
// Make newform 800.2.y.e in Magma, downloaded from the LMFDB on 22 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_800_y();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_800_2_y_e();".
// 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_800_y();"
function MakeCharacter_800_y()
    N := 800;
    order := 8;
    char_gens := [351, 101, 577];
    v := [8, 1, 8];
    // 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_800_y_Hecke(Kf)
    return MakeCharacter_800_y();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_800_2_y_e();".
// 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_800_2_y_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_800_y();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1963464721, 5870321280, 26267044208, -105348329640, -334842971008, -200253152344, 4412303675232, 16973669729712, 22166601085856, -2044131860272, -28600706332208, -32084119269064, 30993308625920, 175781122152808, 535716706298880, 1226356031057504, 1729638610653996, 1130057184304928, 85098632344112, -788987373459816, -162862624432512, 413631276456296, 250587132824928, 116420885130320, 171080752805632, 110661700960048, 77112462134416, -87261641445128, -9654254619648, 42563440717864, -7605943969792, -5212264613824, 12547241644406, -209094323392, -290433368240, -135208524856, -45947731584, 171785324856, -15803115744, -26531934384, 65661286176, -3624624080, -1483490960, 530316648, -22499328, 93648504, 32623616, -5531296, 62997324, -831712, -43760, 550472, -3200, -17992, 7200, -336, 15936, -48, 48, 104, 0, -8, 0, 0, 1]>],Snew);
    return Vf;
end function;