
// Make newform 6300.2.ch.e in Magma, downloaded from the LMFDB on 13 October 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_6300_ch();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_6300_ch_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_6300_2_ch_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_6300_2_ch_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 := [15876, 0, -23778, 0, 1736173, 0, 466066, 0, 420152, 0, 245850, 0, 61094, 0, 5094, 0, -124, 0, -32, 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, 0, 0, 0, 0], [33920116161181250714761357044, 0, 80316655696592074404916703082, 0, 22900518506595840767694204505, 0, 15961035753004667520247318068, 0, 8562457261719110841053877965, 0, 2033233199798072635875531526, 0, 163702792925477506521364603, 0, -4398123134367578735835514, 0, -1022112320867999807552049, 0, 32308541851692959997970, 0], [-11400728353980435096690759624, 0, -34056664004942830546864272210, 0, -13968597626140635539417802534, 0, -10236627401801184921994494043, 0, -4998951634351307222025567344, 0, -1133098999022046341336439159, 0, -88814125600681120679344312, 0, 2556572861614303481362131, 0, 553433092631405348385166, 0, -17647540694373970895875, 0], [35341921893250147003511797044, 0, -102522084009982895927066244354, 0, -14006572300031259580920557636, 0, -19664190277171635545468876817, 0, -14662950727041775778600003200, 0, -3912046035732320613233803673, 0, -336800420532949495648592426, 0, 7712087406652837330170197, 0, 2099481165140393739197370, 0, -65161811999474412936557, 0], [23494506120040896, 0, 474927650772237000, 0, 128127078598397573, 0, 120621237508356240, 0, 70050174007781329, 0, 17325769267903124, 0, 1441088274504443, 0, -35102450622962, 0, -9068037900549, 0, 283399217654, 0], [-12179051382045373554578553648, 0, -131628047318455474143662089959, 0, -38576518497538981130426325397, 0, -31761438005709392778481251843, 0, -17549818009578671486410789991, 0, -4280927890333413525120275047, 0, -353945451679336000624694137, 0, 8735491078009863903964015, 0, 2229852493810084612059321, 0, -69794397672881119112356, 0], [-13662227375949135104046291330, 0, 131713976680023098528366705127, 0, 31560537966735405712653435437, 0, 28683501724779477460815174471, 0, 17664401958814269360428226529, 0, 4536795458966548539271286861, 0, 387211801433937722334062441, 0, -8777308233288331118117033, 0, -2443611510826759817225157, 0, 75788088227606802300200, 0], [7460766495113168261309772564, 0, 90562099840701718170665440710, 0, 27512998257327959923280992055, 0, 24644533314115354607466144893, 0, 13555547310059521290607495859, 0, 3242556956007136588590493175, 0, 262866731439802778583104827, 0, -6884226766393035209787191, 0, -1646803589155321120040861, 0, 51893760101612200092493, 0], [12446516752547494516825709448, 0, -352036240961431888454465906820, 0, -102148520774484540563248825754, 0, -87644111052972408699189809229, 0, -49822307907368554520873254294, 0, -12225991631107217499001968605, 0, -1011125928986045319507769832, 0, 25155006657300303012021365, 0, 6347785454232625075985028, 0, -198852778768454219073515, 0], [-2834994575435452910805709440, 0, -117405219048025184641418349456, 0, -29344439445279232167081187796, 0, -27400864947233280169708380387, 0, -16521185162905153725290675650, 0, -4142756047940174764483250087, 0, -346935636781760073636052574, 0, 8387457662374080604798079, 0, 2176198112612900581668312, 0, -67952663884870359152297, 0], [0, 127736201897648237652192028656, 0, -294737046815084761317147920856, 0, -44435557715061903404884045069, 0, -70111885054324570384262899791, 0, -47217178141690902980082780023, 0, -12272605080436652413709970859, 0, -1049960409805094007497461369, 0, 23345352902072708079023461, 0, 6641563184467364891714229, 0, -205565518947978821305627], [0, -169984996732617443890005745440, 0, -165531305474784369383775180069, 0, -65739248086195427196877728613, 0, -48727592531525174541395783153, 0, -21703914210233155100416738407, 0, -4530267974008922234007914641, 0, -329777678909195435581608171, 0, 11114737304873656011827131, 0, 2048800803537484127194577, 0, -66776901580375410493362], [0, 13610046404874299330544896952, 0, 165770640684965929075230977337, 0, 45529135592876041252071237971, 0, 38920129126580662382809668514, 0, 22663353593165576582453793873, 0, 5669894457988594880607726020, 0, 476025927034618843013406753, 0, -11333881094902634599479164, 0, -2997044603458165165610323, 0, 93435628921980773196075], [0, -4015056159028370226795174, 0, 9263609502833796022687305, 0, 1864753965888026456292948, 0, 1884282186254816189439079, 0, 1270961657495367576498023, 0, 337790421955074549646695, 0, 29382110264107092631390, 0, -635971265414773793637, 0, -185230406622438260155, 0, 5717828107784769232], [0, -866761267532337409688053541568, 0, -5316740339908141443230958073176, 0, -1615923959779619029321386718531, 