
// Make newform 4050.2.a.s in Magma, downloaded from the LMFDB on 05 February 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_4050_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_4050_2_a_s();".
// 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_4050_2_a_s();".
// 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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_4050_a();"
function MakeCharacter_4050_a()
    N := 4050;
    order := 1;
    char_gens := [2351, 3727];
    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_4050_a_Hecke(Kf)
    return MakeCharacter_4050_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]];
    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], [-5], [0], [1], [-6], [5], [9], [0], [-4], [10], [-3], [-8], [3], [-3], [9], [8], [4], [6], [-2], [2], [-6], [6], [16], [12], [13], [6], [-16], [6], [-11], [15], [18], [11], [18], [-10], [13], [-8], [-12], [21], [3], [-10], [-24], [4], [3], [-4], [-19], [-8], [-24], [-4], [24], [-6], [14], [3], [6], [15], [-6], [2], [-17], [-9], [4], [27], [-20], [6], [4], [-3], [-16], [-2], [-30], [-22], [18], [18], [-8], [-26], [-1], [21], [-24], [-2], [15], [29], [-24], [-22], [-12], [16], [26], [-6], [27], [28], [0], [1], [6], [-30], [-23], [-27], [-13], [0], [42], [15], [34], [8], [-8], [39], [-12], [-33], [20], [16], [42], [-6], [18], [-19], [16], [-29], [-6], [17], [8], [18], [4], [0], [18], [39], [-40], [-2], [-21], [-18], [-1], [-6], [2], [-18], [1], [-14], [20], [24], [38], [7], [3], [-10], [-18], [-14], [6], [-3], [-16], [-42], [-44], [-24], [32], [-12], [10], [-42], [20], [39], [-11], [30], [22], [-51], [-2], [6], [-16], [6], [28], [-54], [12], [36], [40], [-27], [18], [-48], [-16], [37], [47], [-15], [9], [-4], [-30], [4], [-46], [-9], [-19], [54], [-32], [26], [-32], [21], [-41], [36], [-15], [66], [-47], [-14], [-19], [42], [22], [30], [44], [-42], [-24], [-36], [17], [-11], [-30], [39], [30], [56], [-35], [17], [12], [-33], [-22], [-6], [-6], [20], [-62], [0], [1], [24], [-60], [-37], [-32], [-30], [15], [18], [-10], [50], [-3], [43], [60], [-16], [30], [-42], [-59], [45], [-47], [-25], [-40], [51], [76], [27], [53], [-18], [39], [-24], [24], [29], [31], [74], [54], [-18], [-23], [-21], [41], [36], [58], [9], [-72], [-37], [-51], [15], [38], [-44], [45], [-20], [-47], [-78], [-4], [-17], [-60], [-55], [6], [69], [40], [-57], [2], [64], [82], [-4], [58], [52], [-60], [-46], [65], [-15], [21], [-76], [-63], [44], [-68], [36], [-14], [63], [-64], [18], [-30], [24], [-18], [12], [-17], [18], [74], [42], [81], [-2], [28], [-45], [56], [-6], [47], [-2], [18], [50], [54], [58], [-36], [-18], [-57], [-8], [48], [-7], [63], [72], [-80], [66], [17], [-80], [12], [-56], [78], [-1], [-52], [-44], [-15], [-75], [26], [63], [14], [-42], [-7], [24], [-70], [12], [5], [7], [-83], [36], [-60], [-22], [69], [84], [2], [-62], [-78], [-57], [47], [-50], [0], [19], [68], [-30], [-24], [87], [-18], [-9], [-2], [-27], [-27], [75], [-50], [4], [33], [4], [59], [-81], [53], [-69], [12], [-10], [67], [-69], [18], [-32], [-66], [64], [-18], [0], [37], [-66], [-64], [57], [-28], [-65], [-14], [33], [-10], [15], [-84], [-86], [6], [-14], [-10], [-3], [-103], [60], [32], [-42], [67], [-12], [-18], [-70], [7], [27], [58], [12], [-62], [-18], [-84], [35], [52], [-84], [12], [-56], [6], [-24], [-72], [-11], [-21], [-33], [52], [63], [72], [-99], [-52], [-96]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4050_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4050_2_a_s();".
// 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_4050_2_a_s(:prec:=1)
    chi := MakeCharacter_4050_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_4050_2_a_s();".
// 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_4050_2_a_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4050_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![5, 1]>,<11,R![0, 1]>,<13,R![-1, 1]>,<17,R![6, 1]>,<23,R![-9, 1]>,<41,R![3, 1]>],Snew);
    return Vf;
end function;