
// Make newform 1850.2.a.y in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1850_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_1850_2_a_y();".
// 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_1850_2_a_y();".
// 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 := [1, -7, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-5, -1, 1]];
    Rf_basisdens := [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_1850_a();"
function MakeCharacter_1850_a()
    N := 1850;
    order := 1;
    char_gens := [1777, 1001];
    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_1850_a_Hecke(Kf)
    return MakeCharacter_1850_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, -1, 0], [0, 0, 0], [-1, -1, 0], [-2, 0, -1], [-3, 1, 0], [2, 1, 0], [-1, 1, -1], [-2, 2, 0], [-4, -2, 0], [1, 2, -1], [-1, 0, 0], [7, 0, 2], [-6, 3, 1], [0, 0, 0], [2, 2, 2], [-2, -3, -1], [5, -3, 0], [0, -1, 2], [3, 1, 2], [-5, 1, -1], [2, -2, 2], [6, 2, 1], [2, -2, 3], [-4, 0, 2], [4, -2, -2], [-5, 0, -3], [-4, -5, 0], [4, 2, 2], [0, -2, -1], [-11, -1, -2], [4, 1, 1], [-3, -4, -2], [-4, -4, 1], [-7, 2, -1], [4, 0, 4], [5, 0, -1], [-17, 1, -1], [5, -2, 3], [2, -6, -2], [1, 4, 4], [-8, -4, 2], [-6, 4, -4], [0, 2, 5], [-3, 4, -1], [-8, 0, -2], [-12, 1, -2], [-6, -6, -2], [-8, 3, -1], [-8, 2, -4], [-6, 7, -1], [6, 4, 2], [-6, -1, 0], [-9, 0, 0], [6, 2, -2], [12, 2, -2], [-4, 4, -6], [5, -3, 6], [10, 2, -2], [-20, 3, -1], [-3, 9, 1], [3, 6, -3], [-12, 6, 1], [-1, 8, 5], [-24, -1, -1], [11, -4, 5], [-19, 5, 1], [9, -4, -4], [3, -6, -6], [-1, 6, -3], [-2, 0, -4], [12, 2, -2], [5, 1, 4], [3, -2, 3], [-4, -5, -4], [13, -4, -3], [-1, 3, -2], [-1, -8, -5], [16, -6, -1], [6, -13, 0], [-18, 4, -1], [2, 10, -4], [8, -4, 2], [-5, 0, 0], [12, -12, -4], [22, -2, 1], [16, -8, 1], [4, -8, -1], [20, 2, 8], [1, 10, -1], [-6, -6, 0], [-11, -2, 1], [13, -8, 5], [2, 10, 0], [-4, 1, 5], [24, 0, 0], [7, 6, -1], [-7, -1, -1], [-7, 12, 2], [1, -11, 0], [19, 9, -3], [-22, 2, 2], [-8, -11, 1], [6, -8, -1], [-2, -8, -6], [-8, 11, -2], [-25, 5, -1], [13, -4, 0], [-25, -1, 2], [19, -13, -1], [-17, 8, -5], [-20, -4, -2], [-12, 3, -3], [18, -8, -2], [-2, 8, 2], [6, 4, -4], [-14, -3, 3], [-35, 2, -3], [1, -5, -2], [-2, 7, -2], [8, -8, -4], [-14, 11, -5], [0, 6, 6], [-9, 1, -7], [-4, -8, 5], [23, 3, 10], [10, -2, 0], [-3, -13, -2], [-24, 0, -2], [-20, -12, 6], [8, 8, -8], [22, -12, -4], [2, 6, 0], [-12, 4, 0], [7, 3, -7], [22, -4, -1], [3, 0, 3], [-16, -2, -2], [-15, -1, 4], [-4, 6, 4], [-34, 0, -6], [-15, 0, 3], [15, -5, 0], [17, 5, 5], [-13, 3, 6], [1, 1, -2], [1, -11, 2], [-8, 1, -2], [-13, 5, 7], [3, -15, 6], [35, -10, -3], [-4, 17, -1], [19, -6, -6], [-4, -10, 8], [-20, 1, 5], [-7, -6, -5], [6, 0, -4], [14, -5, -9], [-21, 5, -1], [12, 6, 0], [-12, 4, -4], [29, 1, 3], [-32, 18, 6], [0, -20, 5], [34, 9, -4], [-28, 8, 2], [-13, 4, 5], [1, -17, 2], [39, -7, 3], [-7, -7, -4], [-31, 1, 3], [0, 0, -4], [7, -14, 7], [13, -3, 9], [35, 1, -4], [-6, -13, 10], [-3, -3, -1], [-5, -8, -5], [1, 7, -4], [-13, 6, 3], [1, -2, 5], [-29, -2, -8], [-17, -13, -2], [-8, -15, 8], [9, 3, 0], [24, 0, -6], [-26, 2, -8], [-1, 0, 8], [-22, 1, -10], [9, 13, -4], [-14, 2, -11], [-2, -13, 0], [-5, -5, 3], [6, -6, 0], [-30, 12, -3], [-10, -15, 1], [26, 2, -5], [-20, 6, 6], [43, -10, 0], [-27, 6, -3], [-27, 5, 4], [11, -7, -2], [-11, -6, 9], [-31, -6, 0], [-50, 4, -2], [8, -4, -10], [0, 14, -12], [12, 10, 14], [6, 18, -6], [-24, -7, 3], [8, 7, 7], [31, -5, -8], [23, -2, 7], [14, 4, 9], [36, -18, -6], [-25, 2, 4], [-25, 4, 1], [-48, 6, 