
// Make newform 25.20.a.c in Magma, downloaded from the LMFDB on 04 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_25_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_25_20_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_25_20_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 := [10870990650, -26738538, -323561, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [-1034892417, 10576678, 12712, -50], [-2087927994, 57439676, 25424, -100], [43256659674, 120441404, -224584, -340]];
    Rf_basisdens := [1, 1360737, 453579, 194391];
    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_25_a();"
function MakeCharacter_25_a()
    N := 25;
    order := 1;
    char_gens := [2];
    v := [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_25_a_Hecke(Kf)
    return MakeCharacter_25_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 := 20;
    raw_aps := [[105, 1, 0, 0], [-770, -14, 1, 0], [0, 0, 0, 0], [-53505350, 49970, 81, 720], [2896428192, 1619940, -77230, 40240], [-3703083290, -11351688, -1026500, 19040], [-212258435610, 121164168, 7550788, -1498720], [494541927140, 1219060368, 13782296, -6204944], [6642476738190, -4000888282, 123482299, 14097200], [29066793584910, 37650049232, 1412036264, -102722816], [62762418097172, 143885209220, 1856329010, 396745120], [-13367914429130, 243628254544, 12933056552, 2445430720], [1401555474487062, 712181442920, -23031632140, -4324587680], [-981048278190650, 2731979985890, -21833437543, 766535200], [-2493111081747390, 1921361759922, -43968973391, 14751076400], [1865569067700030, 15326779667656, 31983718116, -67159889760], [46538735490709620, 12678949138904, 1608356028, 42781997328], [62129398277661242, 36325386740800, -360031877600, -98525459200], [20994236520169090, 100961063907158, 4551579249123, -520771602720], [-81713627050007268, -165342580099100, 12014191896450, 563398838400], [-176517759515566610, -114653489212472, 8035911594084, -533085805920], [325058972626185560, -916151083256168, 25529181697644, 518296355904], [223679067331191270, 302602901639754, -35028167349747, 3909329328000], [-1098182629867912470, 2022193747562256, -37763877572088, 19031306062272], [-359467624487945090, -2690599733940328, 33285019698124, 217413745760], [-3908636674219559298, -1423690425647120, 62125785178040, -9092016427520], [5699025745487188630, 2119158272121406, -40910223760729, -62338447292240], [-6982565477341953750, -8478111779008730, -127196694400789, -48025404776960], [12672660606068569370, -20525100772966736, -11027975916712, -53221955945792], [9751396812196737510, -7888532671694928, 149772687294040, 5884458904640], [-51376471521891886070, -33258343158886054, -969350709369115, -156286074110480], [38813006979159682272, 13336325678174940, -824575165631730, -648391222067760], [25639607830104800670, -45483924153082336, -2912140277245808, 269247939234560], [100420623163598435780, -89565893152448184, 7500647813035572, 678328161441552], [56530608320153491650, -453458060598003280, 765825983408600, -1711806200429120], [-3003130071728449348, 207915871465399780, 996254447583490, -2447646703429120], [103244225596860385150, 411125175799076080, 9217458208804856, 1582053174621760], [-162724207982379321290, 184647650806279082, -27934615675071875, -3348862477708480], [202182079350189150090, -373814277126601102, -42426281443445327, 2695165658535440], [340269369236640376590, -1412126506593044272, -2261178290412056, 1322265990941120], [1280733875863913924460, 768514851050792, -59879023434339516, 6760207164872304], [-2561722897562589144178, 455453238758314400, 4115223972845200, -54301413217600], [2537248692931292862012, -4214869180971682820, 58506705699098190, -1579167046986720], [1083028623463365826150, 4429005567086116040, 97984912502719972, -753645029753440], [-3433230707185949419290, 2494765530404421992, -156302859083407436, -7273537254210400], [214642779655364682200, 2283973063759408960, -107457945434211200, 9192575673623840], [3262732980177809073392, 11162436913537180, -201302315111353810, 23018983880475280], [5881740980843197112590, 15721683243770889918, 324999641241422919, -65254466281471920], [-15010590323553069454470, -8844674290747429554, 403201547056214015, 16882432571028640], [3428368663930653751910, 18324704424671507072, 506629748205778304, 54661756531770304], [25683810315859684934070, -10486103623470190696, -105083694331002932, -25777220804263840], [17020845711290809858680, 28061844742579323576, 1058504936209334012, -103250586254080448], [-2364263708227082876338, -40759260823767473400, -874692713299584700, -101753033861866400], [24631074085402053454752, 1496232195202564500, -923670755098449750, 190444354795494000], [-20339242147002183502650, -59893261322593866960, 944299442553537816, -217922162089671360], [-29021932678450191908490, -39027734244992413458, 987784696466342015, -183019989086289680], [-19816725117044658317910, 6617806052079768288, 2217027456056211056, -491381981567345024], [100309776464920287152732, 36733390899747944220, 397538609364476510, 422645409408025120], [-43707398456310439251170, 51810785365339639696, -3230770767784644920, -448993106240324800], [32742045926717974426422, 189766402675728444920, -3086919298948035140, 197326792632108320], [172881220771972642673470, -24708231876144493366, -2825827599777397587, -44083779030217440], [273714118729061076408150, -56755726696497568480, -1893718773141241328, 