
// Make newform 5.17.c.a in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_5_c_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_5_17_c_a();".
// 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_5_17_c_a();".
// 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 := [8484029084015876402584471207936, 0, 7749363027207208037244731392, 0, 1450505914836224854327296, 0, 97982926395671183360, 0, 3062948254140560, 0, 47647865256, 0, 354217, 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], [-2600279260325231550566347833591463936, 135746553974964854480053862400000000, 1806562382916320352001552197156864, 0, 7652456798273282271778114560, 0, -11915381996069728263743120, 0, -433881598453749944040, 0, -5101784131984041, 0, -19395026881, 0], [-2600279260325231550566347833591463936, -135746553974964854480053862400000000, 1806562382916320352001552197156864, 0, 7652456798273282271778114560, 0, -11915381996069728263743120, 0, -433881598453749944040, 0, -5101784131984041, 0, -19395026881, 0], [135754668388390846860513301872069607512473870532608, 0, 16335189004828961281575011564803635137637777408, 0, 1431263943129674722011255835512755052664320, 0, 58957747433427088900053257171606257360, 0, 1100344919748515360239952239086120, 0, 9195159940417656768107794173, 0, 28058340253264616295893, 0], [0, 47578547842678974659740222298405859808765804544, 0, 5145297330466188001440381879902353812946944, 0, 206115998756370451349625440305946741760, 0, 5744514679325030042978957109034480, 0, 108110982248511274839701699160, 0, 1014557318660728928263839, 0, 3487216833586249799], [0, 7777570297778692971188974709013023099223380279461631844417536, 0, 1138450654765652769840900940112779429566832438278540558336, 0, 54043588639842136853049732698769909805295980173757440, 0, 1111002824927610433379823327134703938659899931120, 0, 10171305916701520286777867651692978946174040, 0, 34199400881884988817744138584935607391, 0, 2405870722536653292448513997831], [-1996633667667709663019063939355469730877336732934658989228032, 43059800310567267201774822554928989981789043226943254691840, -689622705908848756734829382656027319315555464681154936832, 10150702221739728709925804196780809757832560243003555840, -67326428213339744693969478043597319534152728533729280, 639619844153782938357419047057103472119842869273600, -2718842384541618826465399098880059367713083677440, 16568137074483598408855410375583642558506182800, -49799700061349292005756311539811914656964480, 201273113472012441954659869957321047162600, -409577358600381368558206625638717647792, 1152694227416835548889678961499758665, -1232733444616685757672206988340272, 2529135187545426514157822119265], [-1996633667667709663019063939355469730877336732934658989228032, -43059800310567267201774822554928989981789043226943254691840, -689622705908848756734829382656027319315555464681154936832, -10150702221739728709925804196780809757832560243003555840, -67326428213339744693969478043597319534152728533729280, -639619844153782938357419047057103472119842869273600, -2718842384541618826465399098880059367713083677440, -16568137074483598408855410375583642558506182800, -49799700061349292005756311539811914656964480, -201273113472012441954659869957321047162600, -409577358600381368558206625638717647792, -1152694227416835548889678961499758665, -1232733444616685757672206988340272, -2529135187545426514157822119265], [4959773392463942189348562695257477821279497901301715686171607040, -35530805153007657289703503493501718114219949834266316309004288, 1125058254771201310960884884778001175914803949088042951639040, -9491367622766212727793087265506615995886868135391745343488, -90984844118916989831435781868314638254564172272518758400, -916388777714806368019656384889579410022196018962283520, -10527345552845864343549104120846339382239778964723200, -36158346965539710459479445955294809526846018246960, -282317326868737552299200295606687669773064134400, -652945645765672509463178399203568828972275320, -2794396398942350486070886806320246487185760, -5341256424589616579639045837776472274003, -9294338364049049028502867716248984160, -16069229694661927508980469881551323], [-2502446753147673875162349146572491463887552019223829093015355392, 36478120759840137168142549589710155893819308785259067912224768, -1187277618465999134076756765111386352601021263580927667208192, 9714683071644486759411454957835793810559184460737823571968, -109293264886885101109093736895036528979228059873427783680, 930460414286189592663519603924835686408832562086302720, -4097375961597092637667807195746332117667574253168640, 36522845981178349624474264983557649663133154268560, -72315533444894232977495289651203659921982714880, 657373654262056783186180916342629892009852520, -585888809354452318078800853626370385549952, 5366615697592786961714618774929466964633, -1753808918988554334858299384552072832, 16124870668787926892291941968175153], [-319196198616598785847789636504610987909956351460565561137692672, -24059507907805277880821279841050402552806623593019415547871232, 