
// Make newform 4050.2.a.bl in Magma, downloaded from the LMFDB on 15 November 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4050_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_4050_2_a_bl();".
// 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_4050_2_a_bl();".
// 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 := [-6, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_4050_a();"
function MakeCharacter_4050_a()
    N := 4050;
    order := 1;
    char_gens := [2351, 3727];
    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_4050_a_Hecke(Kf)
    return MakeCharacter_4050_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 := 2;
    raw_aps := [[-1, 0], [0, 0], [0, 0], [-2, 1], [0, 2], [-2, 1], [0, 2], [5, 1], [0, -1], [0, 1], [2, 1], [4, 3], [-9, 0], [-5, 1], [-6, -2], [-6, 1], [3, 1], [8, 0], [7, -3], [-6, -3], [1, 0], [2, 6], [-3, -1], [9, 0], [1, -4], [-6, -1], [-2, -6], [-3, 5], [8, -1], [-9, 2], [4, -3], [-6, 4], [3, 0], [-4, 0], [6, 0], [20, 0], [10, 4], [-5, -1], [0, 8], [0, 4], [3, -5], [8, 4], [6, -2], [-20, 0], [12, -5], [-4, -4], [-1, -1], [-8, -4], [21, -3], [-4, 7], [-9, -6], [-12, -1], [-13, 4], [3, -1], [-3, -4], [18, 1], [-12, -1], [8, 3], [4, -1], [-24, 2], [1, 5], [-6, 9], [-8, -6], [6, -2], [1, 2], [0, 7], [-1, 3], [16, 2], [12, 0], [14, 0], [-9, 0], [24, -4], [-2, 6], [4, 8], [-4, -2], [24, -1], [-6, -8], [-2, 8], [-24, -6], [5, -2], [9, 7], [-4, -9], [-18, -3], [10, -8], [-4, 3], [18, -4], [9, 8], [1, -6], [-12, 3], [-2, 3], [-3, -3], [12, -5], [22, 1], [15, 5], [-1, -3], [24, -1], [0, 0], [0, 12], [13, 3], [-28, -4], [-23, 3], [12, 12], [3, -11], [-24, -4], [-1, 11], [-11, -12], [-12, -6], [-3, -2], [30, -3], [8, -10], [4, -3], [10, -9], [15, 12], [5, -15], [-16, 4], [-15, 12], [25, -3], [12, 5], [6, 5], [-27, 7], [-22, 12], [-8, -6], [6, -13], [15, 11], [17, 9], [-24, -4], [2, 1], [-30, 9], [16, 0], [-26, 0], [-19, 9], [42, 3], [-28, 9], [16, 9], [-9, -2], [11, -16], [-30, 8], [-20, -6], [-18, -5], [21, -6], [-31, 1], [12, -16], [22, -2], [-9, -11], [-10, -15], [-30, -3], [46, 0], [-45, -2], [-13, -3], [-12, -13], [22, 4], [-24, -6], [-38, -4], [30, -10], [-29, -3], [0, 3], [-4, 3], [12, 16], [-17, 4], [-12, -17], [9, 7], [12, -4], [-32, 0], [9, 13], [-9, 0], [-18, 2], [-16, 0], [-14, 9], [-13, 0], [-24, 11], [33, -5], [8, -12], [0, 11], [1, 10], [8, 5], [-51, 4], [-22, 4], [18, -8], [10, -2], [-16, -4], [10, -13], [21, 7], [10, 12], [-9, 8], [-30, -1], [12, -8], [-26, 0], [25, -1], [-34, -8], [6, 11], [40, -4], [15, 5], [2, 8], [12, -12], [36, -6], [3, -14], [-19, -6], [-8, -9], [24, -10], [42, -2], [30, -5], [14, -18], [46, 0], [8, -12], [-3, -27], [-48, -4], [-10, -11], [-9, 19], [39, 6], [-1, 7], [-17, -10], [-12, -7], [16, -21], [-39, -3], [-36, 8], [-28, 12], [-2, 