
// Make newform 1859.2.a.d in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1859_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_1859_2_a_d();".
// 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_1859_2_a_d();".
// 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 := [-3, 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_1859_a();"
function MakeCharacter_1859_a()
    N := 1859;
    order := 1;
    char_gens := [508, 1354];
    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_1859_a_Hecke(Kf)
    return MakeCharacter_1859_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], [-1, 1], [0, -1], [3, -1], [1, 0], [0, 0], [-6, -1], [-3, -1], [0, 2], [6, 1], [-6, 0], [0, 1], [-9, 0], [-2, 6], [3, -1], [3, 2], [-3, -1], [-2, 3], [-3, 1], [0, 2], [-9, 2], [1, 3], [-3, -7], [0, 2], [0, -2], [-6, -1], [-8, -6], [-9, -5], [-6, 4], [-3, 6], [14, 0], [9, 1], [6, -9], [2, 6], [3, 4], [0, -10], [-1, 0], [-3, -5], [12, 4], [12, 4], [-3, 5], [1, 6], [-9, -3], [-15, 0], [6, -2], [5, 3], [-17, 3], [9, -7], [-6, 8], [0, 6], [12, -6], [6, -6], [-21, 4], [3, -7], [21, 6], [6, -10], [6, 0], [-12, -4], [10, 9], [-3, -8], [-1, -3], [-3, 6], [-15, 3], [-15, -5], [4, -12], [-6, 15], [0, -4], [-22, 3], [3, -9], [-18, -2], [-12, -1], [24, 4], [-10, 0], [4, -15], [0, 6], [-6, 6], [-27, 2], [-6, 0], [24, 5], [-15, -6], [-18, 12], [-18, -1], [-9, -5], [-13, -12], [-5, -3], [3, -13], [0, -14], [3, -8], [-3, -8], [-9, -15], [-15, -11], [21, -1], [9, 11], [9, -5], [27, 7], [15, -7], [0, 19], [-3, 16], [23, -3], [-39, 0], [-16, -6], [33, -6], [-33, -7], [6, -12], [29, 3], [6, -1], [21, -9], [-9, 12], [6, 0], [16, -3], [5, 9], [3, -2], [-36, 5], [-15, 11], [-15, 3], [-27, 2], [-15, -1], [-6, 8], [30, -2], [-9, 1], [-30, -1], [10, -3], [-18, -12], [-24, 10], [-6, 6], [24, 0], [-36, 1], [36, 4], [32, 0], [-3, 2], [12, 4], [27, 11], [4, 0], [-14, 6], [-12, 4], [24, -12], [30, -8], [-15, 3], [-12, 0], [24, 11], [6, 24], [-12, -12], [10, -18], [-3, -3], [5, -24], [-3, -23], [15, -2], [6, 11], [19, -3], [-6, 2], [15, 14], [-9, 6], [-28, -18], [-6, -6], [-43, 9], [-3, 1], [40, 6], [-30, -15], [-8, 27], [18, 6], [-24, -10], [-42, -8], [-15, -13], [-15, 19], [30, 5], [-18, 12], [41, -9], [2, -15], [-18, -2], [-3, 0], [-39, -13], [-12, -11], [45, 1], [27, 18], [-26, -12], [-15, 14], [24, 10], [-12, -7], [-14, 18], [-8, 6], [-30, -16], [-12, 4], [1, -6], [-6, 10], [-6, -22], [-3, 22], [-8, 6], [12, 10], [21, -12], [57, -3], [-25, -12], [18, 16], [8, 12], [18, 4], [-27, 3], [-12, 8], [-33, -8], [11, -24], [3, 2], [27, -13], [39, 10], [-40, -6], [-48, -9], [8, -15], [-30, 4], [27, 6], [33, 7], [-9, 21], [9, 2], [-28, 0], [14, 15], [-12, 16], [25, 3], [24, 14], [3, 13], [48, -1], [-26, -12], [-6, -5], [45, -9], [-30, -16], [8, -27], [66, -4], [6, -27], [-27, 7], [27, 1], [46, -9], [-33, 8], [-21, -1], [44, -18], [3, 19], [31, 0], [-31, 15], [12, 14], [-18, 11], [-34, -24], [12, 6], [12, 12], [-3, 4], [18, -12], [-15, -3], [24, 22], [8, -6], [-14, 6], [-18, -33], [63, -6], [-45, 9], [27, 11], [0, -4], [30, -22], [-24, 14], [-18, 6], [6, 6], [-3, 15], [17, 0], [36, -18], [30, 10], [4, -36], [-48, -12], [-24, 16], [-15, -2], [38, 6], [30, 14], [-9, 28], [-4, 15], [24, -31], [40, 6], [-12, -6], [-36, -23], [6, -4], [-24, -15], [59, 0], [-48, 16], [-6, 7], [13, -15], [-8, 12], [18, 24], [54, -10], [3, -30], [-57, 6], [-9, 35], [45, -17], [27, -25], [39, -19], [-9, 4], [9, 33], [-45, 1], [55, 0], [51, 4], [21, -25], [-36, 11], [18, -6], [-3, 1], [-3, 18], [27, -3], [44, 0], [0, 13], [2, 24], [9, 16], [15, -15], [-69, -7], [-2, 15], [-3, -10], [16, 36], [-45, 19], [5, -21], [-6, -15], [3, 27], [5, -12], [-33, 9], [20, 9], [0, 40], [18, 36], [30, 25], [-11, -3], [-30, 8], [40, -9], [-33, 15], [-81, 5], [-12, -2], [-6, 35], [-20, -30], [-12, 10], [18, -41], [18, 34], [42, 8], [61, 12], [-75, -7], [-42, 14], [-6, 6], [18, 23], [-33, 6], [27, -32], [-68, 6], [6, -12], [-60, 4], [-15, 33], [-18, 17], [-54, 8], [-24, 25], [-11, 27], [42, 16], [-27, -8], [-51, 16], [34, -18], [36, -12], [21, 3], [1, 6], [-6, 10], [-12, 42], [12, 0], [-39, -27], [24, -4], [-18, -17], [74, 0], [-16, -27], [78, 5], [12, 28], [12, -8], [-78, 3], [-69, -5], [21, -26], [0, 15], [9, 15], [3, 19], [-53, 15], [2, 18], [12, 0], [-51, -15], [-2, -21], [-6, 44], [58, 24], [-9, 33], [-24, 1], [37, -3], [23, -42], [-21, 23], [-12, 4], [66, 14], [48, -15], [22, -15], [-12, -6], [-12, 25], [-27, -3], [-72, 8], [15, -9], [-30, 24], [0, -36], [2, -18], [0, 22], [12, -42], [-24, -37], [60, 7], [-33, -15], [-38, 6], [-3, -1], [28, -6], [-12, 52], [-12, -32], [19, -21], [75, -4], [60, 16], [-48, -20], [51, 7], [-18, -45], [3, -18], [-38, -12], [-12, -12], [-15, -34], [72, -8], [30, 26], [-20, 15], [-36, 9], [66, -14], [37, -27], [-56, 15], [-81, 10], [-3, 23], [-50, 12], [-36, -2], [9, -29], [-21, 22], [9, -36], [51, 1], [3, -33], [-12, 18], [6, -33], [-15, -35], [51, -28], [-87, -11], [48, -22]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1859_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1859_2_a_d();".
// 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_1859_2_a_d(:prec:=2)
    chi := MakeCharacter_1859_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_1859_2_a_d();".
// 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_1859_2_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1859_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-3, 0, 1]>,<7,R![6, -6, 1]>],Snew);
    return Vf;
end function;