0, -1359003642439536072043405069725, 0, -742496308940453478442505312915, 0, -178327002999303791744988121213, 0, -14536414281644302446404163427, 0, 374685882122937966399504031, 0, 91219254661405803950355633, 0, -2869214196045874799620129], [0, -375438791639927344752860396256, 0, -9796986758439818079102835845678, 0, -2676348390525585518720054700739, 0, -2413473197880181895684533675299, 0, -1395661319946195027430324480625, 0, -344040023595086933488571131201, 0, -28513612752600380513011987819, 0, 705932654631384378504998161, 0, 178952948596735997342200407, 0, -5602776363847065315533827], [0, -44403084984683876065365542652, 0, 731315130801089997719596886172, 0, 193144669204353422964557611535, 0, 177872344848349618375684533345, 0, 105235267862861971946148567751, 0, 26262427474980934342812476180, 0, 2195172039678450926327446253, 0, -53122652010458995546431809, 0, -13788424905046485215041485, 0, 430606325622996454228988], [0, 26221644576038746043042404932, 0, -5392061131610184559586498471193, 0, -1448975715765804930253127991004, 0, -1296735450174140332449675256452, 0, -757705867359364058874658025864, 0, -188138076887743821377130192364, 0, -15676848862100273463309696052, 0, 382444226743271587105999222, 0, 98459359084757726288491326, 0, -3077593291959846971809627], [0, 590575768120326175748827522344, 0, -5392779137240729238660865862997, 0, -1388345378285846772418708519078, 0, -1267313059959306795973916912535, 0, -760584185508161323320769192262, 0, -191556956339682839316929626501, 0, -16115593606476543685605091798, 0, 383101658113358522868955321, 0, 101304090484519769403738564, 0, -3157569473984663059917766], [0, 4324209187731480051319962, 0, -121674095414209444666564416, 0, -32729155652799489185809489, 0, -29482754868668324218405626, 0, -17261006492966728710015761, 0, -4295557123763166391553653, 0, -358660417631778702430633, 0, 8691281798968788639220, 0, 2254091072569379790567, 0, -70405746000355188301]];
    Rf_basisdens := [1, 13604181794436856575343224564, 4534727264812285525114408188, 13604181794436856575343224564, 52305219817862796, 6802090897218428287671612282, 6802090897218428287671612282, 4534727264812285525114408188, 13604181794436856575343224564, 4534727264812285525114408188, 95229272561057996027402571948, 15871545426842999337900428658, 15871545426842999337900428658, 295377987956060509145226, 95229272561057996027402571948, 95229272561057996027402571948, 6802090897218428287671612282, 47614636280528998013701285974, 47614636280528998013701285974, 886133963868181527435678];
    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_6300_ch();"
function MakeCharacter_6300_ch()
    N := 6300;
    order := 6;
    char_gens := [3151, 2801, 3277, 3601];
    v := [6, 3, 6, 5];
    // 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_6300_ch_Hecke();"
function MakeCharacter_6300_ch_Hecke( : Kf := false)
    N := 6300;
    order := 6;
    char_gens := [3151, 2801, 3277, 3601];
    char_values := [[1, 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], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]];
    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 := 2;
    raw_aps := [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 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, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 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, -1, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 1, 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, 1, 0, -1, 0, -1, 0], [0, 1, 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, 0, 0, 0, 0, 1, -1, -1, 1, -2, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0], [0, -1, -1, 0, 0, 1, 1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, -1, -1, 0, 0, 1, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 1, 1, 0, 1, -1, 0, -1, -1, 0], [-1, -1, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 2, 0, 0, 0, 1, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, -2, 2, 0, 2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 2, -1, 2, 0], [0, 0, -2, 1, 0, -1, 1, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, -1, -1, 3, -1, 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, 2, -1, 3, 1, -2, -1, -1, -1], [0, -1, 1, 0, 0, -1, -1, -2, -1, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, -1, -1, 0, 0, 1, 1, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -1, -1, 0, -1, 1, 0, 1, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 2, 0, 1, -1, -2, 0, 0, 0], [3, 0, -1, -1, -6, -3, 3, 0, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 1, 2, 1, -2, 1, -1], [0, 2, 2, -1, 0, 1, -1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -1, 1, 0, 1, -2, 2, 0], [0, -2, -1, -1, 3, -1, 0, 2, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1, -2, -2, 0, 1, 0, 1], [-2, -2, -3, 3, 0, 0, 0, -2, 2, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, -3, 