142313800589074175174569139181979626096331077560701632380928, -8530298665568866728756012199690129100306791339142288965632, 32873172124673815616450261581169237212359506856790917120, -696592274313279393136716211384571370025691182075745280, 1679802850148936336294775879246886492669579634997760, -23949472920649222274216818749421538905688251047440, 34136124474628678070217036788374066032185329920, -395704428544755030746626956620082574645293480, 297944584944109895490329203501202093374368, -3061563841565256230323767335163258115617, 933812523074124490731263286490198688, -8880832919709445359197540914968697], [-4964677204622794725204119382501697590294863892210821053485678592, 41045232294970020835493904932894177868798802517518301928620032, -1188827746284436382827434136525712708219438615827504815931392, 19437505994777689925710770467389593112518955284187620114432, -30225115068060332016945039068902437116304979544203591680, 1383642403740008958749989551613893073867796952309857280, 662368453738739779054452619668594067649918568367360, 40831366129579999984620167357058788240635744083440, 29696351802507276909420332372318821670406357120, 594543095416754750611518803078525672534235480, 326246750743101293644793781655475267680848, 4183650934108899383379959492877321307167, 1144401745082514311059348460731029968, 11308200894655857313945680111042247], [-54042405140409814464469078758491058340055572254233511319306240, -14184904516679213134521443343869288111624397864018129916526592, 27254481882558456636118593162035623019782215940398970306560, -5200748467984114170669341930471385211082633762769233313792, 6111593242731009601469804698117965041315488521938534400, -458883773982372548879183767205828202363114346412743680, 310856384796162571524708230788103117402622464627200, -16963693858871821320920977950961275714523397246640, 6306167486418821869505519311874908562438534400, -293311127213617193852765344788067744775501880, 54990897434311652826267177524223270454560, -2329548860126069631868833826112080026027, 172249561993619642557392284247625760, -6863841429709776011709644929064307], [-8050607216813295500157734670305679750846295048260533441301315584, -287439111683350708872278090739699326386412394049686641712300032, -5587872197710629320451059038976190175796699725442669749469184, -89370590604540144984100614049401703731127087338970315948032, -632931125296577720822471539999226862133632615804822159360, -7859180900906550280502114583940023613486369034030945280, -26408172546054579600326779762316288996823706746273280, -288694738548552904834290116648290257316348636679440, -492971129872205574212893514757726009919229141760, -4990437429069580617523910634310490339825181480, -4122069093302945324241736376044738840948704, -39759770519275893891185150732848459206417, -12590447230348395344097696410728074464, -117617565858904449759920266209157097]];
    Rf_basisdens := [1, 135746553974964854480053862400000000, 135746553974964854480053862400000000, 919323204574785191249037119776449331200000000, 1544507147888151708822930513164514204057600000000, 1901807876660639345404406476410402323848950290841600000000, 172891625150967213218582406946400211258995480985600000000, 172891625150967213218582406946400211258995480985600000000, 3803615753321278690808812952820804647697900581683200000000, 3803615753321278690808812952820804647697900581683200000000, 1901807876660639345404406476410402323848950290841600000000, 1901807876660639345404406476410402323848950290841600000000, 345783250301934426437164813892800422517990961971200000000, 3803615753321278690808812952820804647697900581683200000000];
    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_5_c();"
function MakeCharacter_5_c()
    N := 5;
    order := 4;
    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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_5_c_Hecke();"
function MakeCharacter_5_c_Hecke(Kf)
    N := 5;
    order := 4;
    char_gens := [2];
    char_values := [[0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, 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+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 := 17;
    raw_aps := [[0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [565, -3, 0, 0, -565, 0, -1, 0, 0, 0, 0, 0, 0, 0], [13777, -280, 259, -1, 62039, 0, -8, -5, 0, 1, 0, 0, 0, 0], [-28046, 2, 2764, -13, -28044, -13, 0, -180, -1, 2, 3, 0, 1, 1], [-38555658, 171956, 171726, 1098, 80, 0, 6400, 6608, -12, 303, -205, 5, 0, 75], [-48653379, -156377, -102, 4427, 48653361, -4427, -21178, 345, 114, -282, -567, -9, 114, 120], [936561521, -3587, 5628943, -2022, 936557659, -2022, 925, -267483, 456, -4237, -8093, -275, -456, -1931], [2020, 5624087, -5623957, 0, 910093816, -28686, -842245, 839617, 0, 15605, -8677, -1075, 3692, -3095], [16887812190, -31090554, -16160, -316765, -16887815680, 316765, -1829324, 35500, -2055, -67985, -71310, -1745, -2055, 19650], [20880, 697792115, -697688145, 0, -43237322276, 1471510, 17883591, -18081727, 0, 520495, -72659, -62425, 72444, -83305], [-79150769198, 664623628, 665232308, 7049994, -272030, 0, -24134770, -24521446, 