
// Make newform 1449.2.a.o in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1449_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_1449_2_a_o();".
// 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_1449_2_a_o();".
// 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, 5, -4, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-3, -1, 1, 0], [1, -4, -1, 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_1449_a();"
function MakeCharacter_1449_a()
    N := 1449;
    order := 1;
    char_gens := [1289, 829, 442];
    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_1449_a_Hecke(Kf)
    return MakeCharacter_1449_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 := [[0, -1, 0, 0], [0, 0, 0, 0], [-2, 0, -1, 1], [-1, 0, 0, 0], [0, -1, 0, 1], [3, -2, 0, -1], [1, -2, 1, 0], [0, -1, 0, 1], [-1, 0, 0, 0], [1, 0, 3, 0], [3, -2, 1, 0], [3, 0, -1, 2], [-4, 3, -2, 1], [3, -2, 0, 1], [3, 3, -1, 1], [-5, 1, -4, 0], [7, -1, 0, 0], [2, -1, 3, 2], [0, -2, -5, -1], [7, -2, 2, -1], [3, 2, 3, 0], [-3, -2, -5, 0], [3, -2, 1, -4], [3, 6, 2, -3], [7, 0, -1, 2], [1, -7, 2, 0], [3, 1, 5, -3], [-7, 6, -2, 1], [-2, 0, -1, 5], [-8, 2, -3, -1], [4, 3, 1, 0], [8, -1, -2, 3], [-2, -1, 2, 3], [-4, 2, -1, 3], [1, -7, -1, -1], [5, 1, 1, 3], [15, 1, 1, -1], [7, 3, 0, 4], [6, -5, 0, -1], [9, 4, 5, -6], [6, 6, 3, -3], [-1, 8, -5, 2], [-3, 1, 1, 3], [5, -1, -1, -3], [-13, -2, 4, -1], [1, 1, -2, -2], [-6, 1, 4, -3], [2, 4, 5, -3], [-10, 0, 5, 1], [-3, 3, -4, 4], [-8, 0, 7, -1], [-4, 10, 3, -5], [-6, 7, 0, -3], [-2, 2, -4, 4], [15, -1, 9, -7], [10, -5, 0, -5], [-1, 2, -2, -3], [-1, -4, 3, -4], [-2, -5, 1, -4], [-4, 0, 4, 2], [2, 2, 1, 7], [6, 2, 4, 6], [-9, 4, -11, -2], [18, -9, -3, -2], [-9, 1, 7, -3], [6, 2, 7, -9], [1, 1, 3, 1], [-9, -2, 2, 5], [-7, -2, -7, 2], [-18, 2, -5, -1], [-9, 0, -1, -2], [4, 2, 3, 3], [4, -7, -3, 6], [-21, -4, -9, 6], [6, 6, 10, -6], [9, -2, 3, 4], [-13, 8, -1, 6], [-14, 8, -6, 10], [12, -3, -10, -1], [-7, -12, -9, 8], [-10, 4, -13, 7], [9, -8, -2, 5], [4, 9, 9, -6], [-15, 3, -5, 1], [-7, 12, -1, 2], [12, 0, 0, 8], [-2, 2, 3, 7], [-8, 14, 3, -7], [12, -2, -5, -3], [8, 1, 0, 3], [-19, 2, -13, 8], [21, 4, 5, -8], [7, 8, 9, -6], [3, -8, -6, 5], [-14, 10, 1, -9], [-20, 10, 3, 3], [-9, 7, 1, -3], [-10, -8, 4, -4], [-1, -5, 5, -13], [9, 1, -1, -5], [-6, -10, -3, 7], [-8, -8, 0, -2], [7, -2, 6, -9], [11, -3, 3, -11], [9, -16, -1, 2], [-10, 12, -4, -4], [2, 1, -3, 0], [24, 4, 4, 2], [19, 0, 2, 1], [14, -8, 5, -13], [7, 2, -6, 3], [19, 0, -3, -4], [-17, 10, -2, 1], [23, -6, 6, -5], [21, 2, 9, -10], [-20, 3, -7, 2], [2, 3, -13, 0], [-2, -2, 1, 3], [-24, 4, -1, 7], [-17, -8, 5, 2], [2, 5, -8, 3], [-14, 7, -6, 11], [-3, 8, 2, 9], [-23, -4, 5, 0], [17, -14, -2, 5], [-5, -11, -14, 6], [3, 10, 10, -7], [-5, 20, -3, -2], [8, -7, -4, -9], [-16, 0, -12, 6], [7, 0, -9, 0], [-7, 5, 14, 2], [-6, -2, 3, 5], [-5, 6, 3, -8], [11, -3, -1, -7], [-4, -16, -8, 