
// Make newform 8016.2.a.f in Magma, downloaded from the LMFDB on 18 April 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_f();".
// 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_f();".
// 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], [2], [4], [4], [0], [-4], [4], [4], [-2], [-4], [-12], [12], [8], [0], [14], [-2], [-2], [4], [-8], [6], [14], [-6], [-6], [14], [-2], [6], [16], [-16], [-4], [8], [-18], [-18], [16], [18], [-2], [6], [-24], [-1], [2], [4], [14], [0], [-2], [-18], [-16], [20], [4], [2], [10], [6], [0], [18], [16], [-16], [-24], [6], [-6], [24], [6], [20], [6], [-12], [-24], [-26], [-6], [-8], [-22], [6], [-16], [-18], [0], [-12], [-32], [20], [8], [34], [10], [24], [26], [-32], [-10], [-24], [-2], [26], [18], [-10], [-14], [18], [22], [-4], [36], [10], [28], [0], [-24], [2], [-32], [28], [36], [8], [-26], [24], [-28], [40], [34], [-30], [-20], [-16], [30], [-22], [26], [-14], [-4], [32], [-40], [-4], [8], [-30], [30], [28], [-2], [-10], [-34], [-28], [18], [-40], [8], [14], [50], [-20], [0], [14], [22], [-54], [30], [-18], [32], [18], [46], [16], [-50], [-2], [54], [44], [-32], [-22], [18], [20], [48], [-22], [-4], [-12], [20], [4], [-8], [-52], [-6], [10], [18], [-4], [-52], [0], [2], [0], [-36], [-38], [-42], [-14], [-22], [14], [-22], [-40], [54], [18], [42], [-12], [-14], [4], [-56], [-4], [-44], [-36], [36], [32], [14], [-62], [20], [14], [-32], [-42], [42], [-44], [-62], [12], [-26], [-14], [-26], [2], [16], [-6], [-28], [8], [38], [54], [-66], [18], [-12], [-44], [52], [-2], [-54], [-18], [-42], [-40], [22], [-2], [54], [-56], [-22], [-4], [-8], [-24], [36], [6], [14], [14], [12], [-54], [36], [-48], [64], [30], [-24], [20], [-48], [58], [54], [2], [0], [-46], [28], [10], [-40], [-14], [8], [-12], [-22], [28], [-16], [22], [-22], [-16], [6], [22], [32], [-80], [-76], [54], [-38], [-6], [-42], [12], [60], [-18], [-20], [18], [-32], [-44], [54], [28], [4], [-26], [56], [62], [-62], [-30], [84], [-30], [16], [-8], [-34], [-12], [-10], [76], [0], [30], [18], [-80], [72], [18], [-46], [42], [-52], [-6], [54], [44], [46], [-78], [36], [-58], [-54], [-16], [-10], [-28], [-70], [66], [-78], [-60], [46], [-52], [50], [-10], [-32], [-52], [22], [62], [-32], [6], [44], [76], [-42], [-66], [-46], [-12], [6], [-20], [12], [-8], [46], [82], [54], [62], [56], [76], [-64], [-70], [64], [62], [-92], [-82], [-38], [54], [90], [-6], [-80], [-50], [52], [12], [-58], [28], [18], [54], [-38], [-56], [-24], [-24], [66], [-12], [-32], [70], [-44], [-48], [-32], [60], [-30], [-58], [54], [50], [-54], [84], [40], [-6], [-90], [44], [6], [-48], [-74], [-36], [6], [-42], [-50], [-46], [-66], [76], [0], [66], [20], [-44], [-24], [102], [6], [32], [-44], [4], [58], [88], [-26], [-88], [-62], [-50], [-94], [-82], [-100], [86], [26], [-16], [28], [80], [-104], [-78], [-38], [44], [-44], [-22], [-18], [60], [-72], [2], [-32], [-6], [-56], [72], [-16], [-10], [30], [4], [18], [-20], [96]];
    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_f();".
// 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_f(: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_f();".
// 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_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8016_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-2, 1]>,<7,R![-4, 1]>,<11,R![-4, 1]>,<13,R![0, 1]>],Snew);
    return Vf;
end function;