
// Make newform 6019.2.a.d in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6019_a();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_6019_2_a_d();".
// 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".
// To make the character of type GrpDrchElt, type "MakeCharacter_6019_a();"
function MakeCharacter_6019_a()
    N := 6019;
    order := 1;
    char_gens := [2316, 1392];
    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_6019_a_Hecke(Kf)
    return MakeCharacter_6019_a();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_6019_2_a_d();".
// 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_6019_2_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6019_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![586577920, 51934698496, -1081887158272, -78270674033152, -378835975601408, 10985620214443008, 77800730042612800, -607024585772488528, -5370289569747195008, 16609939149504613412, 192986664166624790966, -249771337042780186774, -4352773932594219386755, 1872284498931539031707, 68361792739419561087508, 2287836104518535494197, -797722673187857415325995, -232607065367276013428076, 7225854252118249616053772, 3113877223844116462463828, -52414239032521590633250134, -26005698959947404218634144, 311562324599620085426333290, 159856851087826057804662029, -1544637277692773047522991432, -769295239058745145808075111, 6475850444351186108952275658, 2992017606788584264626174376, -23215471267879513492324892892, -9580776014788978206063046019, 71816395740987907236468167100, 25526055729688467484399910700, -193170169972705336861768065952, -56805006461034073522817461055, 454708538588603535602040433835, 105135957547589162548879187295, -941926916754944049391529078200, -158851804227194379596577965065, 1725406268460790076699217270979, 185448504704678942785657631057, -2806688688779134116386639360465, -135794249953473581691607468998, 4069530188717463856438828195916, -33888450700185179773843426189, -5276784139948154146486831028730, 330147210930589751319676797975, 6136612020322591972021234885465, -702172024130606702126237257198, -6416958460479196879572023397638, 1053109517524697476632070500855, 6046965699819916960466757745034, -1281610139907736783562165912268, -5145028465580652370864348794812, 1328918019192599480838105555274, 3959015102079272947011471295471, -1201990935206040119090664560793, -2758782543235635252967807352962, 961193618090522015055290093360, 1742745223018910140027304820201, -685432849993987515706851898024, -998756835576556831227865994472, 438460768573264398472687895119, 519489045876777283347094369516, -252671220228471260332882877948, -245248524702634620601521326118, 131584858334584104771161816635, 105047883358709504286811653901, -62072795518982904296351490767, -40787805507446279316434460165, 26570378417519048552835004493, 14333600257420093829378940924, -10333253724073497054302249202, -4547618828622574546815937231, 3654086806706370283024721752, 1297653046565868106974134719, -1175491346512025304395952389, -331061652892457678367169970, 344035141425596799016902202, 74798567084855539104704190, -91585071088137904913751145, -14720527330449351028713448, 22163134094967473764011370, 2442554409030627103552927, -4870933695659762525446683, -315303862541019696533875, 970933998402194218497180, 22694122298145265736153, -175226325700528827185328, 2569244585320905211672, 28567012706148381901579, -1430780435656984317082, -4195286777960085449128, 350552799419700460891, 553052947765394872053, -64014747323843930460, -65160823942454958862, 9602956798473821756, 6824219402650186238, -1223523045502737047, -630905982972823182, 134148575526687692, 51031083963839623, -12707960156044983, -3568289887518995, 1038827291539979, 212086755742579, -72903608239014, -10443036263809, 4353716996941, 407370444660, -218393975299, -11404703067, 9034363090, 155621371, -300185272, 3844475, 7702961, -299516, -143302, 8849, 1720, -141, -10, 1]>],Snew);
    return Vf;
end function;