
// Make newform 370.2.o.d in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_370_o();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_370_2_o_d();".
// 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_370_o();"
function MakeCharacter_370_o()
    N := 370;
    order := 9;
    char_gens := [297, 261];
    v := [9, 2];
    // 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_370_o_Hecke(Kf)
    return MakeCharacter_370_o();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_370_2_o_d();".
// 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_370_2_o_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_370_o();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![64, 960, 5136, 11624, 23976, 52482, 62103, 58971, 196524, 317983, 332352, 224343, 240070, 45285, 26349, 276, -579, -1620, 536, 111, 48, 10, -6, 0, 1]>],Snew);
    return Vf;
end function;