
// Make newform 8619.2.a.bp in Magma, downloaded from the LMFDB on 16 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8619_a();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_8619_2_a_bp();".
// 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_8619_a();"
function MakeCharacter_8619_a()
    N := 8619;
    order := 1;
    char_gens := [5747, 5917, 2536];
    v := [1, 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_8619_a_Hecke(Kf)
    return MakeCharacter_8619_a();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_8619_2_a_bp();".
// 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_8619_2_a_bp( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8619_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-13, 516, -1187, -25984, -55804, 30909, 166878, 49783, -173888, -107543, 81690, 77160, -14479, -27758, -1844, 5228, 1188, -448, -178, 5, 9, 1]>,<5,R![-4794259, -29562043, -6011215, 84895943, 58360004, -79006222, -86943934, 18302366, 50766605, 10063344, -12326763, -6017014, 731744, 1113521, 179294, -68477, -29076, -1955, 962, 257, 26, 1]>,<7,R![-3109499, -11214339, 28483023, 60733839, -89305910, -76735514, 114066998, 29459752, -63874581, -1398901, 18446283, -1605566, -2991801, 426193, 280993, -48262, -15091, 2829, 429, -84, -5, 1]>],Snew);
    return Vf;
end function;