
// Make newform 6040.2.a.a in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6040_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_6040_2_a_a();".
// 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_6040_2_a_a();".
// 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_6040_a();"
function MakeCharacter_6040_a()
    N := 6040;
    order := 1;
    char_gens := [1511, 3021, 2417, 761];
    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_6040_a_Hecke(Kf)
    return MakeCharacter_6040_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], [-3], [1], [-4], [-1], [-1], [-6], [-7], [-9], [5], [-5], [-10], [8], [6], [-6], [-14], [1], [-14], [13], [6], [7], [-8], [-15], [-12], [-12], [-8], [-6], [13], [-14], [-10], [-8], [0], [10], [-4], [-16], [1], [5], [21], [0], [-6], [2], [-10], [0], [26], [6], [10], [10], [8], [-24], [23], [19], [-5], [13], [-21], [-3], [-17], [-1], [-22], [-10], [-12], [-21], [14], [-26], [19], [32], [-13], [-4], [-19], [6], [-1], [11], [10], [-4], [3], [-26], [20], [-12], [12], [-14], [28], [26], [2], [38], [38], [9], [-4], [14], [-8], [-18], [8], [-28], [30], [-26], [28], [-16], [-2], [-32], [-39], [-33], [25], [-2], [-17], [36], [-30], [41], [16], [-28], [-15], [24], [25], [-16], [-34], [-37], [-38], [6], [5], [-8], [-42], [-4], [39], [-36], [14], [-22], [-21], [-18], [-18], [1], [-12], [48], [-33], [-30], [44], [20], [28], [30], [30], [52], [-24], [12], [0], [-44], [-18], [54], [6], [-15], [-36], [-6], [-39], [0], [-32], [31], [0], [32], [-33], [-28], [23], [6], [40], [2], [8], [-21], [-30], [-27], [-4], [42], [16], [-13], [28], [-50], [-1], [-24], [-42], [-5], [-56], [-58], [-18], [-1], [33], [9], [-50], [-31], [35], [-26], [-44], [15], [-52], [-6], [-4], [-25], [-19], [37], [44], [-48], [-30], [9], [-2], [-17], [8], [40], [4], [47], [-14], [22], [4], [-8], [10], [-20], [39], [41], [-20], [-39], [-28], [-60], [-8], [-56], [-14], [7], [27], [-8], [21], [-33], [4], [-15], [-58], [-12], [-46], [6], [24], [8], [-24], [18], [19], [-30], [-12], [-28], [26], [6], [-3], [-44], [-3], [19], [13], [-57], [17], [-34], [39], [-37], [-28], [-15], [-23], [55], [19], [18], [-9], [-46], [-32], [-2], [-31], [-8], [-38], [34], [-36], [-15], [-44], [60], [4], [28], [-18], [-14], [54], [-6], [-74], [-74], [69], [-2], [63], [-21], [-66], [-60], [72], [-64], [-40], [3], [-34], [-28], [24], [-19], [-9], [68], [18], [38], [52], [57], [-20], [-6], [81], [-12], [6], [45], [-20], [63], [38], [83], [28], [-46], [11], [-72], [-32], [-71], [-50], [26], [58], [-38], [12], [-60], [-78], [9], [71], [71], [12], [44], [-77], [-12], [-8], [-40], [70], [-20], [-20], [-33], [6], [-24], [22], [-35], [11], [28], [4], [-9], [-84], [73], [8], [-17], [4], [-49], [40], [-8], [-8], [-89], [13], [0], [-3], [6], [23], [-45], [18], [-43], [-76], [0], [-18], [54], [69], [8], [-58], [18], [33], [-4], [71], [67], [31], [75], [-24], [-65], [46], [6], [-36], [21], [-8], [16], [10], [19], [66], [-64], [-19], [-12], [-90], [-68], [28], [26], [-88], [-16], [13], [-80], [-1], [14], [-96], [-51], [38], [-81], [-15], [68], [12], [25], [47], [16], [-32], [-64], [-56], [-10], [-12], [8], [-48], [79], [45], [36], [-32], [-42], [38], [-59], [-86], [-67], [26], [-45], [57], [42], [46], [-16], [14], [96], [47], [-22], [80]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6040_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6040_2_a_a();".
// 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_6040_2_a_a(:prec:=1)
    chi := MakeCharacter_6040_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_6040_2_a_a();".
// 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_6040_2_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6040_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![3, 1]>,<7,R![4, 1]>,<11,R![1, 1]>],Snew);
    return Vf;
end function;