10], [15, -14], [0, -4], [-30, -12], [14, 13], [14, -6], [0, -22], [-26, 6], [3, 9], [62, -1], [-21, 18], [-6, 6], [-2, -13], [33, 7], [28, 7], [5, -3], [50, 2], [-3, -2], [4, 0], [12, -1], [23, -12], [-6, -9], [12, 0], [-24, -11], [9, -11], [5, 15], [-50, -6], [-4, -27], [-30, -8], [36, 3], [-26, 2], [39, -1], [23, -5], [-6, 10], [-32, 0], [39, -4], [30, 7], [5, 24], [-42, 11], [-33, 13], [2, 3], [7, 15], [-6, -1], [-8, 0], [-2, 26], [-27, 3], [38, -3], [-62, 4], [15, 2], [20, 0], [18, 16], [51, 6], [-26, -12], [12, 12], [8, 11], [1, 3], [-32, 18], [-4, -24], [37, 14], [-32, -12], [-51, 5], [20, -20], [-43, -8], [-18, 20], [24, 11], [26, -21], [12, -8], [38, 12], [28, -18], [18, 10], [22, 16], [18, 1], [-4, 9], [-39, 14], [-54, 12], [-3, 11], [-21, 10], [72, -2], [16, -15], [12, 13], [2, 7], [-6, -27], [36, 0], [13, 15], [-56, 6], [-12, -5], [-22, 14], [69, 3], [71, 3], [61, 4], [-36, -6], [14, 9], [12, -24], [-8, 12], [24, 1], [-30, 8], [15, -4], [-20, -4], [54, -3], [5, 14], [12, -10], [-6, -34], [-38, -16], [-9, -28], [5, -15], [43, 4], [-36, -12], [-8, -25], [0, 18], [59, -12], [20, 0], [1, 15], [-30, -25], [-24, 16], [26, -24], [6, -12], [50, -14], [-54, 12], [-7, 21], [-12, 12], [-46, 12], [9, -18], [47, 14], [52, -3], [4, -28], [33, -6], [54, -3], [32, -13], [12, -11], [-51, 1], [-4, -27], [7, -5], [-30, -7], [-18, -16], [17, 5], [-41, 0], [18, 20], [-2, 21], [-58, 3], [-15, 8], [-12, -1], [57, 7], [0, -26], [6, -3], [76, -7], [30, 0], [24, -3], [-57, -7], [37, 9], [40, -20], [-6, -3], [-14, -15], [29, -22], [-12, -2], [-61, -1], [-60, 16], [-18, -29], [14, 6], [-32, -16], [-21, 11], [30, 26], [1, -14], [24, -2], [-56, 18], [-48, -16], [12, -8], [10, -30], [75, -10], [5, -15], [48, 8], [-46, -6], [-2, -12], [4, 24], [12, 0], [11, -8], [18, -16], [15, 13], [10, -28], [-36, 0], [1, -36], [-22, -14], [21, 0], [8, 2], [30, -17], [8, 9], [-93, -2], [16, 32], [33, 20], [-36, -8], [-34, 6], [40, -16], [-39, -20], [19, 15], [39, -15], [-41, 8], [18, 16], [-12, -20], [-43, 15], [-35, -2], [6, 17], [48, -1], [-14, 2], [33, -16], [0, -35], [54, 4], [-20, -5], [90, -6], [-12, -26], [37, 12], [-78, 12], [21, -3], [21, 2], [5, 27], [36, 16]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4050_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4050_2_a_bl();".
// 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_4050_2_a_bl(:prec:=2)
    chi := MakeCharacter_4050_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_4050_2_a_bl();".
// 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_4050_2_a_bl( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4050_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-2, 4, 1]>,<11,R![-24, 0, 1]>,<13,R![-2, 4, 1]>,<17,R![-24, 0, 1]>,<23,R![-6, 0, 1]>,<41,R![9, 1]>],Snew);
    return Vf;
end function;