
// Make newform 387.10.a.c in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_387_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_387_10_a_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_387_10_a_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 := [529838441422848000, 2103623681144094720, -1349125586394823680, 139803541742443008, 10892588268404224, -1516975415483904, -32493903147264, 5986976782464, 47465836576, -11474166224, -37600244, 11288030, 14888, -5425, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-724, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1110527448043560235826201055638316769378741582409903970666219405312, 1356198803054744058880725885871710929129050454222878047067288141824, -186327181218290901073747973484539596505883407001061956082172183040, -3762932457742440301384762221432514190941083153087214714185874944, 2067534908039729428110834447028604182925835831227205429369049600, -27089576933960873936848057662235560694600489750451165613307904, -8905443425394453543593003103816714237142925959168682540120960, 118620734045953406704668670987462501928492061476165614414336, 18697312115211009705272681307726718408088833870647582627824, -161569501491201777577531756044683346939395699520488752032, -19720475246457293192425076434009405267522823079638502326, 92603739004608564642471741918761267701589476749500454, 9954292327005961039452847231312554199007948246746655, -19627755240831137658665568701533785248645884882496, -1901841900496055915643906245540679324583056769231], [-11079058360317895211235450902937844791975468861144848680310540697600, -20730155145536635969597373397288470519450550203040865789964499517440, 2687950061151847488799873542370413356044202383829219820028905733632, 241850797331785625734490349465582009494136563840359833756001583616, -21711710135376913253053319678223929266036181121380272533236051456, -1192624272682889491680374738176689328292111650402965681689348096, 68240048436046740233191574269910638203057208344282782327149696, 2839520485052722893134714719131678637478227921092806345895424, -104585964691360306833014401237676102711549878771477876147216, -3271510575051164866415844769551363627312690498880039441056, 84481814241137150964558901502602301155393833064127832010, 1747637612374536876442971929865085048435696075564125222, -34544246114389407585862627115083602454723912549326625, -344618622461123669949201999713174215769427943924928, 5551989946177116308108559072515228322313129698929], [5434382468522185825114515169023595231602450246515875473377855766528, 1006123851015959443219723613499478358919886160992587354588854452224, -576396317322683238909394807635064346878244506778707306707656676864, -31235142904816696733034608067768657941017733916749128065636354560, 5417704779855254218469380826987417081508325175098420443311871488, 184797917891881793740525994524982302496234993134886441412608000, -20191825333086791580700446597825008568072394208059672180329088, -446574266021681778467460807018571873714633671266937781788672, 37119936985888288112605443519936241975295454308230382756432, 505754149702289160470628201554228173553846410424636637088, -35427296278293116038903046255811654518967379104745627730, -263774816083800183060843125418066857499319848947258014, 16653149768162297296563746155491111007490375592824853, 50207687936548854658620190939064520440036393134848, -3011925103805084953459790250243660827745395157541], [15885128855385420551016271506553964509498507148525871717341985669120, -31974687093722774438028054094910066484179158707314040162949048074240, 3760271048440522655139893817707938795835500103540408762057069243904, 351169486993015717068454025644562207111449480427863562165359683072, -37000568408626715097782047579185346811794688807190416368859072512, -1454359620342371747695085611202087232115752133513769882159395072, 130999623180733084462759752341588369727186944322315632851407872, 2907709273650235038681441257506931664759432396909521598496448, -222403045392219857728879488028480609975021626751238915032112, -2943973824084915692947641434333459809594881075427186279632, 195790934647835336651552358386980569368125109972480574530, 1432447584660241897968339831967235410718800520832053574, -85753028476919467454406047928728305056193288774933805, -261653727299656711371592842751087005252236906413336, 14647328733937523133282173254928879207821325397133], [-2870350031706211487470245902428330270615792222760951486994429542400, -20334506755955888362720491943900506249608779863199950782635868160, 878593294998482858040587712150757459657931063660430104950434789888, -68382753185006627712144187584060511979826362983881672122431863296, -9958687659823870076622774152064687244665509488342833616342444544, 543336643041307996004100232408441368105066682646221127782844416, 44418932219851152122040922108057841592107002579275074520294784, -1483314315052579346906930521168574104103238983963101943678464, -95557419024572890192285599697838842714310780898475882515504, 1766974812031886079567573915251150942074424814612824374816, 