
// Make newform 169.8.a.i in Magma, downloaded from the LMFDB on 29 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_169_a();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_169_8_a_i();".
// 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_169_a();"
function MakeCharacter_169_a()
    N := 169;
    order := 1;
    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_169_a_Hecke( : Kf := Kf)
    return MakeCharacter_169_a();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_169_8_a_i();".
// 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_169_8_a_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_169_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,8,sign)));
    Vf := Kernel([<2,R![618039895230250221568, -698740375987929153536, 54792691948098420736, 125069178493300281344, -25076671497098250240, -5958983572887843840, 1431482843960505344, 124434751468174592, -34866523470570752, -1205808926314368, 452991529756544, 3685037378032, -3402270784328, 25361989564, 15146495220, -264006077, -39220914, 947492, 54321, -1568, -31, 1]>],Snew);
    return Vf;
end function;