
// Make newform 96.4.a.e in Magma, downloaded from the LMFDB on 19 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_96_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_96_4_a_e();".
// 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_96_4_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 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_96_a();"
function MakeCharacter_96_a()
    N := 96;
    order := 1;
    char_gens := [31, 37, 65];
    v := [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_96_a_Hecke( : Kf := Kf)
    return MakeCharacter_96_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], [2], [12], [60], [-42], [10], [132], [-48], [226], [-252], [-362], [-94], [-228], [-408], [346], [-300], [-466], [204], [1056], [330], [612], [564], [-1510], [594], [554], [1284], [-1356], [390], [-766], [-2388], [396], [-110], [-732], [-1934], [-1092], [-578], [2532], [-648], [3338], [3804], [1854], [1344], [-1262], [-4294], [4308], [1212], [2172], [-3948], [-3522], [-2774], [2784], [-4686], [-2484], [6658], [2904], [-1006], [876], [2718], [5354], [-780], [-3350], [-9636], [-7560], [-3526], [7634], [7572], [162], [6636], [4430], [8402], [11520], [-7404], [1910], [10332], [6624], [10210], [-4066], [-5510], [15450], [-3084], [10446], [-2184], [-110], [-2412], [6540], [-9670], [-6774], [14602], [-13620], [-8508], [6312], [10572], [-4332], [-3684], [-11184], [12946], [-17150], [7884], [5910], [-972], [2458], [11316], [1810], [10500], [-19438], [-15084], [5794], [25152], [-11846], [8940], [-4570], [17786], [-15804], [-18468], [-7814], [5364], [-3936], [7610], [-13620], [8710], [-12094], [16466], [16428], [13332], [-19118], [798], [-8856], [-13764], [-20538], [15900], [-20856], [-10332], [13806], [15554], [13106], [18874], [15444], [-39286], [20018], [-8388], [-37942], [11628], [32388], [9846], [16848], [18214], [-2446], [-26244], [25248], [-34], [-19022], [12852], [40104], [42540], [-18528], [15756], [-15542], [-29702], [2890], [9180], [37906], [-41916], [-7764], [32666], [-53016], [17844], [-55834]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_96_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_96_4_a_e();".
// 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_96_4_a_e(:prec:=1)
    chi := MakeCharacter_96_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_96_4_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_96_4_a_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_96_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![-2, 1]>,<7,R![-12, 1]>],Snew);
    return Vf;
end function;