
// Make newform 4014.2.a.q in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4014_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_4014_2_a_q();".
// 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_4014_2_a_q();".
// 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 := [2, 1, -5, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [0, -4, -2, 1], [3, -2, -3, 1]];
    Rf_basisdens := [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_4014_a();"
function MakeCharacter_4014_a()
    N := 4014;
    order := 1;
    char_gens := [893, 2233];
    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_4014_a_Hecke(Kf)
    return MakeCharacter_4014_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+1)]];
    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 := 2;
    raw_aps := [[1, 0, 0, 0], [0, 0, 0, 0], [-1, 1, 0, 0], [-1, 0, 0, -1], [-1, -1, -1, 1], [-2, 0, -1, 2], [2, -1, 2, -2], [-1, -1, 2, 1], [3, -3, 1, -1], [-3, 0, -2, 1], [-1, 1, -2, -1], [-4, 2, 0, 1], [-3, 0, 3, 1], [-5, 0, 1, -1], [1, 1, -2, 0], [-3, 3, -2, 1], [2, 1, 0, -4], [-1, -2, 0, -1], [-5, -1, 0, 0], [0, -3, 1, 2], [-1, 0, -3, -2], [-10, 4, -2, 1], [2, 0, -1, 3], [-5, 4, 0, 3], [-9, 3, 0, 1], [-3, 4, -4, -1], [-5, -5, 3, -2], [1, 5, 0, 3], [2, 0, 3, -3], [1, 0, 2, 3], [-9, 4, 3, -3], [3, -2, 2, -4], [2, 1, -2, -1], [-8, 1, -4, 0], [1, 5, -4, -3], [0, -4, 1, -3], [3, -5, -1, 1], [-9, -3, -3, 2], [3, 0, 1, -3], [-7, 0, -5, 8], [10, -1, 9, -7], [-3, 5, 6, 2], [-9, -1, 4, 3], [9, -2, 10, -3], [0, 3, 6, 0], [-3, 6, -6, -1], [6, -3, 5, 2], [-1, 0, 0, 0], [0, -4, 4, -5], [11, -3, -4, 5], [2, -2, 1, 8], [-2, 1, -2, -9], [7, 4, -2, -4], [3, 4, 1, -6], [2, 0, -3, -4], [-13, 4, -8, 7], [2, -3, -2, 5], [-2, 4, -6, 6], [-2, 11, -7, 4], [-7, -1, 1, 7], [10, -10, -2, 4], [-8, -2, -5, 6], [1, 0, -2, -2], [2, -2, -3, -6], [9, 2, 1, -1], [-1, -1, 1, -1], [1, -10, 6, 1], [12, -8, -1, 4], [1, -8, -2, 1], [-6, -5, -5, 11], [15, -8, 8, -5], [6, -8, 2, -6], [3, 1, 7, -9], [-1, -2, -11, 3], [-6, -4, 1, -4], [-7, 3, -10, 5], [1, 5, 3, -5], [-15, -2, 1, 9], [-2, 2, 9, -2], [2, 3, 4, 0], [8, -6, -7, 4], [-8, 12, -8, 4], [-14, 11, 6, -6], [-6, -1, 3, 9], [8, -11, 9, -3], [-1, 16, -9, -6], [8, 3, -1, 4], [-14, 11, 5, -10], [8, -11, 6, -4], [4, 4, -5, 2], [6, 6, -5, 10], [-1, -5, 8, -7], [2, -1, 2, -2], [-4, -5, -1, 8], [5, 3, -6, -11], [12, 5, -6, 2], [6, -8, 0, -8], [4, 5, -4, -7], [-14, 2, -11, 14], [4, 2, -6, 10], [8, -6, 7, -4], [-17, 9, -4, -4], [2, -5, 7, -4], [2, 6, -5, 5], [12, -10, 2, 5], [10, -5, 4, -5], [13, 2, 8, -3], [-1, 7, -4, -12], [-19, 8, 7, 1], [21, 2, 2, -7], [-2, 6, -4, -3], [2, 0, 2, -12], [-3, -5, -15, -1], [-4, -11, 10, 1], [-25, -9, 3, 4], [-11, -13, 10, 8], [-5, 1, 10, 7], [17, 3, 10, -13], [-3, 8, -7, 11], [8, 7, -4, -7], [12, 2, -3, -10], [17, -3, -2, 1], [18, -4, 5, -6], [1, -11, 4, 2], [-17, 14, -11, 0], [15, 1, -9, 7], [-20, 1, -2, 2], [2, 1, 1, 9], [-25, 6, -10, -1], [21, -14, 2, -1], [2, -2, -4, 9], [3, 2, 7, 12], [0, 9, 