
// Make newform 775.2.ck.c in Magma, downloaded from the LMFDB on 23 December 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_c();".
// 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_c();".
// 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_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_775_ck();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 0, 404, 0, 386236, 0, 12976677, 0, 2448920734, 0, -70643366428, 0, 913390199352, 0, -6816106097762, 0, 37334535619093, 0, -173323772486442, 0, 691562359480664, 0, -1978705429880352, 0, 3960804447925992, 0, -5723879912416957, 0, 6335989121598845, 0, -5733428638512223, 0, 4493548166850261, 0, -3118533339166011, 0, 1926302570101848, 0, -1069991109948959, 0, 545140696943613, 0, -252401075629697, 0, 107120667803602, 0, -41835206904378, 0, 15040356353752, 0, -4956243472278, 0, 1505639101791, 0, -420817925021, 0, 107836402894, 0, -25297767661, 0, 5446159561, 0, -1069677690, 0, 190595955, 0, -30718703, 0, 4463236, 0, -573961, 0, 64256, 0, -6191, 0, 501, 0, -30, 0, 1]>],Snew);
    return Vf;
end function;