
// Make newform 775.2.ck.a in Magma, downloaded from the LMFDB on 10 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_775_ck();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_775_2_ck_a();".
// 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_775_ck();"
function MakeCharacter_775_ck()
    N := 775;
    order := 30;
    char_gens := [652, 251];
    v := [15, 26];
    // 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_775_ck_Hecke( : Kf := Kf)
    return MakeCharacter_775_ck();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_775_2_ck_a();".
// 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_775_2_ck_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_775_ck();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![6561, 0, 30618, 0, 443961, 0, -62208, 0, 1463346, 0, -2565216, 0, 2443194, 0, -1696509, 0, 1137037, 0, -580754, 0, 206839, 0, -51961, 0, 10266, 0, -1733, 0, 241, 0, -22, 0, 1]>],Snew);
    return Vf;
end function;