14594, -969821, 90155, -103705, 0, -168325], [-644377695811, 1561787, -3479158470, 6867657, -644376556379, 6867657, -2011230, -99128687, 596804, 3995372, 1608603, -422355, -596804, 569716], [2824733727617, 165709928, 164112703, -8126686, -533475, 0, 369754135, 373145904, -1403106, -1938181, -9591350, -1598825, 0, 1065350], [-4503369141745, 14201307275, 4081360, -29165750, 4503365444805, 29165750, -341334127, -3267420, -650110, -6890730, 7733200, -1848470, -650110, -384420], [-201933359770, -2535740, -383507612, -168388775, -201941179080, -168388775, -4175720, -595144678, -2481765, 6923550, -33971095, -5283570, 2481765, -3909655], [-19648739943089, -20693340354, 5630143, -337624073, 19648725609641, 337624073, 4270361561, 25122880, 14882994, -39893817, -28196042, -7166724, 14882994, 8703305], [-930360, 262336291189, -262340478579, 0, -44890246432432, -1110865282, -5886836391, 5890693411, 0, -21633565, 7541045, 2558875, -7308840, 3489235], [48857644711997, -86546405400, -86500229235, -272669010, 6938235, 0, 3282904185, 3226077750, 56340600, 50598105, 213653340, 33495435, 0, -26557200], [-12942705313089, -202631752, -238610577997, -1373545142, -12942803709901, -1373545142, 263829840, -32877690643, -6161554, -570696532, 109749642, 104234940, 6161554, -49198406], [-95366921727398, -651965026900, -652169159380, 2445083770, 96285130, 0, -24182895770, -24100937750, -37451550, 305327515, 7674795, 42361455, 0, 53923675], [55389673229827, 1372860704244, -558886974, 724797654, -55389219029683, -724797654, 39455536920, 466909530, 30435798, 624775596, -1130483334, 227100072, 30435798, 104686830], [367796060, -1013816534784, 1012749979124, 0, 259164232167084, 5348705512, -39287009116, 40407643528, 0, -974474360, -1543679252, 349379800, 90911272, -18416260], [953810209605995, 2081960536863, 937630150, -1724659500, -953810806781695, 1724659500, 82934977767, -1004770400, -25273650, -198261700, 2282855000, -298587850, -25273650, -340454450], [-733060180, 3848391555230, -3846096626550, 0, 1941296575394192, -2835937060, -165139030282, 162401413314, 0, 2534299030, 3327054938, -780934250, -346940148, -47874070], [-238339782526713, 2133889776, 16003715460520, -1674643564, -238336559735577, -1674643564, 2035922910, -299413685782, -1469515758, -990934494, 8242770594, 1088901360, 1469515758, 1611395568], [1820649185449172, -4872948210928, -4872910190798, 14408795516, -1571174730, 0, 17734951810, 24531597696, 2159161956, -1787784934, -11650740020, -2337752030, 0, 766577300], [-1417057281638586, 6688167613378, -1934904448, -1545293737, 1417060970998384, 1545293737, 140467424332, -1871442600, -207211799, 12434873847, 1690993502, 1844679899, -207211799, -1754455350], [-1552826747154827, -4201062436, 22759665244593, 8312913184, -1552833707828963, 8312913184, -1709611900, 1012991339267, -329274432, 268161164, -22529457804, -2759611700, 329274432, -3480337068], [5693613945, 15945267991182, -15958941209277, 0, -2489093722959597, 16486872324, 553639324533, -548016061374, 0, -5154158745, -16428312714, 3989802075, 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[-473500731547508193396398, 366647496132648152756, 366718977420040764326, -5707413720763973702, -52224245004317120, 0, 2775682052503955200, 2491938978651368308, -297453234080315112, -463754490287413697, -141497331658237905, -42595723810169895, 0, -9628521194147225], [97646478055243053184235, -192549578328537800, 1850402673826441028884, 3072151036506247760, 97646284919120185236755, 3072151036506247760, 85991841248209870, -41584853231703717890, 9403131106944510, -277954875037337990, -378628748247179490, -586544539409680, -9403131106944510, -96568061433973740], [129939017535818807980628, -936156446026948402940, -427465772828582626, 3420039011319286571, -129938878418291573003522, -3420039011319286571, -35668958100874403492, 674139408847424120, 27884154042724527, -558925528268137821, -1389953427269612266, 69558763617488553, 27884154042724527, 288348245593605520], [-274083720767397433604498, -462272862305355227828, -462687139267954635388, -4443793835372442774, 138783664223445610, 0, 42084240849713064950, 42688916981000361446, 192473198758818206, 746570840617136011, -409091887422084385, 1037015035464635, 0, 137746649187980975], [21124647509734455344415, -83808674809151130, -815721903921811650017, -6745030536961979725, 21124506991535266328445, -6745030536961979725, -24589854757434465, 103428369178597054502, -18570094407787230, -2509295671851825, -469735065925413690, -56709524379864840, 18570094407787230, -70259099594507985]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5_17_c_a();".
// 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_5_17_c_a(:prec:=14)
    chi := MakeCharacter_5_c();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 17));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_5_17_c_a();".
// 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_5_17_c_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,17,sign)));
    Vf := Kernel([],Snew);
    return Vf;
end function;