
// Make newform 4056.2.a.c in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4056_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_4056_2_a_c();".
// 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_4056_2_a_c();".
// 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_4056_a();"
function MakeCharacter_4056_a()
    N := 4056;
    order := 1;
    char_gens := [1015, 2029, 2705, 3889];
    v := [1, 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_4056_a_Hecke(Kf)
    return MakeCharacter_4056_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 := [[0], [-1], [-2], [1], [-6], [0], [8], [-4], [2], [10], [7], [2], [-4], [-7], [6], [0], [-6], [-5], [-7], [6], [1], [1], [0], [4], [-19], [6], [-19], [-2], [-3], [6], [-15], [18], [-4], [-5], [-12], [0], [-17], [17], [-12], [-6], [-4], [14], [-18], [7], [-16], [-13], [5], [-16], [-24], [-10], [-10], [-8], [10], [-20], [-8], [20], [-12], [-5], [14], [-18], [-19], [-14], [13], [-12], [19], [20], [-17], [-27], [-16], [15], [-6], [-32], [-13], [31], [-7], [24], [-8], [27], [-4], [-35], [20], [-11], [-12], [-9], [-7], [-20], [30], [21], [20], [-13], [18], [4], [-20], [30], [24], [10], [-34], [-4], [20], [-3], [-29], [20], [34], [38], [-4], [6], [38], [2], [6], [10], [8], [31], [-18], [-5], [-15], [-30], [1], [38], [-8], [-20], [35], [-17], [-12], [-10], [35], [-12], [-39], [-26], [-7], [-43], [4], [-2], [-32], [-42], [-28], [-26], [38], [1], [-28], [-4], [51], [18], [-16], [-4], [-19], [30], [19], [-18], [-43], [54], [50], [-40], [-13], [-22], [-28], [8], [48], [36], [14], [-38], [-44], [-18], [-12], [32], [16], [-30], [24], [-25], [-43], [6], [-28], [42], [-14], [-15], [56], [30], [25], [2], [63], [-49], [-16], [42], [41], [38], [-18], [6], [-47], [28], [-59], [-36], [21], [-52], [-23], [18], [2], [16], [27], [-39], [2], [-22], [-44], [-17], [13], [-7], [-54], [12], [49], [16], [-60], [-52], [47], [-12], [-47], [38], [-56], [10], [-9], [58], [-2], [48], [57], [41], [24], [-55], [12], [-17], [18], [56], [37], [4], [33], [-75], [32], [-58], [5], [42], [41], [66], [48], [-30], [-4], [-56], [-24], [-17], [2], [32], [-21], [54], [-15], [48], [-7], [26], [38], [-11], [-18], [-10], [-66], [-64], [36], [58], [-68], [60], [-35], [-1], [-40], [-63], [26], [-30], [-4], [0], [7], [-33], [2], [76], [-71], [16], [-18], [-22], [27], [56], [18], [-40], [30], [-65], [-36], [78], [-39], [-58], [-17], [-34], [-40], [-18], [-56], [-42], [-5], [-60], [-8], [-58], [42], [65], [-21], [-42], [80], [14], [-7], [65], [-30], [-66], [-78], [-46], [-44], [-22], [32], [11], [42], [21], [-8], [22], [-27], [-78], [-77], [31], [-20], [-37], [-6], [81], [19], [36], [-66], [-22], [18], [78], [-17], [-12], [-65], [18], [-61], [-6], [26], [-73], [2], [-24], [-30], [37], [-58], [34], [13], [-80], [26], [-56], [5], [22], [-8], [13], [-6], [0], [16], [36], [-16], [-16], [-26], [12], [-58], [-38], [67], [-13], [48], [19], [14], [40], [63], [6], [-100], [-32], [-82], [0], [0], [-25], [30], [17], [0], [18], [64], [-66], [75], [-8], [72], [33], [33], [2], [-22], [42], [60], [56], [76], [-19], [-32], [74], [20], [-32], [-45], [66], [99], [98], [-24], [-57], [33], [-40], [-67], [0], [-33], [-8], [18], [-96], [41], [-72], [-12], [75], [72], [-24], [66], [47], [-52], [32], [-21], [-54], [94], [4], [79], [42]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4056_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4056_2_a_c();".
// 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_4056_2_a_c(:prec:=1)
    chi := MakeCharacter_4056_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_4056_2_a_c();".
// 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_4056_2_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4056_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![2, 1]>,<7,R![-1, 1]>],Snew);
    return Vf;
end function;