
// Make newform 8016.2.a.b in Magma, downloaded from the LMFDB on 06 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8016_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_8016_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_8016_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_8016_a();"
function MakeCharacter_8016_a()
    N := 8016;
    order := 1;
    char_gens := [3007, 2005, 5345, 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_8016_a_Hecke(Kf)
    return MakeCharacter_8016_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], [-6], [-4], [4], [-8], [-2], [5], [3], [-6], [8], [-3], [-7], [-11], [4], [7], [4], [6], [-4], [15], [9], [-13], [1], [-6], [10], [2], [-4], [17], [12], [9], [7], [9], [-20], [-18], [-9], [-1], [-4], [-2], [20], [-15], [16], [6], [8], [2], [-29], [-1], [-20], [-9], [-24], [12], [-20], [-22], [-15], [27], [30], [-9], [15], [8], [12], [27], [27], [10], [6], [-20], [29], [-3], [29], [-21], [24], [0], [13], [11], [-16], [34], [-26], [-24], [26], [16], [2], [39], [-11], [8], [-9], [5], [-2], [-6], [22], [38], [-24], [16], [-8], [24], [12], [-16], [-14], [10], [-3], [17], [-38], [24], [-4], [31], [31], [21], [22], [-19], [27], [38], [-4], [-26], [32], [35], [-10], [-40], [44], [-36], [-9], [-14], [-8], [-22], [-16], [20], [0], [35], [-4], [-28], [-34], [-11], [-21], [-4], [-38], [27], [18], [-45], [-4], [54], [49], [-23], [10], [-26], [0], [-31], [25], [-52], [27], [50], [-12], [26], [-28], [-6], [-46], [34], [1], [11], [-33], [-14], [-6], [-58], [26], [24], [47], [-12], [24], [-20], [-24], [25], [-34], [-25], [-10], [-25], [3], [6], [-42], [24], [37], [-32], [25], [-49], [-14], [-15], [36], [26], [26], [-38], [-52], [-55], [4], [24], [27], [-62], [16], [42], [31], [-23], [-62], [-43], [7], [30], [11], [-37], [-28], [0], [18], [0], [24], [40], [40], [25], [48], [-6], [-36], [-28], [-35], [-2], [30], [-59], [-1], [-70], [7], [60], [12], [12], [14], [-25], [-48], [18], [6], [15], [-20], [6], [-42], [-10], [-60], [-32], [24], [29], [0], [-52], [64], [40], [53], [34], [32], [15], [47], [34], [-22], [58], [-70], [-58], [42], [-14], [32], [-17], [50], [-57], [-11], [-12], [-3], [-63], [27], [-66], [52], [42], [-44], [31], [54], [46], [58], [82], [-7], [50], [4], [27], [-33], [24], [-8], [4], [80], [-42], [-58], [43], [-48], [-9], [42], [-71], [84], [-30], [-55], [-3], [32], [-54], [0], [-13], [7], [15], [24], [62], [-27], [17], [-4], [8], [2], [-3], [36], [-30], [52], [2], [-28], [26], [52], [68], [-26], [-25], [-47], [18], [20], [-56], [45], [78], [-64], [72], [69], [-2], [12], [67], [-8], [-62], [-84], [-64], [14], [-68], [-64], [-4], [-56], [50], [-5], [-40], [-2], [-15], [24], [-42], [70], [-62], [-56], [-18], [-22], [-92], [0], [-21], [70], [-47], [18], [-27], [45], [-66], [61], [22], [28], [-6], [-38], [69], [18], [83], [-84], [-85], [75], [12], [-32], [-36], [30], [89], [60], [-54], [79], [-90], [-6], [18], [37], [-64], [-81], [-62], [6], [12], [-7], [-80], [60], [-90], [-54], [-40], [-77], [10], [-80], [-41], [37], [-55], [-95], [16], [-34], [62], [50], [11], [-52], [47], [-20], [83], [16], [24], [22], [38], [64], [-70], [96], [-30], [-39], [11], [91], [-6], [-95], [-15], [74], [-43], [30], [-20], [21], [-80], [72]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8016_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8016_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_8016_2_a_b(:prec:=1)
    chi := MakeCharacter_8016_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_8016_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_8016_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8016_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![1, 1]>,<7,R![-1, 1]>,<11,R![2, 1]>,<13,R![6, 1]>],Snew);
    return Vf;
end function;