
// Make newform 350.2.m.b in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_350_m();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_350_2_m_b();".
// 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_350_m();"
function MakeCharacter_350_m()
    N := 350;
    order := 10;
    char_gens := [127, 101];
    v := [1, 10];
    // 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_350_m_Hecke(Kf)
    return MakeCharacter_350_m();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_350_2_m_b();".
// 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_350_2_m_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_350_m();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2835856, 96897360, 1090582040, 3757772480, 5524830620, 6349806400, 15275905110, 27034332730, 25265736375, 23705788450, 37597639519, 42975710230, 26995758220, 14151808940, 16292887935, 15088824550, 3140446105, -5154626260, -3110700025, 939660400, 1219573971, -38703120, -293329430, 25479940, 102511010, 12958650, -19151120, -4181460, 2881075, 688200, -375316, -82240, 46265, 9730, -4430, -750, 385, 30, -25, 0, 1]>],Snew);
    return Vf;
end function;