
// Make newform 690.2.m.h in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_690_m();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_690_2_m_h();".
// 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_690_m();"
function MakeCharacter_690_m()
    N := 690;
    order := 11;
    char_gens := [461, 277, 511];
    v := [11, 11, 3];
    // 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_690_m_Hecke(Kf)
    return MakeCharacter_690_m();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_690_2_m_h();".
// 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_690_2_m_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_690_m();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![7930971136, -101206088192, 790923217920, -3498172617984, 9233871404736, -13634882057280, 21471147242512, -20772402750000, 12057869111376, -4816906302472, 1935492063121, -985703727971, 430420546825, -124606373560, 26116712906, -4306910515, 829930716, -237640981, 98910142, -27094140, 6740184, -802964, 33268, 16460, -3492, -523, 532, -190, 48, -8, 1]>],Snew);
    return Vf;
end function;