102080560131022005593077687706083077679435823880623412510, -954504096536085547575733508538744755569931685153593422, -51852398249550380166452885563390577460335990485900355, 193167025451001150495773582530488568554189730186688, 9952721170712397222530360805183437940986499513651], [51059093907701967125984133672411754348812300469756618115193940377600, -26018912753340128679439039477371482378549503098807391055682786263040, -7772094323187519313473452557515766506091982175489910203835755999744, 1172075099384437522233796165566574406162165733223330746291673256448, 123728729727521197231301339621349877075204684833566485804243784192, -8540734962780109424093366351597766478466734426737272071489343488, -674918848651580146199686218201309063374080165566871470463503232, 23017680788353414972683137305071521213809311352540750878313472, 1613863374582779978752433171553591056000807682970888558414192, -27330034321537450075442295759749130864947933135065037437728, -1823017332385664662163136909599687709794279461165000746310, 14760612741285841977611059046041299884758068815548556406, 955688608782292712290310590606888635993629985243567175, -3003039720366819404632507502744316852146920500402304, -186841917534807503207570319097396304810992550884343], [1622288691707504800726202464461669070123229871788561939534696448000, -8383791696359477167286847618320286769006645313901708918953556541440, -1172474426782093025994551081924750425669913316689076793411641494016, 243295617155336509425951628353569325024588419525642246278259999232, 20336527234404991879065230954786807334451220856943733446556098048, -1624238428726294478895886173008623780424444043478949562026143232, -113174810006589903814291236983256774240388508948757907446912128, 4203895274092952400813919255718570044968482866002822222425728, 272907561840514713357370563733046441806968487353337886098928, -4879547341103860064345560848513035893701227662805434343232, -310079819971867914888667947406086426530148720097858908830, 2597919946821959953752260647234606740720480189422074974, 163348176433577024875966261337740345569668730104759555, -523759940600990911836036781470059752502270921143376, -32065829982562344870671716528964611266615638707987], [-66781281339360559960684982804099945561959597862664629660762489978880, 3213452300681245282922048994936832408497257164332154610678642933760, 9541926600190016345235688291612170158977930985769762865227741292032, -865957760429633708728336999536605115548552057013929827520288065024, -119757546558340353291922139650380979692683467771951170893199074816, 6854163234254094134986515429311992128251216154415376951179853824, 587947199564308850094502167141760638769878300638618848554342016, -18864726262337000398250199741978587488200920544671145141313536, -1344064767957348591622747933297819151406649346727931241544656, 22597639805050304447796091447275423533663866363664610461024, 1488618189776194683585710304101198662891297786514226139970, -12266574091875281685901525749568959801130097453606298578, -773336860982568251190934212633711371494385800528807725, 2502509482448108471179720605232212463426125050971392, 150570197122404124311993731325214564662155240166589], [-5027230581682130323613458475713294330702615895814147668417492582400, -115031997748532163483919734275294937547849631256602004269551333048320, 35251489935604051520705822307749291089635345768071218589943250622976, -299494517460765013285368516879886869226005703001698995076471675392, -408804352281798204862407346667410780935269965688929573288058464768, 7003648719909439414274896521059205660510941866468620433878505472, 1823745034006882786903617114559966636360560194813151577011836288, -22750266877214878308072097624166557767828235181101007547294208, -3894285479984205073490911720981546107627486827634724858108208, 28533148333968809348432062417281089036719923457200619375392, 4141086832024862486221350544265456258097870137224612730750, -15802456352462025340166042105638213782793163413744057134, -2099523286291658025586015334715158322139783641262050995, 3325384657726970751832987387397408556422017912836416, 402039766973354451222385415412162245564175334497187], [-266414635305149054614521837956110802168001234260477026115681304576, 273699114052456738245454652922155088016432653925376091185983168512, -70663642878398205503984535480032057909529081938050216496474237184, 1201134170719397007152072831942823623075164288415623861304282496, 860033748637011396923953900324065361424322925109700287412711104, -20159595770453293327761529463011231859641722519159210077736352, -3919994408520864764171858722823955609161488709725174174979728, 63095262765671257733165793074504564875327424732206765049720, 8445253768686217432393306290197363828150725080686955556084, -78623263604837484577320139902740974420343998444149464334, -9002314516041848516942142602130561939503113306399402330, 43495602827493245302594122093209564508419548994825697, 