
// Make newform 315.2.be.c in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_315_be();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_315_2_be_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_315_be();"
function MakeCharacter_315_be()
    N := 315;
    order := 6;
    char_gens := [281, 127, 136];
    v := [1, 6, 5];
    // 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_315_be_Hecke(Kf)
    return MakeCharacter_315_be();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_315_2_be_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_315_2_be_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_315_be();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![81, 891, 324, -32373, 76644, 135045, -485316, -429999, 2165308, 174711, -4412199, 278307, 6196824, -220980, -5328454, 95421, 3298788, -31752, -1451373, 8106, 484083, -1329, -120765, 144, 23178, -6, -3298, 0, 348, 0, -24, 0, 1]>],Snew);
    return Vf;
end function;