
// Make newform 6040.2.a.g in Magma, downloaded from the LMFDB on 24 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_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_6040_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_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], [1], [-1], [0], [-5], [-1], [2], [1], [-1], [1], [3], [0], [-2], [6], [6], [-6], [13], [8], [-7], [0], [7], [-6], [5], [10], [-2], [10], [-16], [13], [-16], [18], [-4], [-2], [4], [16], [24], [1], [13], [1], [0], [8], [20], [14], [12], [-2], [-10], [6], [-4], [22], [-2], [-5], [-21], [15], [-3], [-9], [-27], [-9], [19], [12], [2], [-30], [-13], [-6], [10], [27], [-20], [23], [-20], [9], [-12], [27], [-1], [10], [-20], [-13], [-6], [18], [-6], [12], [-6], [-10], [14], [-28], [-30], [30], [33], [36], [-6], [2], [-6], [-24], [20], [36], [34], [-24], [22], [16], [4], [17], [11], [-7], [8], [-9], [12], [-30], [-15], [16], [-12], [21], [14], [-31], [-20], [28], [-21], [-24], [34], [-15], [-14], [28], [26], [-1], [-44], [0], [42], [15], [-10], [-30], [29], [2], [8], [-13], [34], [10], [10], [-20], [2], [16], [-28], [52], [4], [-48], [-26], [-38], [14], [-36], [-35], [8], [4], [33], [-8], [-52], [7], [36], [52], [-29], [22], [11], [12], [26], [4], [12], [15], [-18], [13], [24], [2], [-24], [-17], [-46], [54], [7], [30], [14], [11], [-6], [26], [30], [27], [13], [-31], [10], [-7], [15], [22], [10], [-45], [22], [26], [44], [-49], [45], [49], [-4], [-68], [66], [-35], [-36], [-65], [36], [-52], [-56], [-9], [10], [-4], [-28], [-28], [22], [8], [51], [9], [-32], [-11], [-48], [50], [42], [20], [0], [7], [-17], [6], [-71], [35], [32], [49], [4], [6], [-32], [-36], [24], [62], [-48], [-26], [7], [12], [-18], [14], [-2], [-40], [45], [-64], [-35], [39], [-15], [67], [-15], [20], [-69], [-21], [-64], [-63], [41], [19], [-49], [-54], [7], [-6], [-16], [-68], [-11], [26], [2], [32], [72], [17], [18], [48], [52], [-4], [-14], [-70], [-24], [26], [0], [-26], [65], [32], [-17], [-21], [-30], [-56], [38], [-20], [-32], [-21], [6], [46], [-60], [-55], [19], [20], [40], [24], [-62], [25], [28], [12], [45], [64], [58], [49], [-88], [11], [34], [-41], [-64], [-38], [31], [56], [58], [9], [-10], [-62], [46], [-18], [20], [44], [-60], [69], [3], [-41], [-54], [-28], [23], [-40], [-56], [56], [-62], [32], [4], [87], [36], [-10], [-94], [33], [87], [68], [0], [35], [76], [25], [-40], [51], [90], [-17], [36], [66], [0], [19], [73], [0], [57], [44], [51], [3], [46], [-67], [26], [-60], [-30], [60], [81], [-4], [-22], [-52], [-39], [28], [-85], [87], [-5], [-5], [-42], [91], [34], [34], [-14], [25], [66], [-68], [-22], [-5], [86], [72], [-39], [-70], [28], [-44], [-24], [-80], [-24], [56], [-39], [-36], [63], [46], [52], [9], [2], [83], [41], [-34], [-44], [5], [-9], [8], [36], [6], [14], [86], [36], [-78], [48], [-29], [-35], [56], [18], [-28], [48], [21], [-64], [77], [30], [83], [69], [-42], [-12], [-98], [92], [-84], [51], [46], [56]];
    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_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_6040_2_a_g(: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_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_6040_2_a_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6040_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-1, 1]>,<7,R![0, 1]>,<11,R![5, 1]>],Snew);
    return Vf;
end function;