
// Make newform 8619.2.a.bw in Magma, downloaded from the LMFDB on 27 April 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_bw();".
// 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_bw();".
// 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_bw( : 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, -33, 365, 768, -3912, -6112, 20663, 19821, -60042, -25807, 96382, 8096, -87948, 11811, 46628, -13305, -14226, 5900, 2302, -1343, -136, 154, -8, -7, 1]>,<5,R![343, -53165, 491862, -1162427, -1635522, 7942116, 749287, -19167246, -1098265, 23562034, 6416016, -13151413, -5908531, 3387472, 2227486, -336736, -421795, -15884, 41405, 6344, -1853, -498, 13, 13, 1]>,<7,R![-810641, 76202421, 203219452, -435303307, -841096004, 695096590, 1223090845, -462320714, -897980149, 124711761, 376142132, 3663404, -94266349, -10375500, 14325310, 2667478, -1291597, -333768, 62844, 22840, -1075, -818, -27, 12, 1]>],Snew);
    return Vf;
end function;