4, -1, 3, -1, 0, -4, 0, -2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 1, 4, 0, 3, -3, -1, -3, 1, -2], [3, -1, -1, -1, -6, -3, 3, 1, 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, -1, 0, 1, -1, 0, -2, 0], [0, -2, -4, -1, -3, -4, 3, 1, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-6, 1, -2, -1, 3, 2, 3, 3, 1, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [3, 1, 3, 2, -3, -1, -2, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -1, 2, -2, 0, -2, -1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, -4, 1, -1, -1, 0, 0, 0, 2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, -2, 4, 0, -1, 1, 2, 1, -2, -2], [-1, 3, -1, -1, 2, -1, 1, -3, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 4, 0, -1, 4, -1, 0, -3, 0, 0], [0, -1, 4, 1, 3, 4, -3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, -1, -1, 3, 3, 0, -2, 0, -1], [-6, -2, -2, -1, 3, 2, 3, -3, -2, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, 4, -1, 1, 0, 2, 2, 4, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 1, 1, 1, -2, 1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -3, 2, 2, -1, 3, 1, -3, 1, 1], [0, 0, 1, 0, 0, 1, 0, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, -3, 0, 1, -4, 1, 1, 2, 2, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1, 5, 1, -4, -1, -2, -1], [6, 1, -4, 3, -3, 4, 1, -1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -8, 3, 4, 1, 1, -1, 0, -3, 2, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, -2, 6, 0, 3, 1, -1, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, -1, -4, 0, -1, 1, 1, 1, -1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 2, 1, 0, -1, 0, -1, 1], [2, -1, -3, 4, 2, 1, 4, -3, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 1, 3, 9, 1, 2, -2, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -5, 6, -2, 0, 2, 2, -5, 1, -2], [-6, 3, 3, -3, 3, -3, 0, 4, 3, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 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-12, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 12, 2, -8, -8, 0, -1, 0, 2], [-15, 11, 0, -7, 15, -7, 7, -3, 8, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, -7, 3, 0, 1, 4, 2, 4, -2, -10], [-4, -4, -10, -10, 8, 5, -5, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, -6, 4, 10, -7, 0, 1, 0, 1, 5], [0, -2, -10, 4, -17, -10, 14, 1, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, -4, -6, -6, -4, 4, 0, 4, -6, 6], [4, -3, 6, -6, 0, -5, -5, -3, 5, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -3, 2, 3, 3, 11, 0, 8, 0, 6], [-17, -11, -9, -9, 34, -3, 3, 11, 12, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-6, 3, -3, 5, -6, 2, 5, -3, -6, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 11, -4, -7, 3, -4, -3, 4, -11, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -7, -2, -2, 5, 5, 0, -7, 0, -2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, -3, -2, -1, 3, -3, 0, 3, -2, 1], [-16, -2, -4, 4, 0, 4, 4, -2, -11, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -14, 0, 2, -11, 9, -3, 5, -6, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, -1, 3, -8, 6, 14, -4, 7, 3], [26, -1, 7, -2, -13, -7, -5, -5, -1, -5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [19, -4, 6, -6, -19, -12, 6, 1, -3, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -6, 4, 3, 2, 4, -4, 0, -4, 9, -2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, -9, -6, 2, 10, -1, -1, -10, 0, 4], [-3, -15, -3, 4, -3, 1, 4, -4, 11, -11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -4, -1, -9, 25, -1, -8, -1, 4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, -5, -3, -5, 9, 14, -4, 7, -3], [-18, -3, -4, 14, 9, 4, -10, -8, -3, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [28, 7, 10, 2, -28, -8, -2, -6, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -5, 0, -1, 1, -1, 0, 5, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, -4, 5, -3, -2, 2, -1, 0, 10], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -6, 0, -6, 2, 8, 0, -8, 0, -3], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15, -5, 0, -4, -1, 8, -4, -3, -8, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -11, 0, 17, 1, -16, 2, -8, 0], [-16, -1, 1, -1, 0, 12, 12, -1, -6, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 3, 6, -2, 11, -11, -14, -8, 0, -4], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -17, -1, 17, 0, 13, -10, -2, -10, 2, -11], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 1, -3, -1, -6, 0], [-36, -3, -13, 3, 18, 13, 10, -1, -3, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-9, 2, 2, -4, 9, -6, 4, 14, 16, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, -1, -5, -4, -2, 2, 0, 1, 2, 4], [0, 3, 10, 8, 23, 10, -2, 2, -3, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-13, -8, -7, 7, 0, 7, 7, -8, 1, -9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -14, -1, 