
// Make newform 4050.2.a.r in Magma, downloaded from the LMFDB on 30 April 2026.
// 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_r();".
// 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_r();".
// 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 := 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], [4], [0], [1], [-3], [-4], [0], [-9], [-4], [1], [-6], [-8], [-12], [-6], [0], [-1], [4], [12], [-11], [-16], [-12], [3], [-2], [6], [4], [12], [11], [-15], [16], [12], [9], [20], [-9], [8], [13], [-8], [12], [-3], [-12], [-10], [-12], [13], [-3], [-4], [8], [-8], [-12], [23], [21], [-12], [-13], [24], [-15], [12], [-21], [-16], [10], [27], [4], [9], [-20], [-24], [-23], [21], [20], [-2], [12], [14], [-18], [0], [-8], [10], [-28], [-12], [6], [25], [3], [-25], [24], [-13], [12], [-11], [-28], [-12], [18], [1], [-18], [-8], [-24], [12], [4], [0], [-40], [0], [30], [-6], [16], [-1], [-44], [-3], [12], [15], [-16], [25], [-24], [33], [36], [35], [-20], [-38], [-3], [8], [8], [-45], [4], [36], [18], [-12], [23], [-11], [-6], [36], [-28], [51], [47], [0], [28], [-14], [20], [12], [20], [-38], [15], [-37], [-27], [-32], [-51], [39], [-16], [15], [-44], [-48], [14], [-24], [10], [-39], [-52], [24], [25], [-30], [4], [24], [16], [-24], [-16], [3], [37], [27], [-12], [9], [4], [36], [-18], [48], [-52], [37], [11], [42], [-36], [23], [12], [13], [8], [-54], [8], [27], [-32], [-1], [-32], [-12], [13], [54], [-48], [51], [61], [40], [26], [-60], [-23], [24], [8], [-21], [-12], [54], [-1], [34], [-15], [-48], [15], [-16], [10], [-1], [60], [-30], [-4], [24], [42], [20], [46], [-9], [28], [48], [24], [26], [-32], [3], [12], [9], [35], [-4], [3], [16], [48], [11], [6], [-12], [-32], [-36], [-11], [-52], [-40], [3], [-68], [-72], [35], [-54], [24], [-12], [-24], [56], [40], [-34], [-18], [-72], [-68], [48], [-40], [0], [-23], [-9], [-72], [-37], [-39], [-60], [-25], [-44], [-63], [-74], [16], [-48], [59], [-62], [-66], [-28], [-69], [3], [40], [21], [-61], [-80], [-26], [-4], [-23], [-20], [-48], [35], [11], [-48], [24], [32], [-36], [71], [40], [-36], [13], [-54], [-28], [-9], [30], [12], [45], [-84], [-71], [-45], [-16], [-51], [-72], [16], [73], [18], [-52], [-48], [56], [-2], [0], [14], [-36], [49], [36], [54], [75], [28], [-12], [74], [72], [72], [1], [-30], [80], [-26], [-21], [-56], [-42], [35], [56], [64], [-12], [-6], [-82], [-27], [-40], [-48], [20], [48], [-25], [-39], [-22], [88], [61], [54], [-21], [-4], [93], [60], [-25], [-8], [-84], [57], [-16], [49], [-45], [-8], [-13], [21], [-84], [48], [-18], [36], [-47], [27], [0], [-48], [-32], [13], [-78], [76], [-22], [-72], [-64], [24], [15], [80], [-59], [24], [72], [49], [-78], [73], [-9], [-63], [28], [-15], [80], [-48], [-28], [-47], [4], [-24], [71], [21], [-24], [-32], [48], [49], [44], [39], [32], [-33], [59], [-66], [-68], [-15], [99], [-16], [97], [27], [-104], [-84], [73], [-63], [84], [8], [61], [-33], [60], [88], [-51], [-48], [-18], [-38], [-96], [24], [25], [81], [36], [-45], [56], [24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4050_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4050_2_a_r();".
// 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_r(: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_r();".
// 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_r( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4050_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-4, 1]>,<11,R![0, 1]>,<13,R![-1, 1]>,<17,R![3, 1]>,<23,R![0, 1]>,<41,R![6, 1]>],Snew);
    return Vf;
end function;