
// Make newform 83.13.b.c in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_83_b();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_83_13_b_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".
// To make the character of type GrpDrchElt, type "MakeCharacter_83_b();"
function MakeCharacter_83_b()
    N := 83;
    order := 2;
    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_83_b_Hecke(Kf)
    return MakeCharacter_83_b();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_83_13_b_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_83_13_b_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_83_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,13,sign)));
    Vf := Kernel([<2,R![33957156264246802063850972322870213574576532805246862689082908495494424243349475246158371001344885273809953858138377686752723203627417600000000, 0, 1127519630099737142620364703753868411437498308643778202010484046405948477482819151227888591072172645962324559433771248323532118728638464000000, 0, 16146860316725826773165736132109335355073980902314544973574587065334127956584803619650282562184141718198025020611807943411389508144332800000, 0, 134416677910722432882767356637158219009812105604701500748852139356505265017481549979038392800507458193548478556446317202997205469757440000, 0, 740105128748167751216869100942839951438489174493420400761812641821549464193410368389238755742616112489529214339294860708507140751360000, 0, 2901526903086035519567971165146837150818013871977451276013557155119185205116827745744719359209221442466202464306523157092404035584000, 0, 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    return Vf;
end function;