154506088, -365530518185361819, 1700012156067937384, -503095680267227459, 7533833085618394, 25127507241894272, -4451688264821559], [-10707309166144657288, 14177921096540618264, -7465061177803119004, 3774840891023763161, -7133526464299110688, 9095599701847085878, -4992065728638574148, 2171271284087370666, -1220058645333238992, 418108081976384011, 14598765205215543, -62099095508720498, 18191737755581248, -234643800713243, -914285067944359, 159676076414873], [841040759914296213499749035351828, -841493771606572273244589013367914, 469050913240786226818641810492469, -311123165279843762123603883071576, 671340658958474387624063398089298, -606525839556527507331836690060668, 285792422081724175202651611174818, -139299926660594083063273974071306, 70809200680807455050347273012897, -16833749373144736470106561408481, -3756949765890348309914803642458, 3450582661927746766625895654618, -724304812614252267144934908173, -44883398995602467838053509377, 44374361179901173003936910609, -5801137088075990058597377208], [-4491037380731800344, 5630202289892593608, -2654280103990674836, 1569763229381074715, -2957736086704258848, 3499946812108971074, -1907848441766773260, 847789580697016670, -464651479228027440, 152074997410441361, 8018880147198549, -23442443101543350, 6649936334645056, -48306108152577, -340284802097573, 58391035384283], [9602706660859587148, -12143280040176614426, 5772656403744448237, -3386141928544278285, 6354311637778988206, -7547470420479440998, 4135221539762614090, -1830563395597285560, 1007420739876044865, -334913245560656472, -15248372178403533, 50877737255825240, -14667605669140197, 152615443805654, 742677923572096, -129747753771601], [-485242507977632831758890407727516, 628268830981466886602318219214498, -314657526816117049340947137495153, 186255104169812205162781390301077, -321077379179897551259792566310246, 375014009929425782037736229497526, -206004897347915019890446886537026, 90504373318898356145728652143392, -49482189584922906880304938042469, 16546048960741402398653376278292, 705678060942340881443119030061, -2505549508002572986239494438496, 729017311695202525026913107801, -8916017759648770940655769866, -36731994542040480053631854668, 6495401683289560023231256361], [-707060558216223711587654659366388, 940964504430551428600779815705334, -505307502525204141913801916517979, 270591868157004875851840483771391, -468110463704090733566190668090978, 574943361845736590525921220586018, -313766650554163404615481777818038, 136228883009466998583735477064016, -75822297126330780379825194275447, 25735324481995524284557469285956, 999893867647730361523768490903, -3862127973739119738137582016208, 1124530879532590541146607868403, -13311434045385460842017073558, -56765053011922900737077595404, 9886410451601803380054743323], [-125269909874403727304614321123316, 179885483255873202080730641865494, -108402208483099788959453235922155, 49536617188283954366896124326759, -84494809563511629641301686742466, 113535380980664453553014554654418, -63262411693099497521982236735414, 26828643690815967246585069560640, -15177947660947706743129143444711, 5407989788793236239883566708844, 104168440995576864871694408079, -779629403757712622319862648448, 236542101773868245936164472995, -4536256580487904242556157150, -11666564149750407272616872980, 2096030362470587214732357507], [-1120890612014692344100001067958196, 1436643537905337999856641208557398, -715863633441692442384129425674123, 425709744352383420216912276947487, -736906340273934190965508124713346, 857888076217355997412270905211106, -466624144102926510265160123473366, 205739132080234792794107623718192, -112377371348017334021239349852839, 36935199612116934272954923940052, 1883455371668741087786416420951, -5686351138702356708607555834416, 1621674946892391711069806635491, -13408974053234193973161752006, -82769818937713497542951858908, 14296154769641482994832516731]];
