
// Make newform 4050.2.a.g in Magma, downloaded from the LMFDB on 09 June 2024.
// 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_g();".
// 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_g();".
// 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], [-1], [2], [-6], [2], [6], [-1], [-9], [-2], [2], [11], [-4], [7], [0], [4], [-7], [-11], [6], [-4], [-12], [11], [-1], [-8], [-2], [-8], [-3], [7], [-12], [-19], [-12], [-12], [16], [1], [10], [4], [4], [3], [4], [-2], [-13], [-6], [-10], [-8], [-18], [18], [23], [8], [-7], [10], [-28], [1], [-18], [-12], [-16], [-3], [-14], [-22], [3], [1], [18], [-9], [-6], [-22], [2], [-8], [8], [12], [-11], [16], [-30], [-16], [12], [-16], [-32], [-19], [-4], [10], [38], [-34], [22], [-16], [2], [24], [9], [-18], [-10], [21], [36], [-36], [-28], [-12], [0], [-24], [-27], [15], [37], [-29], [-17], [35], [24], [37], [2], [-20], [32], [-3], [30], [12], [-22], [1], [-34], [32], [-10], [8], [-13], [-33], [-33], [26], [-20], [-10], [6], [-22], [-4], [-44], [-13], [-27], [44], [21], [4], [40], [15], [26], [-10], [9], [-15], [-12], [44], [2], [30], [-20], [31], [15], [35], [15], [-38], [46], [18], [14], [45], [46], [15], [3], [0], [33], [-12], [-36], [18], [52], [41], [51], [-2], [-17], [42], [-54], [19], [4], [0], [17], [-18], [48], [-9], [-34], [-44], [-32], [-42], [-28], [30], [-7], [35], [32], [58], [22], [-6], [49], [39], [-16], [-11], [23], [-8], [66], [5], [-50], [15], [15], [-66], [-15], [48], [-6], [-65], [30], [6], [-24], [1], [-40], [-42], [-6], [-25], [-5], [-44], [-32], [66], [55], [-12], [-38], [49], [43], [46], [31], [68], [57], [58], [63], [25], [27], [6], [62], [10], [-35], [-10], [40], [22], [2], [-45], [-61], [41], [-69], [-50], [-18], [-30], [3], [50], [-1], [49], [-58], [-6], [-69], [-42], [38], [59], [-42], [-55], [-68], [-41], [-56], [60], [10], [-23], [-62], [-38], [-21], [-49], [-50], [-68], [-18], [20], [-6], [17], [47], [24], [-46], [19], [-14], [-8], [-64], [31], [19], [-10], [-31], [-78], [0], [-10], [4], [-5], [55], [-52], [78], [4], [-40], [5], [-82], [-12], [54], [-30], [26], [45], [-66], [72], [56], [-12], [-50], [48], [2], [-48], [-16], [-70], [-47], [53], [-62], [-36], [0], [3], [66], [-39], [56], [29], [-8], [-50], [-64], [78], [20], [22], [42], [-28], [-72], [17], [30], [21], [-85], [-88], [37], [74], [-4], [-92], [-36], [-45], [31], [36], [1], [-16], [-24], [52], [-19], [-76], [-34], [0], [54], [-53], [28], [-24], [-28], [54], [63], [9], [65], [78], [-48], [84], [44], [-67], [44], [-15], [36], [86], [-84], [-10], [66], [-9], [-78], [46], [8], [-49], [-27], [-60], [12], [50], [-44], [12], [-3], [82], [81], [-86], [-35], [-50], [-30], [-24], [18], [-44], [33], [77], [-37], [42], [48], [-68], [64], [-40], [-16], [46], [-40], [-42], [-53], [20], [88], [-42], [21], [62], [-36], [21], [49], [24], [-8], [-36], [-27], [52], [38], [-67], [-58], [48], [-28], [-15], [93], [-20], [-16], [12], [-86], [84], [73], [58], [-18], [32]];
    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_g();".
// 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_g(: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_g();".
// 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_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4050_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![1, 1]>,<11,R![-2, 1]>,<13,R![6, 1]>,<17,R![-2, 1]>,<23,R![1, 1]>,<41,R![-11, 1]>],Snew);
    return Vf;
end function;