9, -8], [-8, -3, -3, -4], [-2, -7, -12, 9], [21, -1, 9, -15], [29, -14, 2, 3], [8, 5, 1, -19], [28, 1, 12, -8], [12, 10, -5, -7], [-6, 12, -5, 6], [1, 3, -4, -15], [-5, 0, -3, 12], [0, 3, 6, -18], [-12, -1, -7, -2], [16, 5, 0, -4], [18, 4, -2, -2], [-1, -1, -3, 11], [-16, 7, -11, 1], [-1, -16, 9, -5], [19, -1, 9, -1], [-14, 5, 7, -2], [-17, -2, 7, 6], [-3, -2, -12, 10], [20, -8, 18, -14], [-25, 10, 7, 2], [-13, -11, 1, 1], [28, -26, 3, 2], [16, 6, 7, -10], [-5, 20, -2, -3], [-5, 9, 13, 5], [15, -10, 19, -6], [-18, 16, -11, 10], [18, -8, 14, 4], [6, -14, -3, 5], [-26, -2, -12, 14], [21, 6, 8, -11], [9, -18, -1, 10], [9, 1, -10, -1], [1, 4, 3, 9], [-17, 1, 4, -15], [5, -12, 17, 3], [-6, 19, -14, 5], [16, -14, -5, -2], [14, 7, 8, 8], [16, 0, -4, 18], [-28, -2, -11, -1], [-33, -2, -4, -3], [23, 5, -5, -15], [-14, 2, 13, -12], [12, -17, -2, 13], [-32, -2, -8, -5], [13, 0, -5, -6], [25, -17, -9, 10], [-16, -11, 1, -11], [-15, -5, -15, 19], [-25, 0, -8, 16], [-6, 13, 6, -2], [27, -5, -4, 4], [-23, -12, 2, 8], [-12, -2, 17, 10], [0, 10, -6, -2], [-26, 1, -19, 10], [-8, 0, 8, 4], [-14, -7, -17, 5], [4, -9, -5, 2], [29, -9, 12, 4], [-19, 4, -1, -6], [-28, 10, -4, 5], [-11, 13, 4, 3], [-3, 9, 10, 6], [5, 11, -6, 15], [2, -7, -14, 22], [-4, 3, -10, 4], [16, -16, -2, -4], [-29, 9, -6, -3], [-28, 0, -2, 4], [-14, -7, -9, -10], [15, 6, 13, 2], [14, -6, 12, -13], [5, 8, -3, 7], [-4, -17, 1, 17], [-34, 17, -7, 0], [-4, 0, -2, 2], [-23, 7, 15, 5], [14, -7, 17, 2], [41, -6, 6, -5], [-35, -9, 1, 6], [-4, 5, -3, 5], [7, -9, -6, 3], [5, 3, 11, -14], [4, -5, 7, 14], [8, -3, -15, -1], [6, -4, 13, -10], [-2, 3, 11, 3], [-15, 24, -2, -7], [5, -15, 20, -2], [-47, 13, -11, 3], [3, -4, 3, 1], [3, -14, -2, -7], [-5, 14, 10, -7], [-10, 9, -8, 16], [-7, -7, 10, -9], [26, -17, 22, 2], [-28, 19, -3, -5], [13, -21, 12, -7], [-50, -4, -9, 10], [9, 11, 2, -14], [25, -7, 0, -5], [-22, -10, 4, 2], [20, 6, 7, -6], [-10, 15, 11, -21], [-13, -19, 0, 8], [12, 3, 4, -7], [-8, 3, -23, 8], [34, -19, 2, -2], [29, 6, -1, 13], [-10, -9, 13, -14], [-33, 30, -2, -3], [-32, 20, -7, -8], [-21, -6, -13, 12], [3, 2, -15, 16], [-23, 21, 6, 5], [-33, 9, 7, 5], [-8, -10, 4, -5], [30, 1, -10, 23], [-31, -7, 7, 18], [-21, -9, 1, 13], [-5, 7, -12, -11], [-9, 4, 3, 17], [-31, 19, -12, -2], [-1, 2, 1, -12], [10, -10, -16, 2], [-37, 16, 3, -7], [43, 6, 10, 3], [-39, -9, 9, 15], [16, -19, -1, 2], [-2, -16, 1, 7], [-16, -14, 1, 20], [-9, 14, 11, -9], [-21, -4, 6, -15], [12, -7, 19, -13], [16, 7, 19, -8], [5, -13, 6, -17], [-19, 4, 7, 13], [-1, 7, -8, 2], [16, -10, -1, 18], [-7, 11, -22, 17], [-10, 21, -15, 19], [-11, -1, -9, -15], [4, 11, 15, -15], [-28, -2, 6, -14], [-21, 12, 0, -9], [2, 0, 8, 12], [-13, 6, -2, -9], [-65, 8, -14, 7], [37, -10, 16, -4], [-27, -11, -3, -8], [45, -23, 4, -2], [3, 15, 6, 1], [-53, -5, 8, 6], [19, -9, -4, 1], [39, -19, -11, 12], [-7, 1, 14, -33], [-21, -1, -16, 3], [16, 0, 11, -2], [-4, -3, -17, -14], [7, 20, -9, 14], [33, 10, 6, -17], [1, -6, 5, -5], [-18, 5, -9, 2], [15, -12, 