3], [16, 4, -8], [34, -2, -2], [5, 12, -3], [40, 0, 4], [1, -2, -14], [7, 4, -1], [18, 4, -7], [23, 1, 2], [-7, -6, -14], [-16, -8, 6], [4, -2, -4], [-11, -8, 10], [21, -16, -1], [38, 1, 8], [12, -18, 2], [-11, 5, 3], [-26, -4, -14], [43, 9, -4], [35, 3, -3], [34, -2, -2], [-17, -17, 7], [-30, -8, -4], [15, 16, -4], [-17, 15, 7], [-9, -19, -4], [-9, 5, 0], [19, -17, -11], [33, -8, 5], [-3, 14, -1], [-6, 18, 0], [-22, 19, 2], [-46, -4, -4], [37, 3, 0], [-13, 11, -13], [0, -12, 12], [-38, -3, 4], [-11, -9, 8], [-28, 11, -4], [-15, -8, 7], [-8, -8, -7], [4, -8, -2], [22, -7, 4], [14, 8, -6], [-1, -7, -1], [47, 4, -1], [-33, -10, -1], [-8, 4, 7], [32, 7, 2], [-49, 14, 5], [14, -10, 14], [-8, 2, 13], [11, 3, -2], [10, -2, 12], [37, -10, -2], [-20, 2, -17], [-13, -10, -11], [12, 0, 10], [25, 5, 10], [-12, -6, -15], [-21, 11, -12], [-10, 1, -7], [-33, -5, -7], [13, 1, -8], [-13, -10, -5], [10, 8, 0], [14, -16, -8], [17, -14, -2], [-7, 1, 12], [20, -9, 11], [6, 0, 6], [-28, 14, 4], [4, -1, 3], [2, 14, 14], [-12, 7, 11], [7, 15, 5], [-34, 5, -4], [8, 16, -14], [15, -16, -5], [21, -7, -12], [39, -6, 3], [-50, -2, 4], [28, 2, -5], [10, -10, -5], [-50, 10, 4], [-25, -3, 0], [1, 11, -3], [29, 4, 2], [-16, 15, -4], [40, -27, -7], [-37, 15, 6], [32, -26, -6], [-38, 6, 6], [18, 2, -2], [-2, 4, -8], [-41, -8, -2], [-8, 6, -6], [34, -12, 8], [2, 4, 8], [-28, 16, -3], [33, -12, -3], [41, -9, -1], [-20, 22, 13], [14, 15, 9], [1, 7, 5], [17, -7, -4], [40, -4, 4], [6, -1, 7], [-16, -18, 12], [4, -6, 13], [14, -15, 11], [19, -8, 13], [-25, 12, 1], [-3, -22, 9], [-31, 11, 2], [31, -18, 1], [-35, 10, 4], [43, -12, -2], [-52, 12, 4], [-30, 20, 12], [20, -5, 21], [-24, -6, -7], [-23, 19, 8], [-23, -18, -9], [-27, -9, -3], [-28, -22, 2], [-18, 8, -12], [-3, -15, -6], [-60, 0, 0], [-34, -6, 0], [-6, 6, 13], [37, -14, 1], [44, -10, 8], [24, -8, -5], [34, -1, 7], [13, -3, 8], [-24, 0, 10], [19, 12, 7], [51, -12, -6], [10, 4, -8], [-3, -14, 2], [-28, 12, -14], [-46, 12, 4], [-15, 13, 6], [5, -19, -1], [-35, -10, -9], [-26, -21, 5], [-26, -5, 8], [32, 5, 12], [35, 11, 2], [73, -2, -1], [-4, 8, 1], [0, 21, 9], [33, -10, -6], [1, 7, -14], [-12, 12, 0], [29, -24, 3], [-45, 4, 9], [-22, 21, -2], [61, -5, 4], [23, -8, 10], [26, 2, -7], [20, 11, 9], [-28, 22, 0], [58, -14, -5], [4, 24, 0], [-35, 9, -19], [14, 9, -9], [-30, 24, 12], [39, -3, 2], [-41, 9, -12], [-2, -10, 4], [-10, -6, 16], [-27, 21, -1], [-17, 19, 4], [-31, 17, 11], [1, -20, -16], [-14, 4, -14], [-21, -10, -10], [-26, 20, 14], [35, 11, 17], [34, 2, -7], [43, -4, 3], [-10, -16, -2], [-48, -2, 5], [17, 6, -1], [-12, -15, 6], [25, 13, -8], [26, 12, -12], [-25, -13, -6], [-4, -19, -13], [-18, 22, 12], [-32, 22, 1], [-59, -9, -9], [26, -4, -4], [59, -16, 2], [0, -7, -15], [28, 11, -14], [16, 22, -14], [-7, 19, 0], [-55, -7, -12], [-55, 2, 8], [-52, 2, -16], [4, -32, -2], [1, -21, 18], [50, 8, 8], [34, -17, -2], [-5, -3, -15], [-18, 20, 4], [-8, 6, -10], [-5, -11, -11], [6, -21, 8], [-24, 12, -12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1850_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1850_2_a_y();".
// 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_1850_2_a_y(:prec:=3)
    chi := MakeCharacter_1850_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_1850_2_a_y();".
// 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_1850_2_a_y( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1850_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-1, -7, 1, 1]>,<7,R![-6, -2, 4, 1]>],Snew);
    return Vf;
end function;