
// Make newform 2850.2.a.bn in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2850_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_2850_2_a_bn();".
// 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_2850_2_a_bn();".
// 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, -3, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [-1, 2, 0], [-4, -2, 2]];
    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_2850_a();"
function MakeCharacter_2850_a()
    N := 2850;
    order := 1;
    char_gens := [1901, 1027, 1351];
    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_2850_a_Hecke(Kf)
    return MakeCharacter_2850_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], [1, 0, 0], [0, 0, 0], [0, 0, 1], [-1, 1, 1], [2, 0, -1], [3, -1, 0], [-1, 0, 0], [2, 0, -1], [3, -1, -1], [5, 1, -1], [0, -2, 1], [0, 0, -1], [-3, 1, -1], [0, 2, 1], [4, -2, 0], [5, 1, 0], [2, 0, 0], [-2, 2, 0], [-2, -2, -2], [-2, 2, 2], [7, -1, 1], [-1, -3, 0], [2, 2, 1], [-3, -3, -3], [-3, -3, 0], [-11, -1, 0], [2, 2, 2], [3, 5, -1], [-2, 4, 2], [-11, -1, -2], [5, -1, -1], [7, -1, 2], [4, 0, 2], [5, 1, 0], [3, 3, 3], [-7, 1, 0], [-7, 1, -3], [-2, 2, 0], [0, 2, -2], [9, 1, 2], [5, -5, -1], [-10, 2, -1], [-9, -5, 1], [-6, 2, 3], [-2, 2, 4], [2, -2, 0], [-11, -1, 0], [-8, -4, 2], [-2, 0, 4], [-7, -3, -8], [-2, 2, -1], [0, -2, 4], [9, -1, 1], [2, -4, 2], [2, -8, -1], [7, -1, 1], [14, 2, -4], [-5, -1, -4], [6, 2, -3], [-5, -1, -7], [2, 0, -6], [-22, 2, 0], [-10, 2, -1], [-4, 0, -4], [6, 0, -6], [16, 4, 2], [-9, 3, -1], [3, 1, 0], [-16, -2, 2], [-1, 7, 0], [12, -4, 1], [-18, -2, -3], [0, -2, 3], [-2, -6, -6], [-16, 4, 0], [-21, -1, 2], [-3, -7, 2], [16, 0, -3], [-6, 8, 2], [1, 3, -7], [-7, -5, 3], [0, -8, -6], [-1, 11, 5], [-1, 7, -3], [21, 3, 6], [-8, 4, 1], [-4, 4, 4], [13, 5, -2], [-12, -4, -3], [1, -9, -6], [10, 6, -5], [1, 3, 4], [-15, -5, -1], [-8, -4, -4], [-6, -4, -5], [-1, -5, -3], [16, 0, 7], [4, 0, 10], [-32, -2, -2], [-22, -6, 0], [8, 4, 5], [0, 4, 8], [-16, -4, -3], [-8, -4, -10], [-4, 4, 4], [19, 1, 8], [19, 7, 0], [0, 0, 0], [16, -2, 2], [1, 3, 4], [-1, -1, 0], [29, 1, -2], [-6, 2, -8], [2, 2, 0], [-16, -8, 1], [23, 3, -3], [-30, 0, 1], [-12, -8, -3], [11, 7, 8], [-23, -5, 5], [9, 5, 5], [14, 4, 8], [-6, 10, 0], [-16, 4, 6], [1, 13, 2], [-18, 8, 6], [-16, 0, 7], [-32, -4, 5], [-13, -9, 4], [-2, -2, -6], [0, -8, -2], [-23, 1, 5], [21, -11, -6], [18, -4, -2], [-18, 0, -4], [-24, -2, 6], [0, -8, -4], [14, 4, -2], [16, -2, 8], [14, 6, 6], [35, 3, 4], [24, -4, -5], [-18, -2, 2], [19, 5, 7], [-6, -6, -8], [-13, 11, 4], [20, 10, -4], [-22, -6, 4], [-34, -2, -4], [-12, 6, -5], [10, 8, -6], [-13, -5, 9], [6, 14, 4], [18, -10, 0], [-2, -10, -2], [10, -6, 0], [6, -4, 8], [12, 0, 2], [11, -1, 7], [-23, 7, 2], [-4, -14, -4], [-2, 6, -5], [19, 7, -4], [26, -4, -6], [-20, 4, 8], [-23, 5, 5], [5, 5, -4], [20, 2, 0], [0, 4, 3], [-9, -1, 8], [-33, 1, 5], [4, -4, 1], [-28, 8, 4], [-15, -3, 5], [40, -2, 0], [-4, -8, 2], [-25, -1, -6], [13, -9, -8], [-10, 0, 0], [10, 10, 11], [35, -5, -6], [20, 6, 3], [-4, -6, 4], [12, 6, 9], [27, -1, -4], [8, 6, 5], [44, 4, 0], [8, 2, 4], [22, 2, 15], [1, -11, -9], [23, 5, 12], [38, 6, -2], [5, 1, -1], [25, 11, -4], [30, 0, 8], [6, 8, 12], [29, 1, 0], [-15, -7, 6], [-36, 2, -5], [23, -1, 9], [-3, -7, 5], [34, 4, 3], [6, -12, -2], [27, 1, -9], [-2, -6, 5], [44, 0, 2], [-10, 6, -2], [6, 4, -2], [-22, -6, -12], [-22, -6, 0], [-1, -9, -4], [-34, 6, 1], [-14, -6, 4], [26, -6, -2], [24, 10, 6], [6, 2, 9], [12, -4, 5], [-28, -8, 2], [12, 4, 