437959117432090880], [27591811065428255356450, -132552038897330847930, 3971345262209412571, -999615490522657120], [98634127270292065702692, 74157638651765286220, 9330732889599983510, -291340758102178880], [89400933827337476420710, -224861611170238261648, 8544276926931187000, 1440283626670040000], [58257525598405083974190, -58586720474254950632, 5379159100186988428, 1250281808144889440], [201168228418494871917272, -438729380555148253140, 8730140589635860630, -1897197641825951440], [-423461114775500076992930, -57547597518217621616, 29068655040601184712, 4505111326618105920], [270645357970878953340210, -124010024811175212898, -30803328208849299961, 1962582636858701440], [847139649020089732706750, -520916145592592901280, 2744891547687348720, 3140238913785802560], [-936614884209995589173970, 272880210221632336496, 43393237411560508216, -6049258044508763200], [-1170853574476571285074680, -249364215178953241216, -18806122018500021152, -4481104804865882272], [518789251393272993175090, 38664729346636692178, 13025559086516220673, -15811315765010231440], [1002724735028144521341790, 2376602698171273157808, 14863383346323100504, 3305591050851654080], [-1228215047772918287960740, 3215316391028578960912, -23593008562804513736, 6549473392390972304], [2415392653523515700893470, 1684077506149445457614, -59044889974962090737, -950025378530601520], [-292246513352318649301470, -1310322029423754973984, 95975514919117417072, 677966917041940352], [-1402221291561697383915290, -3620995838287433102728, -68470345238923655236, -6339866354906360480], [-1937529303153885112447998, -3104135178529744506560, -16716125224667872480, 10784987388311578240], [-2637136732708171244022130, 703920343032096402424, 211720759398271484348, 6778648123919507488], [-252909788919140937894660, -4092698264538957788632, -199943275346205498364, 6381108871466848816], [2307807540540095159614082, -3512117633971607587280, -89773565866273451240, 30924730556297213120], [-6857041617043982771879028, 1498781091380338048940, 377974404590396215270, 24942208759758206240], [-10193355155421749929760930, -4041612255541391989736, -147496568422829055732, -26595255211831332000], [14252730452651923444869080, -969486636153222210944, 88859428928995639072, -40524738957427864288], [-2188094024816379411439050, 4914532788039612488770, -49559947970059766263, -56070930751291168160], [-1439412306400276228071450, -4744499268807246595480, 171714572279716152180, 24159232734665359200], [4541325743572966499008150, -17671464744595367840000, 37546295061649904256, 39889356401133047040], [-9767227248755895522528858, 15694452268803100258480, -302251891590523197160, -50834402560666449920], [-24423466350978761147072090, -8031298258303949287658, -77621930256551510765, 19789360082125418000], [-5806233025840901117098110, -17632221590693610190202, -95813019550918376837, 81766507418156016800], [-14015987023653106211852040, -13064053037959233272648, 51085631369613800444, -11244085642793633216], [-32518572286854904011162830, 23529013135881713183754, -431850532202253528483, -60548029366396487280], [-2758873661637898747819488, -335712787445350268940, 147070349375691264730, 132235972538063083760], [6864761607139273322473700, -37339309659371787009280, -524005360060668955440, -99449611871948233680], [16216873918742117249201430, -14359419849676676342194, 1360527399090098289711, 74125716708021502320], [-10887045596635558012698930, 18589160074347637603264, 608548023833820855328, -321819254635889703232], [-13425908060054011653166518, -17522645600106398343680, 826480710355106146560, -102925206217682465280], [-31925298487567708353299810, -23348079230897387194182, 899862094987567562149, -353482369498908052000], [45801855331349749340542262, -3739463595880117595760, 142011894795282038920, 421433949434955175040], [-33342303510899012753420390, -5901622381647529468178, 1653418375163814202959, 446207918784796586400], [23431791586368080094284550, 8272398035720328668080, 2120846863325584578776, -417572787319681924160], [79500664049085003296898510, 72124570785816413524882, 3618391523530009851665, 204233643451468982080], [-112107820242525050666610, 145945521130961302867928, -1014727255847975759444, 601661789866556344736], [87282776449670210822431832, -57697777633797335053140, 2488534612819968156630, 44405729763968512560], [-74257493549773986014918570, 43864646625858519095776, 1427346815136472795760, 140464997804930659840], [11916032868534757954288170, -74769850910598846385306, -2097121608639216962933, 391265624128864887680], [128339108842070110198639950, -89897653310260175412160, 1431197461260302471712, 912874491159142129920], [-173149757694035637307992000, 66140565295967346634520, 4461573605354316963020, -718439319995425315040], [135775603829869167810996902, -63954288652279781580200, -733788455309501070100, -111415382363244783200], [84893365296464095013996050, -27934564874265911788750, -1488169776931610920399, -1294036177461803697680], [-89160920325634133240942090, -81151062978801649298088, -11348594373229657350740, -95490049527857314720], [6179064949576544598059190, 26964299716687761151288, 2233112316090617142428, -2175345346148128942880], [-89192356690456500324650260, 16795015488130842607648, -2188027277160403976704, 10992915170281954576], [-130871530139940874065487828, 233417043554610167680980, -7067375541970730067910, 