10], [-25, -2, -3, -2], [19, -5, -4, -4], [-6, -4, 2, -6], [19, -14, 2, 9], [15, -2, 3, 6], [20, -2, 4, 0], [15, 1, -6, 0], [11, -3, 4, -14], [19, 0, 1, 0], [-5, 0, -6, 1], [14, 10, 6, -8], [13, 15, 3, -3], [-26, 8, -10, 12], [18, -14, 0, -8], [11, -15, -9, 5], [12, -12, 0, 2], [9, -6, 18, -7], [-13, 22, 8, -3], [-6, 18, -5, -3], [14, -4, 8, -6], [-7, 18, -3, 6], [-10, 6, -1, -1], [4, -17, 12, -5], [22, -6, 12, -2], [-18, 8, -6, 8], [0, -3, 1, -12], [6, -4, 18, -8], [3, -16, -8, 7], [-6, 1, -5, 0], [-3, -6, 11, -4], [26, 11, -7, 6], [1, 6, -5, 6], [-16, 6, -9, 13], [-8, -3, -11, 4], [10, -3, -9, 2], [10, -8, 24, -4], [8, 5, 5, -2], [11, 6, 15, -8], [-10, 10, -10, -2], [-7, 12, 7, -4], [-8, 13, -1, 0], [-18, -6, -17, 7], [21, -3, 20, -8], [2, 2, 3, -13], [11, -7, 4, -6], [7, 12, 14, -7], [5, 5, -4, 10], [14, 5, 5, -10], [10, -29, 2, 1], [-19, 10, 6, -5], [-21, 11, -1, 11], [-5, -8, -9, -4], [9, 1, 4, -10], [-1, 8, -9, -4], [-14, 14, -13, 11], [-24, 10, -1, 7], [12, -4, -2, 6], [18, -1, 0, 5], [-8, -11, 9, -10], [-27, 16, 0, 7], [-35, -6, 2, -1], [-8, 20, -4, -6], [16, 2, -10, 2], [11, -11, 12, 6], [3, -9, -9, 11], [-4, 16, -4, -8], [26, 6, 21, -3], [-18, -9, -13, 2], [23, 6, 11, -2], [-27, -7, -23, 9], [13, 0, 1, -8], [-22, 8, -11, 13], [0, 16, 11, 3], [-15, 2, -7, 6], [-37, 7, -3, 5], [-7, -13, 18, -10], [-1, -2, 1, -4], [-10, -14, -2, 2], [1, 10, 0, -1], [-6, -10, -9, 7], [12, -1, -2, -5], [11, -2, 0, 7], [1, 32, -11, 0], [-41, 8, -1, 2], [16, -1, 3, 0], [27, -13, 1, -1], [15, -8, 2, -1], [16, 0, -8, -8], [14, 11, 18, -7], [3, -3, -6, 10], [23, 0, -15, 4], [-12, 4, 4, -4], [11, 2, -18, 3], [-15, 8, 1, 0], [-14, -9, 10, -1], [9, -2, 6, 1], [-20, -18, -13, 7], [-22, 22, -4, -6], [-26, 3, -15, 6], [25, 10, 18, -3], [-5, -9, 9, -15], [-23, 28, 11, -4], [-19, 32, 9, -4], [35, -8, 11, 4], [14, -19, -6, -1], [-28, -14, -13, 7], [40, 8, 10, -2], [0, 17, -14, 3], [18, -1, 8, 1], [18, -13, -15, -2], [-10, 30, 6, -6], [-38, 10, -14, 2], [3, 14, 11, -6], [18, 21, 0, -7], [-7, 2, 9, -12], [20, -6, 2, 2], [-36, 1, -24, 7], [20, 9, 16, -11], [1, -20, -17, 8], [6, -10, 8, -12], [-5, -14, -17, 0], [16, 5, 20, 7], [-20, 27, 1, 6], [9, -4, -1, 8], [-13, 6, 14, -1], [-1, 3, -7, 7], [2, -2, -5, -1], [-30, 21, 14, -3], [16, -13, -3, 2], [-22, 11, 12, 3], [19, 15, 11, -1], [-19, -12, -5, 4], [-35, 12, -1, -6], [-4, 20, -13, -1], [25, -10, 3, -2], [-33, -2, -1, 10], [-7, 26, 10, -13], [36, 6, 10, -12], [-23, -2, -5, -6], [32, 14, -8, 2], [-5, -8, -9, -4], [-48, -7, -2, -1], [-15, -26, 5, 0], [15, 4, 21, -10], [47, -5, 7, 13], [49, -2, 10, 1], [13, -6, 3, -12], [-7, 4, 1, -14], [8, 8, 9, -17], [9, -6, 9, 10], [-49, 16, -5, -10], [-21, -5, -17, 15], [13, -10, -2, -7], [16, -12, -14, 4], [8, -5, 11, -24], [15, 12, -10, 9], [33, 6, 20, -23], [-21, -16, -13, 8], [32, -28, 2, -12], [-17, -14, -19, 10], [16, 12, -8, 12], [-26, -1, -1, -10], [54, -15, -6, 3], [-55, 26, 11, 4], [19, 10, -4, 