
// Make newform 74.5.i.b in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_74_i();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_74_5_i_b();".
// 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_74_i();"
function MakeCharacter_74_i()
    N := 74;
    order := 36;
    char_gens := [39];
    v := [23];
    // 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_74_i_Hecke(Kf)
    return MakeCharacter_74_i();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_74_5_i_b();".
// 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_74_5_i_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_74_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,5,sign)));
    Vf := Kernel([<3,R![199637497961770120403824402831394569884192072975637056, 1101370654737662767529910396053034111813758478927776832, 57540713662950758047199086028745829294373641762397984736, -459813927523822154914447321594012251433111984669407585024, 1902773827678027787775075265557434706924026932961086079040, -7372696888801671314840147879100538476943255055174424550752, 26046728355688072329192907930504034612325359945510792705040, -65264356956697482046500334680358208175402336855700668912888, 82688534461571288585344225065994259011506605433295480398492, -18273682054647644629962392912686110317593186897276798298076, -10679627071178292987125041649596320949251394941496799397162, 16992931874085030051814093531505722090165883336912322436986, -4612173981490924078036614420076803511959628748622258987957, -1758826385340395854219374129485376140175024989564563759878, 1195984523665092748011413580927064781920604800110786623217, -134995050506900405144644607646805506700759505667108743382, -66467128894373367959250004893540131780285812779930089001, 12367506979017695200599001144358649680734949185890233334, 4043700111485362194651984782336868789945466131818011257, -960638001609029949487960335072370095643917492899977314, -257597851576375592949598791140483472127654815435671013, 123953515981460668336292205313736642998219695281079630, -1418004860652548855403377693489472300397156661564331, -4857530042905912690059271271587727586427126520825760, 691343726854679108252489790637944581204707742872533, 69743340222808745753111234807834345734009008265290, -8005030693727555948957466926190169952653875776628, -8855334005221574995622435239740141057052334609448, 955694300498593447227081148034139469051675103202, 53775772084930161951373524112573750049243084208, 1825924871700249375737323387318824373468305825, 643855830457001813278662122935128909083585070, 77706147929577796874271744722036875665298518, 26083474091554646766757961426624821893746286, 102382000186036569786220575586385546431326, 23704065187319741405238319947471315028260, 5695632725520190648210326334786562064981, -186702787318423733151440761459107115866, -24805813915178149487901261093172748556, -22360033093269402747158941455616609200, -69474860549817589264089468425571567, -26525172550298384628302612432231412, -5957707264333129334356002469618392, 99337150482254626968037083566574, -2972201340076076595278003336055, 12123993532617165496719536788260, 216443136884580379342450982532, 13333702928547199080927886260, 3008706276531072512022300322, -53767061722371386564082234, 4295506500226064960213529, -3705698650489591127433612, -58795095963602351644047, -4457338327857580683264, -866318642733686758376, 25814212143288664140, -1022433599917171968, 786631037953482270, 16458929352606831, 655280414717904, 140512718898660, -6285136311630, 89630887236, -92054468700, -757662114, -50799468, -10991093, 494640, -9102, 4860, 105, 0, 1]>],Snew);
    return Vf;
end function;