15, -19], [24, -17, -14, 15], [-33, 22, -7, -7], [13, 2, 6, 23], [48, -29, 7, 4], [18, -17, 20, -4], [3, 19, -2, 1], [18, -4, 24, -9], [29, -10, 14, -11], [25, -34, 3, 4], [22, -30, 6, 12], [-2, 6, -21, 12], [-1, -10, -20, 27], [-17, -18, -7, 7], [-66, 0, -1, 1], [3, -7, 4, 15], [-16, 11, -22, -2], [35, -12, 15, -15], [10, -4, 1, -9], [-4, 4, -22, -12], [-13, 7, 14, -24], [4, 4, -6, -17], [3, -11, -20, -13], [18, 5, -12, -16], [7, -8, -19, -3], [-6, -7, 7, -8], [-57, 1, -6, -9], [-52, 6, -5, 1], [10, -24, -1, 9], [-26, -3, 7, 22], [-3, -12, -8, -17], [-26, -1, 7, -8], [19, -13, 10, 15], [7, 5, -4, -11], [11, -9, -19, 26], [-7, -13, 14, 13], [-17, 25, 1, -3], [-9, 16, -10, -1], [-27, -11, 2, 10], [20, 3, 9, -3], [-21, 8, -24, 14], [-7, 0, -19, 10], [28, 19, -10, 6], [-38, 6, 7, -7], [-16, 5, 25, 11], [28, -26, -15, 22], [19, -15, 27, -4], [-21, -9, -2, -18], [-11, -7, -6, 9], [10, 1, 8, 1], [56, -3, -10, -2], [28, 12, -9, -11], [14, -4, 13, 4], [-6, -12, 11, -14], [10, 11, -20, 6], [-25, 27, 0, -16], [18, -13, 25, -29], [25, 5, 5, 14], [10, -5, 20, -12], [-3, 27, 1, -3], [-5, -3, 18, -33], [-15, 14, 10, -23], [-21, -2, 5, -9], [7, 12, 25, 11], [-22, 29, -2, -8], [-15, 5, 15, 8], [14, -9, 26, 12], [22, -35, 13, 3], [-14, -9, 6, -23], [15, -30, 2, 1], [-10, -12, 1, 14], [-48, 32, -2, -3], [-28, -15, 6, -4], [-31, -7, -21, 1], [-41, 27, -12, 1], [4, 2, -10, 21], [-36, 35, -20, 2], [63, -1, 18, -19], [10, 24, -8, -3], [-45, -4, 2, 18], [45, 17, -5, 9], [34, 11, 7, -25], [-39, 4, -23, -4], [8, 34, -12, -13], [22, -5, -17, 7], [2, 15, -14, 20], [26, -12, 1, 4], [5, -5, -23, 1], [41, -3, 24, -9], [-17, 25, 0, -11], [25, -10, 8, -19], [-25, 0, -12, -1], [-29, -21, 0, 7], [-23, 7, 5, -11], [19, 15, 2, -12], [50, -24, 8, 2], [50, -18, -10, 4], [-51, 1, -1, -7], [-22, -10, -14, 14], [2, 6, 15, -9], [-17, -9, 2, 18], [37, 4, -2, 21], [-22, 14, 27, 10], [2, -21, 15, 2], [-3, -22, 9, 19], [-24, 1, -6, -8], [-52, 11, -14, 8], [-4, 32, -9, 12], [42, 9, -3, 5], [7, 26, -5, 25], [4, -3, 11, -3], [-15, -6, 6, 7], [-44, -9, 3, -4], [-17, -2, 17, -36], [47, -9, 29, -9], [23, 9, 15, 14], [64, -14, 27, -16], [7, -19, 16, -8], [-42, 5, 5, 15], [-18, -2, -18, -8], [46, -8, 21, -7], [-24, -10, -9, 19], [40, -14, -15, 2], [28, -2, -18, 0], [-27, 22, -15, -11], [-15, 19, -14, -4], [13, -19, 22, 17], [-5, 23, -4, -3], [24, 22, 0, -32]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4014_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4014_2_a_q();".
// 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_4014_2_a_q(:prec:=4)
    chi := MakeCharacter_4014_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 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_4014_2_a_q();".
// 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_4014_2_a_q( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4014_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-3, -11, -5, 2, 1]>,<7,R![-12, -13, 2, 5, 1]>,<11,R![16, -9, -13, 4, 1]>],Snew);
    return Vf;
end function;