
// Make newform 4014.2.a.e in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4014_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_4014_2_a_e();".
// 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_4014_2_a_e();".
// 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_4014_a();"
function MakeCharacter_4014_a()
    N := 4014;
    order := 1;
    char_gens := [893, 2233];
    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_4014_a_Hecke(Kf)
    return MakeCharacter_4014_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], [0], [-1], [-2], [-1], [-4], [-1], [3], [-10], [-3], [5], [-6], [-6], [9], [1], [4], [9], [-4], [-5], [0], [-14], [5], [2], [-2], [13], [-12], [-10], [-6], [-2], [10], [2], [22], [-22], [-9], [2], [-8], [1], [16], [4], [9], [3], [-4], [-6], [-14], [-18], [-1], [8], [8], [-4], [6], [13], [-10], [9], [-9], [16], [5], [-8], [13], [-6], [0], [19], [-21], [-26], [-21], [-7], [14], [-16], [15], [14], [36], [-10], [16], [4], [-1], [-37], [8], [26], [6], [-24], [10], [15], [14], [-15], [24], [30], [36], [-17], [-8], [-9], [-6], [-4], [-15], [-8], [-9], [18], [18], [-40], [-10], [-2], [28], [20], [22], [-37], [-7], [-1], [44], [40], [-22], [33], [-32], [29], [21], [31], [2], [-16], [-32], [14], [-12], [24], [-1], [-50], [28], [-20], [-15], [28], [16], [-16], [13], [-52], [-6], [14], [-34], [-24], [35], [-16], [3], [-11], [36], [4], [-21], [4], [9], [-36], [-23], [28], [14], [36], [21], [52], [39], [-36], [6], [-16], [8], [48], [-51], [54], [-59], [-6], [-36], [-20], [-13], [8], [0], [-56], [10], [-38], [47], [-36], [52], [-18], [38], [33], [-18], [17], [9], [26], [-35], [-47], [32], [35], [22], [-24], [6], [-1], [-36], [-58], [-8], [-7], [-36], [-12], [-13], [38], [66], [19], [-37], [0], [-48], [34], [-26], [-38], [-18], [32], [61], [-34], [7], [-4], [-63], [-48], [-22], [-30], [37], [-27], [-50], [-50], [8], [-58], [-32], [-42], [49], [54], [31], [54], [-28], [24], [-24], [-44], [-69], [-37], [14], [-22], [-10], [41], [-64], [20], [-24], [41], [63], [-36], [-29], [32], [-61], [12], [-59], [-27], [17], [52], [-11], [-37], [-56], [-16], [44], [54], [-38], [-7], [18], [43], [26], [-31], [-40], [24], [-41], [71], [-12], [-30], [-66], [-34], [-29], [50], [-29], [-28], [33], [-48], [47], [68], [-32], [13], [-48], [-66], [68], [-36], [-8], [54], [45], [-34], [78], [-53], [47], [-68], [-28], [0], [-73], [40], [-52], [-64], [-56], [21], [-10], [-42], [67], [5], [-56], [-50], [22], [-68], [-56], [38], [-28], [-70], [70], [-14], [44], [-25], [58], [0], [-55], [26], [22], [-10], [53], [-20], [-54], [-9], [-8], [-34], [16], [26], [6], [-38], [63], [0], [4], [55], [8], [26], [-9], [-20], [83], [-61], [-18], [6], [-80], [-54], [78], [-80], [-1], [34], [-42], [-27], [-42], [-54], [-26], [-38], [58], [18], [-22], [-54], [4], [-92], [58], [-10], [-15], [-66], [88], [-30], [58], [-3], [-36], [-8], [25], [84], [-24], [-12], [-48], [-64], [28], [-78], [-82], [40], [-72], [-64], [-90], [-44], [-40], [4], [-67], [54], [-88], [100], [42], [-36], [-42], [-15], [99], [-34], [75], [88], [-39], [5], [72], [27], [-3], [18], [-68], [-15], [6], [34], [-20], [84], [6], [89], [20], [34], [-104], [-66], [-12], [-96], [52], [-30], [63], [-8], [-45], [62], [78], [-48], [56], [38], [-36]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4014_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4014_2_a_e();".
// 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_4014_2_a_e(:prec:=1)
    chi := MakeCharacter_4014_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_4014_2_a_e();".
// 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_4014_2_a_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4014_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<7,R![0, 1]>,<11,R![1, 1]>],Snew);
    return Vf;
end function;