
// Make newform 6040.2.a.b in Magma, downloaded from the LMFDB on 25 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_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_6040_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_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], [0], [4], [-4], [2], [-8], [-4], [-2], [0], [6], [-2], [0], [0], [0], [4], [8], [14], [-12], [-2], [0], [-10], [-14], [-14], [-20], [8], [10], [-4], [6], [8], [-2], [-2], [4], [12], [1], [-8], [10], [0], [14], [-22], [8], [24], [-2], [20], [0], [14], [16], [-8], [10], [-18], [24], [-18], [24], [6], [12], [22], [12], [8], [-6], [14], [0], [28], [0], [22], [8], [28], [-18], [24], [-30], [14], [16], [-32], [-4], [-18], [24], [0], [-6], [-18], [14], [-34], [20], [-12], [-6], [24], [18], [30], [-22], [6], [0], [14], [12], [16], [-12], [22], [16], [-8], [38], [26], [2], [-28], [36], [-12], [-42], [-36], [-2], [6], [6], [-28], [14], [-44], [-14], [6], [-6], [-8], [-30], [-20], [-8], [26], [20], [28], [6], [36], [-18], [-10], [30], [-10], [20], [8], [-40], [34], [40], [4], [34], [38], [46], [-10], [40], [-50], [54], [-38], [40], [-16], [12], [10], [-16], [-26], [-6], [-14], [44], [4], [42], [4], [-8], [-32], [8], [-24], [14], [-26], [-48], [-18], [-6], [-32], [6], [26], [-12], [-32], [2], [-6], [-32], [54], [-40], [8], [54], [-4], [18], [24], [-38], [44], [-20], [32], [48], [34], [22], [12], [4], [-4], [-58], [-10], [0], [-26], [26], [46], [-42], [-2], [-6], [-62], [-6], [26], [16], [18], [-44], [50], [-46], [-34], [58], [-40], [-54], [-6], [46], [-14], [12], [56], [-48], [-52], [-18], [-32], [-50], [-48], [40], [50], [-40], [-50], [-8], [-24], [-8], [-54], [24], [56], [-18], [10], [52], [-48], [-30], [68], [16], [-10], [12], [-22], [-32], [42], [60], [16], [54], [-34], [-48], [12], [14], [48], [-52], [64], [38], [0], [46], [-42], [-22], [-56], [34], [26], [2], [56], [-30], [-10], [54], [-66], [64], [8], [-38], [-28], [-6], [-58], [-48], [-50], [32], [26], [-44], [-6], [30], [-38], [-46], [64], [40], [12], [18], [-68], [24], [2], [-20], [32], [58], [-48], [4], [46], [34], [42], [18], [-80], [10], [-44], [2], [-46], [-62], [-32], [-16], [34], [10], [-40], [-32], [48], [74], [-8], [40], [78], [74], [-16], [-30], [-60], [72], [46], [-30], [8], [-46], [8], [64], [-10], [46], [26], [-74], [-24], [6], [20], [62], [0], [66], [-52], [72], [-10], [-2], [-2], [32], [12], [-6], [70], [-72], [-18], [36], [-62], [-74], [96], [-48], [38], [42], [-30], [-8], [14], [38], [24], [-66], [6], [-48], [38], [-22], [8], [-12], [-74], [-58], [72], [16], [82], [-18], [16], [-8], [-20], [16], [-92], [6], [28], [14], [46], [14], [48], [-6], [-88], [58], [34], [-84], [-32], [6], [-88], [60], [90], [12], [82], [34], [24], [2], [32], [-70], [50], [-20], [2], [-6], [-16], [42], [-60], [32], [44], [102], [96], [90], [-50], [76], [-40], [-18], [-22], [0], [-72], [80], [-22], [-48], [38], [-72], [48], [-42], [10], [26], [24], [90], [-14], [-16]];
    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_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_6040_2_a_b(: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_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_6040_2_a_b( : 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![0, 1]>,<11,R![-4, 1]>],Snew);
    return Vf;
end function;