
// Make newform 2240.2.a.s in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2240_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_2240_2_a_s();".
// 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_2240_2_a_s();".
// 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_2240_a();"
function MakeCharacter_2240_a()
    N := 2240;
    order := 1;
    char_gens := [1471, 1541, 897, 1921];
    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_2240_a_Hecke(Kf)
    return MakeCharacter_2240_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], [1], [-3], [7], [-5], [4], [4], [5], [2], [0], [6], [8], [-9], [0], [12], [6], [16], [4], [-14], [9], [-8], [-6], [3], [0], [15], [10], [-9], [-8], [20], [0], [-14], [6], [6], [-15], [2], [-10], [15], [21], [-24], [-24], [11], [-14], [-2], [-8], [-13], [-1], [25], [-14], [-8], [9], [28], [-6], [6], [10], [18], [-14], [-30], [-9], [15], [25], [-13], [-12], [1], [-18], [-28], [-22], [-12], [-16], [7], [32], [-21], [26], [8], [-12], [-5], [-7], [-3], [22], [-16], [-13], [-33], [-14], [-2], [-24], [15], [-2], [40], [10], [-17], [-32], [30], [35], [21], [9], [6], [0], [-4], [-11], [-22], [32], [4], [-26], [20], [-11], [12], [-19], [-25], [-6], [17], [-26], [-38], [4], [-1], [-42], [-23], [-28], [-24], [-31], [-28], [26], [-21], [-18], [8], [13], [21], [-28], [0], [-13], [5], [-4], [-19], [52], [-48], [-6], [11], [49], [-19], [-39], [-10], [43], [-14], [26], [-14], [4], [-42], [58], [20], [-2], [0], [8], [58], [8], [-12], [-48], [-35], [44], [47], [46], [18], [34], [42], [-30], [44], [45], [16], [41], [-47], [-2], [20], [-2], [39], [34], [8], [-4], [-31], [-13], [64], [26], [-54], [62], [-16], [-42], [38], [-16], [40], [57], [53], [26], [14], [-4], [51], [-18], [-16], [57], [-37], [-22], [-5], [63], [14], [-6], [14], [16], [56], [65], [38], [34], [10], [-50], [-70], [-18], [-34], [-27], [-50], [26], [-44], [-30], [-64], [14], [22], [-30], [-55], [-38], [-19], [-38], [-9], [51], [-17], [-12], [34], [30], [-8], [41], [1], [19], [34], [-4], [-1], [-28], [-16], [12], [-25], [-38], [-39], [-24], [44], [-62], [-73], [-4], [-44], [-70], [-12], [44], [18], [-47], [50], [-28], [63], [-63], [8], [26], [-12], [-5], [26], [16], [47], [6], [-24], [-8], [46], [-38], [-77], [56], [-2], [-9], [-12], [-22], [-79], [34], [19], [-71], [-56], [14], [29], [-73], [52], [-16], [0], [-80], [-29], [43], [34], [-30], [44], [26], [50], [-21], [-56], [-19], [19], [-50], [47], [-24], [-57], [-72], [-84], [-40], [-72], [80], [-8], [57], [-6], [54], [0], [-50], [-34], [-33], [-17], [29], [14], [30], [-70], [-6], [6], [60], [71], [-60], [-78], [-22], [-86], [-2], [6], [43], [-20], [-27], [15], [26], [16], [-5], [-22], [52], [-80], [55], [-28], [-28], [-18], [-22], [36], [-67], [-60], [10], [-45], [-53], [47], [-10], [50], [60], [-12], [66], [-78], [36], [50], [-39], [20], [36], [-35], [-36], [-55], [-3], [46], [-36], [33], [70], [38], [-84], [15], [49], [-26], [89], [-54], [-26], [-88], [70], [-86], [19], [-53], [-17], [8], [-63], [-3], [49], [-33], [-28], [99], [-84], [94], [-54], [57], [70], [-22], [88], [76], [-21], [12], [-38], [-6], [-81], [-20], [68], [71], [-66], [12], [-28], [-16], [-48], [-37], [-8], [74], [63], [-13], [22], [-2], [72], [-51], [25], [0], [-86], [-62], [-90]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2240_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2240_2_a_s();".
// 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_2240_2_a_s(:prec:=1)
    chi := MakeCharacter_2240_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_2240_2_a_s();".
// 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_2240_2_a_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2240_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-1, 1]>,<11,R![3, 1]>,<13,R![-7, 1]>,<19,R![-4, 1]>],Snew);
    return Vf;
end function;