
// Make newform 4014.2.a.b 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_b();".
// 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_b();".
// 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], [-1], [-4], [-2], [-6], [-8], [-6], [-2], [0], [-4], [-1], [-2], [6], [3], [14], [10], [-6], [-13], [6], [-11], [-5], [-9], [-12], [18], [10], [-7], [2], [1], [2], [-8], [7], [-17], [2], [-10], [-23], [-14], [-11], [-8], [-13], [15], [-7], [-22], [2], [12], [20], [-20], [1], [-15], [4], [26], [-27], [23], [5], [-22], [-24], [-15], [0], [-8], [0], [-18], [-30], [-23], [0], [-8], [2], [4], [6], [12], [5], [-26], [12], [-36], [18], [2], [0], [24], [-28], [30], [14], [-36], [8], [4], [31], [-1], [-3], [22], [12], [0], [-30], [0], [24], [-26], [30], [-16], [26], [8], [3], [12], [-10], [8], [-9], [-18], [-13], [37], [33], [-38], [-9], [-32], [-32], [-25], [-2], [2], [43], [3], [41], [-28], [24], [-42], [33], [-2], [6], [2], [-23], [-35], [-4], [10], [3], [22], [34], [3], [-41], [-2], [-14], [-11], [6], [-26], [23], [18], [49], [-28], [-42], [13], [0], [32], [0], [-18], [-54], [-13], [-12], [18], [-40], [47], [-9], [18], [21], [-56], [36], [28], [-30], [12], [47], [-16], [-24], [-17], [-52], [-44], [-37], [-30], [-58], [-40], [-46], [-33], [44], [-52], [34], [-1], [-48], [10], [-14], [47], [1], [-29], [-9], [17], [46], [-33], [-46], [-31], [9], [-2], [-26], [-4], [-4], [63], [56], [-33], [51], [25], [30], [35], [44], [22], [-8], [12], [-28], [28], [30], [-41], [-67], [58], [-9], [-4], [36], [-20], [-18], [30], [27], [-39], [62], [-23], [-8], [61], [8], [17], [20], [-9], [-40], [-68], [-56], [14], [-26], [56], [22], [1], [20], [74], [-33], [60], [50], [-68], [-45], [-57], [-47], [38], [4], [-8], [46], [-50], [34], [31], [-11], [0], [-16], [7], [25], [-39], [-12], [-42], [8], [19], [39], [-38], [34], [26], [36], [52], [-55], [14], [18], [-40], [9], [24], [-4], [0], [-79], [6], [-76], [65], [-40], [29], [-32], [-84], [68], [-68], [72], [-31], [69], [-21], [66], [77], [72], [11], [-32], [34], [42], [-80], [7], [-58], [-2], [58], [60], [25], [-48], [-78], [32], [-10], [-30], [-41], [44], [-39], [-8], [22], [-36], [12], [77], [66], [-58], [-14], [-55], [-38], [1], [-9], [-56], [78], [70], [32], [56], [-21], [3], [-4], [-38], [-38], [-46], [40], [59], [2], [-30], [58], [-37], [5], [75], [-79], [2], [-13], [-17], [22], [-17], [-13], [2], [-35], [-64], [93], [14], [12], [-6], [-15], [-27], [19], [56], [72], [-50], [12], [-34], [-70], [-64], [-39], [-74], [-55], [15], [42], [-56], [-51], [76], [16], [84], [51], [0], [-90], [-59], [35], [-54], [55], [-81], [-5], [-15], [98], [22], [-10], [60], [-86], [-6], [-2], [-68], [38], [29], [20], [72], [-74], [18], [31], [21], [-24], [10], [-6], [-20], [2], [30], [-100], [31], [0], [-25], [-88], [28], [23], [-70], [-97], [28], [19], [-31], [-74], [69], [-35], [-58], [-64], [6], [11], [-102], [-74], [64]];
    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_b();".
// 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_b(: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_b();".
// 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_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4014_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![1, 1]>,<7,R![4, 1]>,<11,R![2, 1]>],Snew);
    return Vf;
end function;