
// Make newform 2040.2.a.k in Magma, downloaded from the LMFDB on 01 October 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_2040_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_2040_2_a_k();".
// 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_2040_2_a_k();".
// 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_2040_a();"
function MakeCharacter_2040_a()
    N := 2040;
    order := 1;
    char_gens := [511, 1021, 1361, 817, 241];
    v := [1, 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_2040_a_Hecke( : Kf := Kf)
    return MakeCharacter_2040_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]];
    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], [-4], [-2], [-1], [4], [0], [-6], [-4], [2], [-2], [-4], [-12], [-6], [4], [-10], [-4], [8], [6], [4], [-16], [-6], [14], [6], [8], [4], [-10], [-2], [0], [12], [-18], [-8], [-18], [0], [-18], [12], [8], [-14], [4], [14], [8], [22], [-6], [4], [24], [-8], [12], [6], [-26], [24], [18], [12], [6], [12], [-22], [-16], [-22], [10], [-4], [-14], [4], [0], [14], [2], [28], [14], [-28], [14], [-18], [-24], [-16], [-18], [8], [-20], [-18], [-6], [22], [10], [12], [6], [24], [18], [-4], [24], [22], [2], [-18], [24], [32], [-8], [8], [12], [-8], [24], [-2], [-2], [4], [38], [-20], [2], [-24], [18], [16], [-6], [-32], [6], [48], [34], [-8], [-10], [-42], [8], [-32], [30], [-28], [-12], [-14], [-28], [38], [-26], [26], [-12], [0], [-26], [-10], [0], [-8], [-34], [-20], [-40], [4], [-34], [2], [34], [-6], [-36], [-22], [-2], [-16], [10], [-8], [-52], [-18], [8], [-6], [38], [28], [36], [10], [-42], [-20], [0], [-60], [48], [8], [-34], [2], [42], [-12], [-18], [-48], [28], [30], [-56], [-44], [-38], [-30], [26], [-52], [30], [32], [-6], [-48], [-26], [-56], [34], [-32], [-50], [48], [-4], [-46], [30], [36], [-50], [2], [60], [34], [48], [-26], [-36], [-36], [-18], [-12], [-10], [-38], [34], [46], [12], [42], [-28], [-2], [-14], [44], [42], [32], [48], [14], [-28], [-34], [-26], [16], [24], [-48], [-14], [-24], [-30], [48], [-46], [-26], [-28], [66], [-16], [-48], [22], [-10], [24], [-16], [-36], [14], [-56], [-48], [-30], [12], [68], [-22], [-6], [-44], [-8], [-44], [-12], [-40], [-50], [46], [-8], [32], [68], [52], [-52], [-26], [-10], [4], [-14], [-14], [-68], [-2], [60], [58], [42], [-24], [16], [70], [-62], [62], [28], [6], [-54], [36], [-54], [62], [28], [74], [-40], [-78], [8], [-24], [-50], [10], [-28], [-4], [-28], [48], [-58], [-68], [72], [46], [10], [32], [-14], [-14], [12], [22], [36], [-86], [-14], [-8], [-30], [-44], [68], [34], [66], [76], [-36], [0], [22], [-24], [-34], [48], [38], [64], [66], [30], [-4], [12], [-38], [12], [32], [-58], [-46], [8], [-74], [78], [-8], [-26], [50], [-16], [-4], [0], [42], [-18], [66], [84], [-16], [-24], [20], [30], [78], [-6], [-24], [30], [-26], [-6], [-88], [-78], [12], [38], [20], [40], [42], [-52], [6], [-82], [0], [54], [-42], [-48], [-12], [-18], [-4], [-6], [-34], [-60], [-28], [52], [-14], [-14], [-40], [-78], [-4], [56], [32], [38], [40], [98], [52], [-40], [34], [90], [-62], [34], [46], [32], [-50], [16], [-8], [-32], [-82], [-20], [-60], [-38], [-6], [-12], [84], [-8], [78], [-16], [-86], [0], [-82], [94], [54], [72], [6], [-18], [92], [22], [26], [28], [68], [-66], [66], [-16], [48], [-30], [-78], [0], [-8], [-82], [-36], [-82], [18], [-96], [60], [46], [-6], [-4], [-98], [-28], [-80]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2040_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2040_2_a_k();".
// 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_2040_2_a_k(:prec:=1)
    chi := MakeCharacter_2040_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_2040_2_a_k();".
// 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_2040_2_a_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2040_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![0, 1]>,<11,R![4, 1]>,<13,R![2, 1]>,<19,R![-4, 1]>],Snew);
    return Vf;
end function;