
// Make newform 8005.2.a.e in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8005_a();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_8005_2_a_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_8005_a();"
function MakeCharacter_8005_a()
    N := 8005;
    order := 1;
    char_gens := [1602, 4806];
    v := [1, 1];
    // 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_8005_a_Hecke(Kf)
    return MakeCharacter_8005_a();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_8005_2_a_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_8005_2_a_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8005_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![61165793, 924942616, -106812859606, -1252869041045, 28873564659222, 358000581919794, -2042338075011273, -32240907108578290, 61112059969862006, 1507708977793002000, -569655569718251714, -44156514896611550774, -17535346419670770072, 893675795568832214152, 762240959462184573013, -13284718042789146579280, -15259996307228519007416, 151203759337549189563398, 205110282947545136762651, -1358017230894942002282827, -2055480715870567943793813, 9846956178698001478282258, 16159667925086367118439884, -58682748544779634689423363, -102822624674503523086802078, 291564097298157863769412457, 541111697008470115817789372, -1221818132229649364917949859, -2393682613779004776991561475, 4359532726816801904346972696, 9014860303262242616570215845, -13346955439142870732438456691, -29204988638313102942869132964, 35278460148575995583304134500, 82091117648406541204034150405, -80883850137630935496366583345, -201666964761691158694350020007, 161367090612213953221494418530, 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    return Vf;
end function;