
// Make newform 6040.2.a.j in Magma, downloaded from the LMFDB on 20 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_j();".
// 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_j();".
// 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], [2], [-1], [2], [4], [-4], [-2], [-4], [-6], [-6], [0], [-2], [-10], [0], [0], [4], [-12], [14], [-6], [8], [-4], [-4], [-10], [14], [-6], [2], [4], [-14], [10], [4], [-4], [8], [-6], [12], [10], [1], [8], [6], [0], [6], [-12], [10], [8], [-10], [12], [8], [-4], [4], [8], [-26], [20], [0], [-2], [4], [8], [-18], [-18], [-8], [4], [-18], [-6], [-8], [-8], [8], [14], [-4], [-20], [-8], [12], [-14], [20], [-8], [-14], [-12], [20], [4], [-30], [18], [-2], [-14], [0], [22], [-32], [-20], [24], [-38], [-6], [22], [14], [-16], [6], [0], [8], [12], [4], [0], [6], [-22], [-6], [2], [32], [-20], [-28], [-38], [44], [42], [-6], [-24], [32], [10], [-22], [18], [-36], [16], [20], [18], [44], [-20], [-6], [-20], [22], [-14], [-20], [-30], [48], [-6], [6], [12], [12], [-4], [-36], [44], [-8], [-22], [18], [22], [-42], [12], [38], [-2], [44], [-10], [-36], [-8], [6], [8], [30], [-32], [-12], [-6], [-44], [-26], [-8], [-38], [36], [0], [-12], [-6], [-42], [10], [-18], [-54], [54], [-36], [8], [6], [16], [26], [-50], [-44], [44], [42], [-16], [-14], [60], [-18], [-28], [26], [-6], [-38], [-2], [12], [-38], [42], [14], [54], [-12], [46], [26], [-16], [-16], [14], [-12], [-2], [-66], [42], [62], [58], [14], [6], [-34], [-68], [50], [-10], [-64], [-38], [4], [14], [70], [-28], [48], [6], [-48], [-40], [-44], [62], [14], [6], [-64], [-32], [-22], [32], [18], [4], [-60], [-22], [-62], [32], [4], [8], [50], [52], [-12], [-42], [20], [-36], [-22], [-24], [-8], [48], [26], [28], [58], [-38], [54], [72], [-2], [-12], [4], [66], [28], [34], [-40], [78], [54], [12], [-62], [-38], [50], [16], [-8], [-62], [-30], [-26], [-66], [20], [-2], [-38], [-16], [18], [78], [-28], [-4], [40], [-34], [-30], [-38], [-66], [-66], [52], [-68], [80], [10], [-38], [28], [32], [-32], [44], [-16], [18], [62], [64], [-24], [-68], [66], [54], [-48], [42], [60], [58], [-68], [18], [40], [48], [-4], [-52], [48], [62], [8], [82], [-8], [-66], [-14], [30], [-32], [70], [36], [-40], [-4], [-22], [-28], [-44], [46], [-88], [-82], [58], [88], [-62], [42], [30], [30], [-58], [-40], [18], [-44], [24], [30], [6], [26], [-32], [4], [-30], [-54], [20], [-18], [-20], [-22], [-34], [-56], [56], [60], [-52], [30], [-72], [-74], [46], [-76], [-68], [-6], [-22], [-18], [30], [-96], [-60], [34], [-36], [12], [-34], [-90], [-12], [28], [48], [-6], [-48], [-24], [48], [-32], [-64], [-62], [42], [2], [-88], [16], [-18], [48], [-42], [20], [-50], [16], [-38], [66], [-36], [-12], [-26], [-48], [-80], [-96], [-74], [-20], [-10], [34], [84], [4], [-42], [42], [-20], [-56], [42], [52], [-8], [-20], [-48], [76], [44], [42], [50], [-24], [-16], [24], [4], [50], [56], [-96], [56], [-94], [26], [60], [-66], [-4], [-52]];
    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_j();".
// 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_j(: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_j();".
// 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_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6040_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-2, 1]>,<7,R![-2, 1]>,<11,R![-4, 1]>],Snew);
    return Vf;
end function;