    Rf_basisdens := [1, 728857387274279840960, 330001530540658574144, 10472574836347715713591894692800, 654535927271732232099493418300, 5236287418173857856795947346400, 18239310734541970015, 14621984778467900800, 366523440079104800, 10472574836347715713591894692800, 98008213974502880, 133441380923735680, 4189029934539086285436757877120, 4189029934539086285436757877120, 837805986907817257087351575424, 4189029934539086285436757877120];
    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_1680_n();"
function MakeCharacter_1680_n()
    N := 1680;
    order := 2;
    char_gens := [1471, 421, 1121, 337, 241];
    v := [1, 2, 2, 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_1680_n_Hecke();"
function MakeCharacter_1680_n_Hecke( : Kf := false)
    N := 1680;
    order := 2;
    char_gens := [1471, 421, 1121, 337, 241];
    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 := 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, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, -1, 0, 0], [3, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-6, 0, -2, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 3, -1, 1, 1, 0, 0], [7, -3, -3, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, -3, -1, 2, 0, 0], [-14, -2, -2, 0, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [10, 2, 0, 2, -1, 2, 2, 0, 0, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 5, 0, -5, -5, 0, 0, -1, 1], [0, 0, 0, 0, 0, 0, 0, -9, -5, 0, -1, 1, 0, 0, 1, 1], [-6, 4, 1, -2, 0, -2, 3, 0, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -6, -8, 0, -2, -4, 0, 0, -2, 0], [5, -7, -3, 1, -2, -2, 3, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -11, 5, 0, 1, -5, 0, -2, 3, 1], [0, 0, 0, 0, 0, 0, 0, 19, -10, 0, -7, -2, -3, -3, 0, 1], [-3, -5, 10, 1, -1, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 6, -12, 0, 2, 0, -2, 0, -2, -2], [0, 0, 0, 0, 0, 0, 0, -2, 4, 0, -6, 0, 0, 4, -2, -4], [-10, 10, 8, -2, 5, -2, 6, 0, 0, 2, 0, 0, 0, 0, 0, 0], [37, 11, -4, -3, -1, 2, 6, 0, 0, -2, 0, 0, 0, 0, 0, 0], [4, 10, 0, -8, 0, 2, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 4, 20, 0, 0, 0, -2, -6, 2, 2], [0, 0, 0, 0, 0, 0, 0, 13, 3, 0, 3, 3, -1, -3, 4, 2], [-4, -14, 6, 2, -2, 6, 2, 0, 0, -2, 0, 0, 0, 0, 0, 0], [-28, -34, 13, 4, -2, 0, 7, 0, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -18, 12, 0, -2, 4, 8, 6, -4, -4], [0, 0, 0, 0, 0, 0, 0, -10, 14, 0, 2, 6, 8, 6, 0, -2], [36, -10, -9, 0, -2, -2, 9, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 10, -23, 0, -6, -15, 0, -2, 4, 5], [53, 11, 9, 7, -9, 4, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -26, 2, 0, -2, -6, -4, -2, 0, 2], [3, -1, -14, 11, 0, 6, 2, 0, 0, -6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 25, 19, 0, 17, -7, -2, -6, -1, 5], [0, 0, 0, 0, 0, 0, 0, -5, -7, 0, -13, 15, -8, -4, -3, 3], [-32, 30, -6, 0, -4, 8, -2, 0, 0, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 19, -26, 0, 11, 14, 12, 5, -4, -2], [-27, 25, 3, -3, 2, -10, 5, 0, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 47, 0, 0, 9, 16, 7, 9, -4, -3], [52, 42, -2, 2, 2, 4, 14, 0, 0, -2, 0, 0, 0, 0, 0, 0], [20, -26, -15, -4, 2, 2, 7, 0, 0, 3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -50, -21, 0, 32, -1, 3, 4, 4, 2], [0, 0, 0, 0, 0, 0, 0, 32, 10, 0, -16, -10, -4, -2, 6, -2], [0, 0, 0, 0, 0, 0, 0, 54, 24, 0, -22, -4, 6, 8, 6, -6], [0, 0, 0, 0, 0, 0, 0, -30, 32, 0, 34, 8, -6, 2, 0, -2], [41, 13, 17, -7, 6, -14, 7, 0, 0, -3, 0, 0, 0, 0, 0, 0], [22, 12, 3, -14, 10, -12, 9, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 9, -4, 0, -17, 28, 5, -9, 4, -5], [34, -8, -14, -4, 0, 0, 14, 0, 0, 10, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -20, 34, 0, 0, 2, -8, -12, 4, 10], [-2, 16, -40, 8, -2, 5, 14, 0, 0, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 13, 49, 0, -33, 9, 17, 13, -4, -2], [-2, -28, 26, -10, 6, -2, -10, 0, 0, -6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -20, 35, 0, 2, -17, 1, -2, -4, -4], [-112, -16, -28, -10, 5, 1, -12, 0, 0, 10, 0, 0, 0, 0, 0, 0], [118, -40, 2, 12, -2, -4, 2, 0, 0, -6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -44, 