
// Make newform 4140.2.a.i in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4140_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_4140_2_a_i();".
// 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_4140_2_a_i();".
// 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_4140_a();"
function MakeCharacter_4140_a()
    N := 4140;
    order := 1;
    char_gens := [2071, 461, 1657, 3961];
    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_4140_a_Hecke(Kf)
    return MakeCharacter_4140_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], [0], [1], [-1], [0], [-4], [3], [-4], [1], [3], [-7], [11], [9], [-4], [-6], [-9], [-3], [-10], [-13], [-9], [-16], [8], [15], [0], [2], [3], [8], [3], [-4], [-9], [14], [-12], [-18], [-13], [-6], [-16], [17], [2], [-18], [24], [0], [-16], [-6], [-4], [6], [2], [17], [-10], [-12], [-4], [0], [9], [8], [-30], [18], [-27], [21], [-13], [2], [12], [5], [9], [20], [-24], [17], [-18], [-7], [-22], [-24], [35], [-6], [-30], [29], [-22], [-16], [21], [30], [-16], [18], [-31], [36], [8], [-12], [11], [8], [-6], [-21], [29], [30], [-16], [-21], [-6], [2], [-21], [-13], [9], [18], [0], [-28], [-34], [-28], [39], [21], [-18], [-22], [44], [30], [-6], [36], [23], [14], [14], [9], [20], [-34], [24], [-13], [24], [36], [0], [-22], [-22], [33], [30], [-40], [-30], [-22], [45], [17], [-1], [5], [0], [-4], [17], [-15], [-22], [-42], [11], [33], [39], [41], [42], [-16], [45], [-7], [-30], [26], [-6], [5], [6], [-22], [-18], [14], [-6], [-13], [24], [-16], [15], [2], [-30], [48], [6], [26], [36], [33], [27], [53], [8], [20], [-18], [0], [-31], [-18], [-1], [8], [-6], [-52], [27], [29], [44], [44], [-54], [14], [-12], [-3], [-60], [-40], [-19], [41], [3], [8], [6], [-16], [45], [-48], [9], [-52], [38], [42], [-30], [-42], [8], [8], [56], [54], [-18], [-10], [12], [-39], [-40], [38], [30], [-16], [60], [-63], [-46], [-22], [-39], [-27], [36], [-55], [-58], [15], [-7], [-24], [23], [-66], [-15], [-28], [15], [-40], [68], [-10], [-21], [71], [12], [-34], [-42], [-51], [-21], [3], [17], [26], [38], [-48], [48], [26], [24], [14], [-9], [-58], [-18], [-27], [50], [-36], [57], [74], [-16], [48], [-58], [5], [21], [-25], [-7], [-72], [-10], [42], [-30], [-43], [-24], [41], [47], [35], [26], [-70], [-64], [66], [53], [-46], [-30], [-66], [38], [-36], [80], [-28], [39], [-49], [-42], [-40], [-30], [-6], [9], [12], [-30], [-46], [-36], [-40], [72], [-21], [-10], [-31], [-33], [-64], [-30], [-40], [-70], [72], [-46], [54], [62], [36], [0], [-78], [32], [-27], [-58], [-87], [-15], [11], [54], [2], [-31], [78], [62], [-57], [-70], [2], [-64], [-24], [81], [-13], [12], [-1], [12], [-88], [36], [-58], [-6], [50], [47], [14], [6], [-6], [-10], [-27], [-33], [-7], [92], [-66], [57], [-13], [-58], [-69], [-19], [50], [-30], [0], [-42], [-24], [54], [-49], [21], [-78], [-36], [-94], [26], [66], [-61], [20], [63], [29], [-54], [-12], [92], [32], [69], [-90], [-49], [24], [-22], [-42], [-66], [56], [-12], [-16], [30], [89], [50], [-13], [-33], [-64], [66], [-45], [62], [-60], [-1], [-34], [96], [-40], [21], [83], [-90], [29], [-75], [-69], [83], [65], [90], [-16], [-36], [-10], [-60], [36], [68], [77], [78], [36], [62], [-33], [0], [-36], [14], [60], [-84], [-82], [0], [0], [-33], [92], [-93]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4140_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4140_2_a_i();".
// 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_4140_2_a_i(:prec:=1)
    chi := MakeCharacter_4140_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_4140_2_a_i();".
// 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_4140_2_a_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4140_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![1, 1]>,<11,R![0, 1]>,<13,R![4, 1]>,<17,R![-3, 1]>],Snew);
    return Vf;
end function;