
// Make newform 4008.2.a.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4008_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_4008_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_4008_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_4008_a();"
function MakeCharacter_4008_a()
    N := 4008;
    order := 1;
    char_gens := [3007, 2005, 1337, 673];
    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_4008_a_Hecke(Kf)
    return MakeCharacter_4008_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], [-2], [2], [0], [0], [4], [-6], [-5], [-1], [2], [0], [-1], [9], [-5], [-12], [5], [0], [-14], [12], [-7], [13], [-5], [-7], [-10], [6], [-6], [-12], [-1], [-20], [5], [-19], [-15], [-12], [6], [5], [1], [-4], [-10], [16], [27], [4], [6], [-16], [6], [5], [-7], [-28], [3], [0], [-8], [0], [-10], [11], [-21], [14], [3], [-5], [8], [-12], [33], [17], [-10], [10], [-20], [-11], [-13], [1], [7], [8], [8], [-7], [33], [24], [18], [34], [32], [10], [24], [-34], [21], [-35], [32], [9], [9], [-10], [-2], [-6], [18], [0], [4], [-40], [-32], [28], [24], [-18], [-30], [17], [-29], [6], [-4], [-40], [-3], [-17], [3], [18], [15], [3], [18], [-40], [38], [40], [21], [-10], [32], [12], [-36], [17], [-14], [44], [-2], [-48], [-4], [48], [-25], [36], [-8], [-14], [7], [-15], [-8], [14], [-25], [-26], [-21], [4], [-34], [5], [11], [34], [-34], [-16], [-19], [-21], [36], [39], [-14], [-44], [22], [-36], [58], [54], [-14], [11], [29], [11], [30], [58], [18], [-2], [-8], [-15], [32], [-24], [32], [-28], [9], [30], [-23], [14], [45], [43], [2], [6], [-24], [-3], [24], [-35], [9], [-42], [-3], [24], [26], [-2], [-18], [-24], [49], [0], [8], [-11], [30], [28], [18], [59], [9], [-22], [-7], [-43], [14], [-3], [-57], [-44], [0], [-18], [-20], [-12], [-12], [8], [-7], [-28], [-46], [28], [-12], [37], [46], [-18], [-9], [39], [26], [-31], [20], [36], [-60], [-18], [-45], [40], [-70], [66], [43], [-52], [-18], [-22], [-54], [48], [24], [-20], [35], [16], [-60], [32], [72], [49], [50], [0], [-55], [-63], [-58], [-18], [-22], [50], [-22], [-50], [-46], [-44], [11], [10], [-73], [-35], [32], [-69], [13], [-41], [-18], [32], [-62], [52], [5], [-50], [-18], [-58], [54], [31], [34], [-8], [-3], [75], [20], [24], [-36], [-32], [14], [-74], [-23], [36], [55], [-54], [15], [-56], [26], [-49], [-55], [-28], [-78], [-48], [-59], [-9], [-31], [-8], [-26], [-51], [7], [-4], [-16], [18], [19], [-16], [-26], [-8], [46], [-16], [18], [20], [24], [22], [-39], [-77], [-46], [-60], [-8], [61], [-90], [-16], [16], [61], [10], [-24], [-87], [28], [22], [72], [4], [-66], [28], [-36], [20], [8], [22], [-11], [-4], [-82], [-15], [28], [-22], [-78], [10], [-32], [-18], [-46], [40], [40], [83], [26], [-35], [-14], [-49], [-53], [-22], [15], [-38], [12], [-2], [-50], [31], [82], [-21], [28], [83], [-51], [44], [24], [-36], [-22], [69], [44], [90], [-41], [-18], [58], [66], [-39], [-48], [-5], [-30], [-90], [44], [-19], [64], [16], [-30], [-6], [-12], [17], [70], [36], [89], [-79], [-5], [33], [-56], [-10], [38], [22], [-45], [64], [91], [60], [17], [-4], [24], [86], [22], [-52], [-6], [-48], [6], [-89], [-25], [-47], [42], [13], [19], [2], [61], [70], [36], [-63], [-88], [68]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4008_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4008_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_4008_2_a_b(:prec:=1)
    chi := MakeCharacter_4008_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_4008_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_4008_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4008_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![1, 1]>,<7,R![1, 1]>],Snew);
    return Vf;
end function;