3], [-42, 16, 4, 0], [31, -6, 9, 0], [-32, 18, 9, -3], [38, -10, 2, 2], [-2, 13, 4, 7], [20, -2, -13, 11], [-6, -5, -18, 7], [-51, 11, -11, -11], [-27, 2, -4, -7], [-44, 5, -12, 15], [-1, 8, -15, 8], [-11, 13, 0, 6], [-1, 8, 13, 2], [30, -17, 4, -9], [33, -1, 5, 11], [20, -19, 8, 7], [-9, 8, -21, 8], [0, 27, 4, -5], [-24, -2, 20, 0], [-39, 21, 3, -3], [-17, 2, -18, 5], [15, 17, 12, -4], [-22, 11, -10, 21], [-14, 11, 8, 13], [-33, 9, 8, 0], [31, -20, -11, -10], [-38, -10, -11, 15], [23, -1, 11, 5], [-21, -11, -7, -5], [52, -4, 18, 0], [27, -10, -1, 0], [-37, -6, -20, 21], [23, -7, -12, 2], [30, 16, 16, -14], [-3, 9, 15, -23], [36, -4, -4, -2], [22, -16, -2, -22], [46, -10, 9, 9], [-6, 6, -17, 1], [5, -12, -21, 4], [33, -15, 1, 17], [-8, -1, -17, 8], [15, -22, -13, -10], [-1, 11, -29, 7], [29, -44, -6, 11], [-4, -12, -17, -9], [29, 9, -8, 12], [24, -32, 5, 15], [-15, -12, 17, -10], [31, -24, 0, 11], [-36, 25, 12, -7], [7, 22, -1, -4], [43, 15, 0, 0], [-15, 24, -5, 12], [-7, -5, -7, -17], [14, -18, 16, -24], [-31, 18, 25, 6], [35, 0, -11, -4], [17, 10, 25, -22], [-21, 22, -3, -10], [10, -31, -8, 7], [-27, 0, 13, 4], [-5, 1, 5, -7], [18, 12, 4, -10], [-5, 13, -1, 5], [16, -5, 17, -14], [36, -6, 10, 8], [-5, 14, -22, -7], [-29, 18, 2, -13], [-33, 16, 5, -14], [34, -14, 22, -6], [-1, -3, -7, 7], [3, 21, 5, -7], [13, -8, -1, -20], [20, 22, 13, -15], [21, -9, -22, 0], [27, -17, 1, 1], [0, -18, -22, 2], [29, 2, 3, -6], [11, -26, 7, -16], [-25, 18, -2, 1], [19, -9, 31, -3], [-13, 28, 2, -17], [-32, 14, -11, -3], [-8, 15, -15, 24], [2, 5, 11, 2], [-8, -14, -6, 6], [35, -24, 3, 0], [12, -12, 10, 6], [29, -14, 23, -18], [-1, 9, 0, 8], [12, 18, -4, 2], [2, -44, -16, 18], [8, -1, -1, -6], [28, -2, 37, 3], [24, -9, 20, 1], [23, 4, 15, -6], [-39, 12, -20, -11], [31, 16, -9, 2], [35, -22, -3, 2], [-37, 28, -10, 19], [-26, 10, 8, -4], [-51, 34, 22, 7], [-18, 23, 13, -16], [-27, 20, -6, 31], [-34, 6, 4, -2], [36, 4, 16, -14], [12, -20, -4, 16], [-28, 22, -15, 13], [23, -10, 24, -17], [6, -12, -14, -2], [25, -33, 3, -5], [27, 14, 6, -13], [-21, 18, 12, 11], [6, -24, -8, 20], [23, 6, -27, 0], [5, 0, 11, 14], [-27, -26, 11, -6], [-30, 20, 0, -10], [13, -16, -13, 12], [46, -21, -16, -7], [34, -8, -3, -7], [35, 6, 8, 1], [-1, 16, -10, -9], [-9, -1, -3, -7], [26, -9, -23, -2], [8, -8, -16, -6], [6, -12, 5, -27], [-10, 19, -31, 18]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1449_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1449_2_a_o();".
// 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_1449_2_a_o(:prec:=4)
    chi := MakeCharacter_1449_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_1449_2_a_o();".
// 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_1449_2_a_o( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1449_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![2, -5, -4, 2, 1]>,<5,R![-32, -38, -3, 5, 1]>],Snew);
    return Vf;
end function;