
// Make newform 119.5.d.a in Magma, downloaded from the LMFDB on 09 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_119_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_119_d_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_119_5_d_a();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_119_5_d_a();".
// 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 ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        poly := [-3, 20, 0, -10, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [-4, 0, 1, 0, 0], [0, -6, 0, 1, 0], [8, -1, -8, 0, 1]];
    Rf_basisdens := [1, 1, 1, 1, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_119_d();"
function MakeCharacter_119_d()
    N := 119;
    order := 2;
    char_gens := [52, 71];
    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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_119_d_Hecke();"
function MakeCharacter_119_d_Hecke( : Kf := false)
    N := 119;
    order := 2;
    char_gens := [52, 71];
    char_values := [[-1, 0, 0, 0, 0], [-1, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..prec]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 5;
    raw_aps := [[0, 1, 0, 0, 1], [0, -6, 0, 0, -1], [0, 0, 1, -9, 0], [49, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [289, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 433, 87, 0], [0, 0, 0, 0, 0], [0, -566, 0, 0, -401], [0, 346, 0, 0, -449], [0, 0, 0, 0, 0], [0, 0, 337, 903, 0], [0, 0, 0, 0, 0], [0, 2554, 0, 0, -161], [0, 0, -2303, 663, 0], [0, 0, 0, 0, 0], [0, 3274, 0, 0, 799], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, -2207, -2553, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 8257, -2457, 0], [-34222, 0, 0, 0, 0], [0, -3254, 0, 0, 4591], [0, 0, 9793, -393, 0], [0, 15706, 0, 0, 31], [0, -12854, 0, 0, 3151], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, -16886, 0, 0, 3199], [0, 1594, 0, 0, 7759], [0, 14266, 0, 0, -5969], [-3022, 0, 0, 0, 0], [0, 0, -11423, 12423, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, -1526, 0, 0, 10159], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 22993, -13593, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, -22934, 0, 0, -12929], [0, 0, -14447, 20727, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [-135838, 0, 0, 0, 0], [76178, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 24577, -26937, 0], [0, 0, -37007, -5913, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, -37727, 29607, 0], [0, 0, 33793, 19527, 0], [0, 0, 0, 0, 0], [0, 0, 30337, 27063, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, -44303, 42423, 0], [0, 77674, 0, 0, 22879], [0, 13786, 0, 0, 36031], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 73537, -32457, 0], [0, 0, 63553, -46953, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, -11087, 63687, 0], [0, -48614, 0, 0, 40591], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 133834, 0, 0, -5921], [-224398, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, -28703, -65817, 0], [0, 0, 0, 0, 0], [0, -151574, 0, 0, 31], [0, 0, 0, 0, 0], [0, 82474, 0, 0, 53839], [0, 0, 0, 0, 0], [0, 0, 35857, -88377, 0], [0, 0, 0, 0, 0], [0, 0, -48047, 92247, 0], [0, 0, 0, 0, 0], [0, -138806, 0, 0, 43999], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [3602, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, -228854, 0, 0, -15569], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 184906, 0, 0, -57569], [-374302, 0, 0, 0, 0], [0, 0, 146593, -111033, 0], [0, 0, -136463, 120423, 0], [0, 0, 0, 0, 0], [697778, 0, 0, 0, 0], [0, 0, -101183, -103737, 0], [0, 0, 0, 0, 0], [0, -202886, 0, 0, -83921], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 181537, 56823, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [-796942, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, -234566, 0, 0, 81199], [-730798, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 246634, 0, 0, -90641], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, -42374, 0, 0, 138751], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 368986, 0, 0, 74671], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [-474862, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 308314, 0, 0, -117281], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [-1059742, 0, 0, 0, 0], [-258382, 0, 0, 0, 0], [0, 0, 52513, -267033, 0], [0, 0, 0, 0, 0], [0, 0, -384143, 44823, 0], [0, 0, 0, 0, 0], [0, 0, 272353, -253113, 0], [0, 0, 213457, -274137, 0], [0, 0, -374687, 180327, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 467146, 0, 0, -121169], [0, 0, -388847, -78153, 0], [0, 0, 0, 0, 0], [0, -240326, 0, 0, -219281], [0, 0, 0, 0, 0], [0, 0, -382703, 258903, 0], [0, -488054, 0, 0, -179969], [0, 0, 0, 0, 0], [0, 0, 351217, -303657, 0], [0, 0, 451393, 63687, 0], [0, 0, 0, 0, 0], [0, -564326, 0, 0, -172961]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_119_d_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_119_5_d_a();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_119_5_d_a(:prec:=5)
    chi := MakeCharacter_119_d();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 5));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_119_5_d_a();".
// 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_119_5_d_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_119_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,5,sign)));
    Vf := Kernel([<2,R![-977, 1280, 0, -80, 0, 1]>,<3,R![72302, 32805, 0, -405, 0, 1]>],Snew);
    return Vf;
end function;