
// Make newform 2116.4.a.j in Magma, downloaded from the LMFDB on 30 May 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_2116_a();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_2116_4_a_j();".
// 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_2116_a();"
function MakeCharacter_2116_a()
    N := 2116;
    order := 1;
    char_gens := [1059, 5];
    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_2116_a_Hecke( : Kf := Kf)
    return MakeCharacter_2116_a();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_2116_4_a_j();".
// 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_2116_4_a_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2116_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![-57758556965013431, 2549268981159581230, 3422353813031257520, -4636351583169801212, -6106609085833062621, 2958317487199815710, 3945469090516615938, -919987255453318390, -1270002897626386615, 168262887192839108, 236396755263612230, -19980305319824360, -27538273735815616, 1619375103901356, 2101408792830729, -91596999065808, -108056859836802, 3642433779828, 3810149690699, -101405368198, -92830030766, 1948064336, 1556384118, -25119470, -17593452, 206494, 127882, -974, -539, 2, 1]>,<5,R![691370322590321899768251097, -997220275694960381995611892, -2666561043457029739855949246, 533999524563076432049654006, 1048776836734030122762665408, -100120731431131267601865138, -163352819029205855970298593, 9775515541763187153527708, 12523599139002240150980643, -644218756925870471264052, -531854466682477081372166, 27221516067166266840650, 13692298313582056766255, -737880332341211510830, -226380439442476453987, 13172841816135806632, 2480055362775377353, -158408840942747772, -18156461302527796, 1297719580147868, 87276383794946, -7225059208026, -258410162263, 26797372214, 380226132, -63141878, 63111, 85216, -1018, -50, 1]>],Snew);
    return Vf;
end function;