4563175310680287732570826794242526272036411426571817, -9073951367411641286571335566824849108043048110295, -873217222539872122743387287209568605159412721479], [-151867208722893184471595898810680124274122696013804309075449534382080, 297698222192430152715486710923707653718795197402485964834352173383680, -87237105393271023104396561583566944129869634189619395617443593600512, 1772604787007294460269970125137634112844324189105761279394647350784, 971935870047495963491089425868842268061135574919035521722340548096, -25665389999641056997398052841093818605090711362898212271698776064, -4190112205327058816434306465585356351271899212931508569672910976, 79243315788205238110745829722386736489610761393368136574278656, 8726044824708867782690125671188514662749518595863515222275856, -99069596559772052369959243252503874294029389079782800395744, -9105589153925163586423437654598344236750407376049539367050, 55038640824215210234530676506851617222170338519358904538, 4551728166268888751817095375714617811415831376475714945, -11431949762874608854174202661414401474871800300166272, -863436205791787753131437597976661742261725617545809], [-60119390818457537079347195724363465144021628608904702182926352384, -9405574690022232355051763687426608557483521798151130808754536349696, 3351237379840623652943281595188795355853597911984797010880349297152, -95572211758316172594103097706535009427403232952946043208266836480, -37961171931006091333920554780712971705575650305617458819109093888, 1110677519707941937823407132140202290078251802432958303834397696, 165934514734014781641895421391732306495450107171371784199645056, -3292072142541374261597079361193783212900848509985750194285824, -348003959002662736716176936971688352945078466125098659550384, 4043487747662761211027415067849796422400028337236990569952, 364373874185285986401106983955513549502314877154438293150, -2223230737789304949436052533787721826941398574307085742, -182447269665018467559802196692579656430898104741060355, 458647090259410208477100495530627686612344608402336, 34635041782844543452364005105104342522949763916211]];
    Rf_basisdens := [1, 1, 1, 2583764471268399651658552761554592937602971958275852833363329024, 4844558383628249346859786427914861758005572421767224062556241920, 2583764471268399651658552761554592937602971958275852833363329024, 4844558383628249346859786427914861758005572421767224062556241920, 3229705589085499564573190951943241172003714947844816041704161280, 38756467069025994774878291423318894064044579374137792500449935360, 4844558383628249346859786427914861758005572421767224062556241920, 12918822356341998258292763807772964688014859791379264166816645120, 19378233534512997387439145711659447032022289687068896250224967680, 40371319863568744557164886899290514650046436848060200521302016, 38756467069025994774878291423318894064044579374137792500449935360, 1291882235634199825829276380777296468801485979137926416681664512];
    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_387_a();"
function MakeCharacter_387_a()
    N := 387;
    order := 1;
    char_gens := [173, 46];
    v := [1, 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;

function MakeCharacter_387_a_Hecke(Kf)
    return MakeCharacter_387_a();
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 := 10;
    raw_aps := [[2, 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], [315, -4, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [-644, -37, -5, -4, 0, 1, 0, 0, -1, -2, 0, 0, 0, 0, 0], [7062, -687, 10, 16, 5, 2, 0, 6, -12, -10, 1, 9, 0, -1, 0], [-7905, 1267, 26, -20, 16, 8, 0, 1, -11, 6, -14, 4, -1, 1, -7], [59338, -2978, 2, -21, -23, 37, -37, -39, 6, 15, -33, 8, -34, -6, 13], [-46704, 6066, 73, 167, 11, 1, 20, 61, -96, -16, 46, -11, -128, 49, 27], [166938, -782, 144, -390, 29, 69, -17, 321, 495, -269, 9, 34, -180, 150, 23], [1224774, 22936, -1939, 2139, 400, 468, -543, -116, -625, -81, 87, 375, 255, 564, -130], [-800733, -10806, -193, -943, 89, 265, -304, 1179, -659, -230, 308, 135, -109, 32, -1007], [-577454, -63537, 1034, -10099, 746, 2291, 735, -1550, -5538, -439, 1501, 1335, 825, 418, 1638], [1233270, 83615, -5275, 1341, 1540, 2411, 3240, -347, -972, -2546, 411, 1600, 1042, 1440, 1139], [-3418801, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [6943537, 413295, 7375, -1917, -4314, 641, 1306, 5183, -7919, 1628, 2615, -2710, -3673, -2471, 113], [14436444, -196345, 34069, 15140, -1040, -3140, -5553, -2452, 2900, -1465, -2718, 3653, -8383, 9990, -3490], [-12296489, -521718, 43251, 65198, 4802, 9324, -2801, -15937, 11591, -18693, -3376, -5983, -11018, 4153, -6447], [16692599, -1267897, 24719, 10432, 689, -7677, -306, -16615, -21614, 14710, -16593, -14441, 9311, 834, -8983], [31466364, -2031243, 41092, -27230, 9897, -1458, -8188, 24470, -22008, -7644, -21447, 16709, -21014, 3113, 4634], [402709, 891494, 62584, -44922, 15225, -14308, 30932, -8823, 17091, -21428, 54755, -16825, -42649, 22558, -4779], [-47462648, -1913733, -146775, 64046, 6468, 26405, 13416, 26456, -30649, -26636, 8084, 5796, 11354, 21498, 40602], [37285066, 1469461, 174610, 119572, -24166, -36295, -37899, 64683, -115787, 