14, 0, 5, -10, 6, -10, -6, 5], [-4, 17, 2, -1, -4, 1, -1, 17, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 1, 0, -6, 2, -6, -1, 5, -2, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -6, -8, 2, -3, 7, 10, -6, 5, 2], [12, 6, 5, -4, -6, -5, -1, 12, 6, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-4, -3, 0, 3, 4, 3, -3, -15, -18, -18, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -13, 1, -2, -9, 6, 2, -6, 2, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 11, 1, 9, -1, -10, -2, -5, 1], [8, -9, -1, -10, -4, 1, 11, 4, -9, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 4, 6, -2, 9, -4, -13, 0, 0, -4], [12, 3, -1, -1, -24, 5, -5, -3, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -6, 2, -6, -10, 3, 1, 0, -1, 0, -5], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, -9, 2, -6, -6, 0, 5, 0, 2], [-3, -7, -9, 9, 0, 14, 14, -7, -6, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -15, -3, 15, 0, -9, 5, 7, 5, -7, -7], [25, 9, 15, 15, -50, 8, -8, -9, -10, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -4, 1, 12, -2, -7, -9, 7, -9, 6], [0, 11, 0, -10, 10, 0, -10, -5, -11, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-4, 13, 4, 1, 2, -4, -5, 11, 13, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [27, -4, 4, 10, -27, 6, -10, 5, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -14, 3, 7, 0, 2, -2, 0, -3, 0, 0], [3, -4, -16, 16, 0, 1, 1, -4, 4, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 7, -6, 5, -8, 3, 1, 10, 0, 10], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -5, 4, 5, 0, 6, -3, -7, -3, 7, -5], [10, -6, 3, 8, 10, 11, 8, -12, -6, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -7, 3, 0, -3, 10, 1, -7, -4, -14, 0], [0, -16, 3, 3, -14, 3, 0, -1, 16, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-12, 6, 5, -4, 6, -5, -1, 4, 6, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 12, 2, -3, 11, -11, 0, -12, 5, 3], [21, -7, -6, 6, 0, 11, 11, -7, -4, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -6, -6, 6, 0, 1, 3, 2, 3, -2, 0], [7, -9, 1, 11, 7, 12, 11, -2, 7, -7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -12, 6, 0, 12, 7, -13, -1, 7, -2, 0], [22, -6, -2, 3, -22, 5, -3, 16, 10, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -9, 3, 18, 4, 0, -15, -3, -12, 0, 8], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 0, 12, -10, 7, 13, -7, 13, 6], [21, 3, -11, 14, 21, 3, 14, 4, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -6, -3, 11, 28, -3, 14, 7, 6, -7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -14, 23, -5, 0, 8, 8, -14, 4, -5], [-46, -6, 7, -4, 23, -7, -3, -3, -6, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 1, -6, 3, -1, 10, 0, 2], [-5, -10, 8, -6, -5, 2, -6, 2, 12, -12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -10, 2, 0, 3, -7, 2, 9, -4, 18, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 1, 3, 0, 0, 0, 10, 0, 3], [6, -13, -9, -1, -6, 8, 1, -5, -18, -18, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [18, 4, -12, 12, 0, 7, 7, 4, 10, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -6, -5, 12, 1, 6, 8, -1, 3, 0, 2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 3, 0, 8, 1, -2, 2, -1, 4, 0], [-16, 8, 10, -3, 8, -10, -7, -4, 8, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -26, 5, 13, -4, 13, -13, 0, -5, -10, 4], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, -4, -3, 17, -7, -20, -4, 0, -6], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -5, 1, 5, 0, 7, -4, -4, -4, 4, -5], [-7, -8, -5, -4, -7, -9, -4, 1, 9, -9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 17, -5, 5, -5, -10, -4, -5, -5], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -10, 11, 5, -4, -2, 2, 0, -11, 18, 4], [18, 6, 5, 5, -36, -1, 1, -6, -13, -7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 14, -3, 4, 9, -1, 4, 1, 4, 2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 18, -3, 0, -9, -2, -4, -1, 7, -2, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 11, 0, 2, 8, 6, 1, 3, 0], [4, -9, -15, 7, -2, 15, 8, -1, -9, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -4, -1, -4, 6, 4, -6, -4, -6, -2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6300_ch_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6300_2_ch_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_6300_2_ch_e(:prec:=20)
    chi := MakeCharacter_6300_ch();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_6300_2_ch_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_6300_2_ch_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6300_ch();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![5184, 0, -45216, 0, 298768, 0, -757952, 0, 1428832, 0, -645992, 0, 219724, 0, -20576, 0, 1408, 0, -44, 0, 1]>,<37,R![21025, 25955, 66551, -59422, 46117, -14307, 3781, -418, 59, -1, 1]>],Snew);
    return Vf;
end function;