4350306577457822080], [620415941648552713662802062, -103416193155599293918520, 4531701786808603234340, 519547316993901098080], [12204230179703799901420750, 10587363884673635620610, 8967329090936954117577, -352349285935854781440], [-117693532057516707805990590, 196118115563834418978042, -1286641055117183660651, 1293480498214281322640], [304541747777625537607910430, 93197508423212705061576, 162417069820670542436, -4074766817228526072800], [338922537802441931784169020, -488749622280966358382536, -1473153094325204194932, -222646012951178925072], [-65625142207043676974246158, 58260135467119812074720, -4840659802646418276240, -2122859094752380298880], [98023528379554972149318190, 210075326739001847974888, -10108416742738715953676, 4376220376815924286880], [27509010124512519693726630, 449787280360571069591736, -17265553665897193993380, 488390979674317732320], [-683762547318094062240704730, -595320935337347468727966, -11547252046411716561447, 2146390926024886741920], [89115143696383558868823512, -160730320737265808074620, -16264456416753963635710, 3258912364670800582480], [-604454675077466493514576998, 90348861121668772508720, 45013915283388428596760, 87987762513281149120], [289191732086635733630758070, 25203754742190906711504, -11179130106080980470232, -2908244128339786982912], [713245038528056138799256440, -614295928846180331233512, -19289940320948893933524, 4763439645300399166656], [1473338844776474354457465130, -75722207293620688569294, 1503904872529201414545, 4504664597871769232400], [679711972597398854598751870, -245930789182128088761216, -20151601635185668003392, -2572079710479310488960], [677672141930956247594768780, 1853368880448498830670936, -894552585920301494308, -230238940510969012688], [98902585145617555894359150, 1416509484609655737886630, 32975750384549051975547, -4537347077451148662480], [-396806918835957007708277548, -458529435217595442425180, 52032731458247548735810, -12965986898237303949280], [176572208214180783767356750, 354043973070959857266080, -18125869440823289257264, -3527698458371537141120], [-360515281791647605199365158, 290739357814101337647680, 61110055795458492113440, 4784156459002754401280], [-2341590906876879841107702910, 1026295082425784914031888, 1631391692037603419656, 14788409778419939907776], [83530983110944064141701590, 159848485341722697772808, 54169714185978292775044, 6201448656392337969440], [-889275069648968078185351430, -1607076791027696146361186, 43080512156896281061287, -9490735105282574038560], [426258774465697041557837310, 1187771048725074227936392, 28920815884081408453444, 15869025869214891756320], [1616935800285213792766386570, -2224445398325173287589376, 136299714255294949663488, 15008776294345793653248], [539456704814909343282652592, -3068467276642425774169740, -13360232943775367077670, -8273197121531700777040], [417254225800597005773355282, 1155226502459626019318480, 48947066013050070868840, -7309467592915782665920], [-233064705158362655461448210, 1115314730515888791414118, -121126268631685532304421, -12473918795087253561680], [349312849078046386329810930, 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8356604357311967276746000], [-20065978807204022325747436055490, 352466809643336430817897686552, -3989930663747293216065412596, 4313677037759121521223900000], [509975469832266277550590541310030, 149986134240258631870175001686, 11599587863152468366328699091, 634493723206596557699973360], [91250702743197030861755521908570, -193602602368592931056133484256, 12522296379799643509418091408, 310894357650779380438862208], [342800785605299251814913626047990, -402445695931483818236038618312, 14668156149996210270054202268, 2109096454334293478248518880], [215092704525455356918065071940930, 95975730927574688340321382866, 217154484921729536547473985, 3162004693014585448082823600], [849773840898577701585836877354180, -59410373140376501802128576184, -11595890176945328264677318348, 2492426578711286784629062672], [-146141059378929303948431299206170, 1165867307229299710266179646656, -6745282866363971085283274144, 3453546816466857033542470400], [590133331339244871916859818978350, -537667774435499456288873539280, -1477041265711038094317393384, -1878104485307931419870773440], [-921842648131672712331509572291890, -517802598401429946302981107438, -13462402618683212902383284655, -618153269294528988932124480], [-45220116260501750588797070587410, 565372410881863149923092131928, 23220591073522415895520575596, -2252932642561406128391124704], [-288634382700978456503416983095368, -114483274165632356989403416020, -12791050905841841335600594410, -4753676495107103847123331920], [294783693079393411391270743198200, 278234830538864850573205007920, 15034211444881772588723463160, -6497681437210768968061768480]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_25_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_25_20_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_25_20_a_c(:prec:=4)
    chi := MakeCharacter_25_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 20));
    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_25_20_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_25_20_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_25_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,20,sign)));
    Vf := Kernel([<2,R![237314973696, 746347520, -2002872, -420, 1]>],Snew);
    return Vf;
end function;