31219, -26528, 2581, 45144, -88833, -12757], [95761721, 2750841, 95733, -181508, 24903, -25832, 45317, 66643, -79257, -17317, -33255, 27420, 3218, -70158, 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4597873839, 134069471295, 35464313644, -108416713081, 114421536714, 436040249518, 103471961817, -285274795781, -194862022197, -181594452462, -418518609395], [-3638004899751, -18442583401219, -876785801457, -2093581587086, -87609964465, 301648487292, -62594158785, -396045756641, -799801882639, -493013093715, -34105471613, 66447523146, 269325843600, 362849177122, -54077022189], [-214844234927105, -53327549900017, -820461954219, 84679439554, -309413455717, -7533796859, -400878126002, 1016708444831, -1639617383082, 310157559050, -303319851501, 347353466069, 148398975349, -23940733910, 191955562251], [783198941532466, 57606410171749, -2624616946839, 421807218898, 195479356676, 188200450500, -249089493215, -307385487866, 17483848820, -241280047717, -570059608892, 33733433199, 338132396147, -344363119594, -470722957238], [-943785278149260, 957766263837, -2861555298162, 1129463035298, 138924424321, 28382635044, -397110579502, 238962556656, -914410830378, 26632916816, -61586022173, -99404455921, 20486298626, 422734727711, 44828111694], [4016507687866, -101597354522821, 263528237204, -92077734020, -146577024278, 50400219263, -385291476613, -36347388455, 562703203539, 303560773015, 162543494276, -327345617265, -376834848722, 31598526663, 184524187927], [-81684278830079, 13027456919056, 219748611597, -2018621307424, -328195409758, 602951104042, -505815833039, 41423041433, -595152837837, -253379057035, 659148781994, -38307193883, 154262858796, 332318345599, 871659670089], [-772740751797912, -56242224327523, -1568349342627, -1096774190430, -285491741348, 114203540939, -395878769050, 2422030958, 277862477451, 132633258056, -981772280282, -99170254328, 435273667944, -480184509280, 193356934536], [1168712321169562, -36028933827992, -1162492047636, -2106154800600, 74823261321, -143461116474, -201933148932, -113417792833, -674491634800, 174111602144, 202907027503, -210399536531, -382154242981, -298305027736, -802856673715], [-147417793289863, -28860653551960, 2429991053021, -1141923425546, 838650500282, 531542613944, 395859186491, -167357214711, -337107970365, -747798085741, -308923186918, 378416921667, -149254169512, 307382600861, -297813278519], [-1310726752855118, -4309924377979, 2083584643847, 2050971008250, 526864714318, 433002367963, 263179722676, -13608960100, -1103444686235, 366412534570, -35552559102, -202302827524, -178752681304, -98282040960, -383668008330], [-320658855289444, 25258489721625, -1395715659210, 1218672768870, -285188790731, 96591418096, -350306282864, 686012361508, 713650082718, 23169675292, -56514057251, 351197582093, 495645019828, 181439400743, 1152154048340], [348995428043059, 82797127954472, -1510632688388, -4163911936792, -195602600891, -235108881148, -239282268368, 263168579657, 1071268557981, -1033567252784, -465116024595, 336228858111, 88473317487, 546218354530, 578628134753], [1434998168122503, 8005682348129, 1571954284391, 1321647725670, 303093656574, 186914124239, 93749764184, 3141951010, 296980155986, 9441155100, 286897964014, 296321969430, -553488045084, 157327178126, -572809763564], [542440590455624, 38449118571627, 2725519800937, 501343112587, -413958340193, -295378044113, -535429306128, 453998807181, -197677380662, -331887995202, -40970698665, 523781169181, 486723539873, 134107192074, 502966260401], [-828297230439043, -28593820795109, 3731612744041, 931706053862, -132788113847, 215222207113, 278348542586, 55424774431, -704889126444, 100347012042, -478306581849, -23228526565, 280272931919, -283013596914, 173070826483], [-625804918415407, -63233324284933, -1524736664941, -291476094336, -275534467479, -256560625036, -2919196539, 144427892093, -1968311614217, 172515808805, -380228871999, 100501730742, 603550761960, -304506848058, 245968366947], [-1693940249321575, 30116615445637, 610309387449, 488218896856, 217293149257, 165972081993, 158424579572, 352650161775, -477697945720, -176656488970, 48152726783, -608593657579, -503014164551, 39414794098, -207554725003]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_387_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_387_10_a_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_387_10_a_c(:prec:=15)
    chi := MakeCharacter_387_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 10));
    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_387_10_a_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_387_10_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_387_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,10,sign)));
    Vf := Kernel([<2,R![-9972342953111715840, 8716693942319579136, -1816851287739662336, 389857664548864, 22534802024577024, -667905460591104, -103575915851008, 3669969902912, 232672713184, -8326611176, -269722200, 9265782, 151570, -4949, -32, 1]>],Snew);
    return Vf;
end function;