9], [11, 13, -3], [1, 1, 13], [12, 12, 1], [-2, -6, -3], [16, -4, 4], [18, 12, 14], [-30, 8, 8], [-32, 0, -6], [-17, -11, -10], [-29, -15, 7], [11, 11, 6], [24, -4, -8], [31, 3, 7], [-38, 2, 5], [-5, -5, 3], [-18, -12, -13], [2, -4, -4], [0, 16, 3], [11, 13, 3], [24, 8, -4], [10, 2, -2], [56, 4, 2], [30, -6, 3], [35, -11, -5], [10, 8, -2], [2, -2, -9], [24, 4, -7], [11, 11, 12], [-14, 6, -12], [-6, -4, -15], [-11, 13, -2], [42, 0, 0], [-24, -6, -1], [12, -6, -4], [54, 6, -5], [21, -9, -7], [-6, 8, 6], [30, 6, -4], [-6, 0, 8], [50, -2, 6], [-27, -1, -8], [-16, 12, -2], [-28, -2, -14], [38, -4, 7], [-5, -13, -4], [16, 0, 8], [-25, -9, 5], [64, 2, -4], [14, 10, -10], [4, 4, 15], [23, -15, -7], [32, -8, -8], [2, -10, -6], [-4, 0, 8], [11, -1, 19], [-10, 10, 3], [3, -11, 10], [-13, -3, 11], [34, 4, -4], [-15, -1, -1], [-18, 10, -6], [-20, 8, 4], [6, 4, -11], [9, -1, 1], [-13, 15, 7], [-30, -10, -11], [-4, 12, 16], [28, 2, -10], [36, 8, 8], [-26, 2, 7], [-23, 9, 10], [-45, 1, -4], [-12, -10, -8], [-1, -1, -4], [-58, 0, -1], [37, 13, -4], [67, 3, 5], [-2, 6, -3], [3, -5, 4], [41, -7, -1], [-4, 4, 6], [-16, 2, 8], [0, 0, 0], [-6, -14, 0], [4, 0, -4], [9, 1, -7], [24, 4, 8], [7, -7, 1], [-50, -6, -5], [-63, -7, 4], [24, 2, -5], [-25, -5, -20], [8, 0, 3], [14, 2, -10], [-12, -6, -5], [-10, 12, 10], [15, 1, 15], [2, -6, 0], [66, -6, -4], [-16, -2, 2], [18, -14, -2], [0, -16, 6], [39, 11, -9], [-27, 7, 2], [51, 7, -4], [36, 10, 12], [14, -10, 2], [42, 10, 14], [12, -4, -8], [-28, 0, 9], [28, -26, -12], [16, -14, 4], [-22, -2, 14], [35, 13, -2], [-48, 4, 0], [19, 1, 8], [-7, 19, -1], [-2, 0, 2], [-28, -2, 10], [-4, 16, -1], [0, 14, 3], [13, 13, -10], [3, -5, -1], [47, 3, 5], [-2, -12, 12], [-3, -23, -4], [32, -6, -10], [-8, 20, 2], [-58, -2, 0], [-2, -14, -15], [0, 8, -6], [55, -1, 1], [-27, -11, -22], [-17, -11, 12], [-61, -7, -3], [-4, -10, 6], [16, 0, 13], [-45, 1, -11], [-55, -11, -4], [-42, -6, 2], [-25, 3, -14], [-16, 14, 6], [-16, 16, -2], [-41, 7, 12], [19, -5, -7], [1, 1, -9], [-22, 14, -5], [-1, -11, -20], [54, 12, -4], [-37, -3, -17], [40, 8, 0], [-70, 0, -7], [11, -13, 3], [14, -2, 6], [1, 17, 14], [-25, -13, 10], [-24, -8, -23], [10, 6, -12], [-62, 0, 4], [25, 1, 1], [-35, -11, -1], [-13, -13, 4], [-26, -2, 13], [-7, 13, 20], [-2, 6, 4], [-38, 14, 4], [24, 8, 16], [-53, -9, -8], [-19, -3, 1], [-18, 10, -8], [-50, -8, 10], [42, 8, 4], [-37, -7, -11], [33, 9, 5], [18, 2, 8], [-17, 7, -3], [-25, -13, 5], [-48, -12, 1], [-22, -14, -2], [-3, -7, -10], [17, -1, -19], [11, 3, -4], [-21, -7, 18], [-14, 8, 4], [-25, 23, 5], [-8, -20, -10], [-19, 5, -6], [-12, 24, 13], [20, -8, 8], [-23, 15, 1], [-21, 3, -15], [32, 8, -3], [24, -12, -10], [-34, 6, 12], [0, -16, 6], [31, 7, 12], [-36, -4, -8], [9, 3, -2], [-41, -9, -2], [-50, 6, 8], [-31, 9, 7], [-16, -18, 3], [0, 10, 21], [-11, -15, 2], [19, 7, 15], [-46, 0, -2], [46, 6, 6], [-2, -4, 4], [-2, -10, -2], [-4, -28, -7]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2850_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2850_2_a_bn();".
// 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_2850_2_a_bn(:prec:=3)
    chi := MakeCharacter_2850_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_2850_2_a_bn();".
// 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_2850_2_a_bn( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2850_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![16, -16, 0, 1]>,<11,R![-32, -16, 4, 1]>,<13,R![8, -4, -6, 1]>,<23,R![8, -4, -6, 1]>],Snew);
    return Vf;
end function;