
// Make newform 1849.2.a.f in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1849_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_1849_2_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_1849_2_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
        poly := [-2, 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_1849_a();"
function MakeCharacter_1849_a()
    N := 1849;
    order := 1;
    char_gens := [3];
    v := [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_1849_a_Hecke(Kf)
    return MakeCharacter_1849_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, -1], [-2, -1], [2, 1], [-1, -2], [1, -2], [5, -2], [2, -2], [1, 4], [0, 3], [-3, 0], [0, -6], [-1, 2], [0, 0], [6, 0], [11, 2], [-2, -2], [-4, 3], [1, -6], [6, -2], [12, 3], [2, 2], [9, -4], [6, -3], [-1, 2], [3, 2], [9, -6], [-6, 4], [-3, -12], [4, 2], [1, 2], [-4, -4], [6, 6], [-3, 6], [0, -6], [-14, -3], [10, 0], [16, -3], [-3, -8], [18, 4], [6, -1], [-4, 8], [-8, 10], [-1, 12], [0, -10], [-2, -4], [-14, 10], [-14, 7], [12, -4], [-9, -12], [-2, -2], [6, 6], [-4, 0], [9, 10], [6, -3], [14, -10], [-3, 0], [-9, -12], [2, -7], [-3, -4], [7, 8], [16, -4], [7, 14], [3, -12], [4, 15], [23, 2], [-20, -1], [-5, 0], [6, -1], [16, 3], [3, 2], [-27, 4], [-10, 14], [0, -6], [19, 8], [12, 6], [-6, 16], [4, 18], [-21, 6], [-10, 19], [14, 7], [10, 4], [-5, 20], [-26, 3], [3, 6], [22, -10], [-12, -15], [12, -9], [20, 16], [18, 9], [-18, -4], [-1, -4], [28, -4], [-12, -1], [30, -3], [-6, 26], [9, -6], [-24, 5], [18, 15], [-11, 14], [9, 0], [-9, 0], [17, 16], [19, -4], [-4, 5], [-26, -5], [-14, -14], [2, 5], [31, 4], [8, 12], [16, 12], [4, 2], [3, 12], [12, 12], [-10, -26], [14, 11], [-18, -6], [4, -2], [-18, 2], [5, 8], [-17, 0], [4, 9], [-14, 2], [-21, -18], [14, 9], [-24, -4], [-7, -22], [0, -16], [8, 12], [0, -12], [14, 18], [16, -22], [16, -3], [-12, 6], [12, -22], [8, 26], [-18, 7], [-10, 14], [24, 8], [0, -4], [-36, 9], [-21, -22], [-49, -4], [4, -4], [-4, -14], [22, 19], [7, 28], [2, 4], [0, -12], [2, 32], [17, 14], [-9, 32], [-25, 12], [-42, -12], [-3, -12], [0, 3], [-21, 6], [-42, 2], [10, 29], [-9, -16], [-3, -2], [0, 17], [-7, -24], [11, -10], [-12, -8], [30, 23], [4, -11], [-10, 5], [-14, -18], [50, -4], [42, -10], [6, -12], [36, -1], [5, 0], [38, 0], [-24, 6], [28, 10], [30, -21], [-2, 32], [4, -20], [-2, -23], [-6, -16], [-8, -9], [-6, -21], [-4, -23], [-24, -2], [-26, 21], [38, -7], [16, -12], [-24, -9], [-29, -14], [2, -1], [-14, 6], [-48, 0], [-10, -11], [-12, -12], [-19, -22], [-29, -20], [-60, -6], [-12, 19], [-21, 24], [-38, 6], [-40, -16], [50, 4], [6, 37], [-20, 32], [-4, 24], [41, 8], [18, -26], [-7, 22], [-18, 30], [27, 2], [37, -14], [17, -8], [-4, 35], [-5, 36], [32, -11], [0, -8], [-14, -13], [-3, 24], [-50, 3], [14, -14], [-12, 25], [-25, -14], [12, 32], [-47, 14], [-57, 0], [44, 7], [-14, 2], [30, -10], [28, -3], [-17, -14], [29, -12], [12, 36], [41, 16], [-10, 28], [-52, 8], [-47, 2], [-24, -19], [-21, 28], [0, 18], [60, -6], [40, 12], [20, -6], [36, -2], [38, 20], [-16, -24], [-37, -16], [13, -2], [-51, 8], [24, -30], [-25, -22], [15, -6], [14, -42], [-10, -37], [-64, 10], [-8, 6], [-33, -24], [-30, 25], [47, 16], [-50, -8], [-54, -15], [2, -24], [32, 6], [-36, -13], [-6, 0], [-26, -8], [-15, -12], [-58, 0], [35, 10], [21, 12], [22, 9], [6, 15], [2, 11], [-16, 18], [40, -29], [7, -8], [-16, 29], [19, -4], [12, -15], [-49, 8], [-24, -12], [21, 0], [-22, -33], [28, -19], [30, -31], [27, 12], [30, -15], [8, -3], [-27, -2], [15, 36], [-3, -42], [16, 2], [22, 26], [-5, -42], [-5, 4], [39, 18], [17, -20], [-18, 6], [-16, 32], [13, -42], [0, 15], [-23, -14], [13, -8], [-6, 27], [-14, 24], [33, -28], [20, 44], [-30, 19], [-20, 6], [50, -14], [18, -25], [3, 0], [-46, 12], [15, -26], [-13, -8], [-2, -10], [-14, 38], [49, 14], [-6, 36], [-1, 24], [24, -12], [-30, 18], [42, -24], [16, 14], [21, 12], [-50, 3], [-67, -8], [24, -28], [-18, 53], [-34, -15], [-27, -6], [8, -9], [-24, -23], [-42, -18], [-25, 22], [60, -12], [-18, 10], [24, -6], [44, -23], [-21, -18], [12, 35], [48, 16], [-28, 3], [39, 12], [13, -22], [10, -7], [20, 8], [0, 28], [-15, 14], [42, 12], [-44, 15], [0, 46], [66, -21], [24, -42], [14, -7], [20, -38], [16, 16], [15, -38], [63, 4], [74, 16], [-26, 47], [46, -11], [14, -20], [19, -30], [-28, -6], [-48, 12], [14, 10], [-10, -33], [-2, 8], [-66, 15], [-82, 2], [30, 13], [67, 0], [48, 24], [-18, -28], [-6, -54], [39, 0], [-8, -31], [34, -19], [-17, 14], [19, 8], [11, -4], [-34, 23], [-51, 18], [-3, 14], [6, -15], [-29, -18], [-14, -8], [45, -6], [-21, 18], [-30, 13], [-24, -18], [13, -42], [60, -21], [16, 20], [-19, -10], [-64, -5], [46, 23], [60, 31], [-11, 6], [0, 48], [33, 24], [12, 30], [30, -12], [40, -15], [18, 24], [-60, -2], [-28, -22], [81, -10], [-60, 6], [-74, 2], [42, 24], [3, -42], [30, -15], [-3, 18], [27, 8], [14, 36], [-8, -40], [0, 23], [32, -25], [4, 32], [44, 13], [33, -38], [22, 12], [4, -2], [-2, -16], [-64, -23], [11, 0], [-30, -39]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1849_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1849_2_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_1849_2_a_f(:prec:=2)
    chi := MakeCharacter_1849_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_1849_2_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_1849_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1849_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-2, 0, 1]>],Snew);
    return Vf;
end function;