12, 0, 36, -40, 10, 10, -6, -6], [-202, 0, 4, 6, 4, -4, -4, 0, 0, 4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -10, -16, 0, 14, -40, -2, -10, 0, 2], [0, 0, 0, 0, 0, 0, 0, -38, -54, 0, 42, 10, -16, -10, 4, -2], [-15, 71, 28, 9, -13, 4, 2, 0, 0, -6, 0, 0, 0, 0, 0, 0], [64, 62, -25, -4, 4, -4, -15, 0, 0, -7, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -40, 6, 0, 16, -38, 12, 10, 2, -6], [-60, -70, -62, -6, -6, -8, 2, 0, 0, -6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 129, 57, 0, -25, 5, 7, 7, 2, 0], [59, 35, 39, 11, -2, 14, -23, 0, 0, 11, 0, 0, 0, 0, 0, 0], [0, -38, 16, 22, -4, 11, -2, 0, 0, 6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 15, -32, 0, 21, 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-74, 56, 19], [-473, 77, -119, -19, -19, 36, -43, 0, 0, -27, 0, 0, 0, 0, 0, 0], [730, 326, -172, -54, 25, -36, -122, 0, 0, -6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -669, 59, 0, -193, -19, -20, -26, -23, -29], [282, -92, -273, -42, 24, 10, 29, 0, 0, -23, 0, 0, 0, 0, 0, 0], [731, 283, 1, 19, -30, -70, -49, 0, 0, 45, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -195, 67, 0, 161, 5, 72, -40, -19, -73], [859, 193, 10, 3, 65, 36, -80, 0, 0, 20, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 82, -92, 0, 78, 12, 42, -30, -4, 18], [-377, -483, 18, 15, -3, 24, -40, 0, 0, -12, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 122, 16, 0, 282, 112, -64, 6, 76, 36], [0, 0, 0, 0, 0, 0, 0, 69, 531, 0, 259, 51, -9, -51, 12, 18], [-58, 320, -172, -56, -30, 46, -84, 0, 0, 20, 0, 0, 0, 0, 0, 0], [-490, -276, -244, -20, -8, 60, -84, 0, 0, -12, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 310, -328, 0, -190, -52, 12, 12, 14, 28], [0, 0, 0, 0, 0, 0, 0, -555, 233, 0, -15, 23, -34, -46, 53, 37], [0, 0, 0, 0, 0, 0, 0, -18, 208, 0, -206, 108, -46, 40, -10, 14], [0, 0, 0, 0, 0, 0, 0, 3, 71, 0, 95, -223, -22, 114, -25, -5], [525, 81, 59, -71, -66, 18, -3, 0, 0, -17, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -262, -300, 0, -290, 8, 4, 12, -38, 40], [0, 0, 0, 0, 0, 0, 0, 495, -428, 0, -139, -180, -19, -43, -8, -5], [174, 528, 38, 16, -58, -42, -38, 0, 0, -50, 0, 0, 0, 0, 0, 0], [302, 296, 97, 90, -36, 90, -13, 0, 0, 31, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 212, -183, 0, 70, 229, -17, -10, -48, 16], [834, 10, -140, -34, -25, 76, -6, 0, 0, -42, 0, 0, 0, 0, 0, 0], [99, -439, 109, -47, -7, -28, -87, 0, 0, -31, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -312, 194, 0, 132, -174, 32, -36, -20, 14], [174, 420, 147, 82, -78, 16, -63, 0, 0, -31, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 194, -494, 0, -154, 6, -60, -112, 78, 38], [-198, -510, -160, 52, 11, -19, -52, 0, 0, 18, 0, 0, 0, 0, 0, 0], [1334, 108, -162, -30, 26, -60, 6, 0, 0, 6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 370, 229, 0, 180, -63, -37, 4, -48, 12], [-136, 158, -148, 28, -44, 138, 0, 0, 0, -48, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -188, 418, 0, -64, -78, 68, -52, 28, -26], [0, 0, 0, 0, 0, 0, 0, 360, 90, 0, 44, -102, 92, -44, -24, -18], [-1299, 367, 78, -23, 57, -102, -88, 0, 0, -20, 0, 0, 0, 0, 0, 0], [-624, 158, 345, 28, -12, 72, -121, 0, 0, -41, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -228, -108, 0, -124, 28, 56, 48, 4, 52]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1680_n_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1680_3_n_c();".
// 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_1680_3_n_c(:prec:=16)
    chi := MakeCharacter_1680_n();
    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_1680_3_n_c();".
// 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_1680_3_n_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1680_n();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<11,R![470560911917056, 0, 514341392613376, 0, 40011680448512, 0, 1159167213568, 0, 16624844800, 0, 129339392, 0, 549632, 0, 1184, 0, 1]>],Snew);
    return Vf;
end function;