
// Make newform 120.4.a.f in Magma, downloaded from the LMFDB on 12 March 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_120_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_120_4_a_f();".
// 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_120_4_a_f();".
// 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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_120_a();"
function MakeCharacter_120_a()
    N := 120;
    order := 1;
    char_gens := [31, 61, 41, 97];
    v := [1, 1, 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_120_a_Hecke( : Kf := Kf)
    return MakeCharacter_120_a();
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 := 4;
    raw_aps := [[0], [3], [5], [8], [20], [22], [-14], [76], [56], [-154], [160], [-162], [-390], [388], [-544], [-210], [-380], [-794], [-148], [-840], [858], [144], [316], [1098], [994], [-834], [1672], [-732], [-970], [1938], [-528], [636], [1754], [-2508], [1486], [2120], [-1850], [-1172], [-1656], [-2666], [1132], [-2866], [1888], [1282], [350], [-3400], [4652], [4016], [2316], [94], [-4230], [2064], [4562], [2532], [3522], [-2232], [2806], [4848], [7790], [-118], [-6508], [-8770], [-4292], [-9464], [9578], [-186], [-492], [2290], [-6092], [5766], [-9374], [-3528], [7616], [3406], [-12284], [5424], [3486], [-3626], [5874], [-12662], [6396], [8286], [-4112], [5330], [11272], [14196], [-5886], [-7526], [8502], [12672], [16540], [8864], [3688], [-16140], [1580], [15000], [20486], [7706], [-3932], [-23930], [11468], [-11498], [16988], [-17366], [-24860], [-26302], [7812], [7986], [-21048], [1738], [18576], [-13602], [19578], [12308], [-8600], [6978], [-7668], [-15384], [-2186], [-1524], [-4242], [24354], [-322], [-7932], [20684], [25222], [23678], [8432], [8312], [-26298], [16956], [-17880], [22032], [11534], [38250], [19330], [-40674], [7004], [12198], [-25734], [15668], [-34450], [-38792], [-20460], [5542], [25240], [-37330], [-3894], [20324], [6288], [-24650], [9426], [-9316], [6968], [-35964], [47888], [-12760], [-25054], [28282], [-30634], [48572], [12906], [-5880], [55444], [32050], [29880], [5216], [6750]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_120_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_120_4_a_f();".
// 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_120_4_a_f(:prec:=1)
    chi := MakeCharacter_120_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_120_4_a_f();".
// 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_120_4_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_120_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<7,R![-8, 1]>,<11,R![-20, 1]>],Snew);
    return Vf;
end function;