
// Make newform 8016.2.a.d in Magma, downloaded from the LMFDB on 23 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_d();".
// 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_d();".
// 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], [0], [-4], [4], [2], [0], [4], [0], [6], [-8], [6], [0], [-2], [-8], [8], [0], [10], [-14], [12], [-2], [-2], [12], [-6], [-6], [8], [-10], [-12], [2], [-4], [-4], [12], [-18], [-10], [4], [18], [-2], [-6], [1], [6], [-4], [26], [8], [-6], [12], [-4], [-4], [20], [16], [-6], [-2], [8], [30], [4], [8], [16], [-4], [22], [26], [22], [4], [6], [30], [-24], [-14], [6], [-18], [10], [8], [14], [14], [8], [16], [6], [14], [16], [24], [-14], [-36], [22], [-4], [-10], [-24], [22], [10], [24], [-10], [-10], [2], [34], [36], [28], [-38], [20], [-6], [16], [-6], [-20], [36], [18], [2], [-38], [-36], [24], [34], [22], [16], [-36], [-40], [10], [22], [10], [18], [-10], [-44], [20], [30], [-12], [-26], [0], [30], [18], [-42], [-20], [18], [-6], [18], [-12], [-10], [-30], [2], [-24], [42], [-10], [2], [10], [-12], [22], [-20], [-42], [-38], [20], [-14], [8], [-6], [-48], [-6], [18], [32], [-24], [14], [36], [20], [-12], [8], [48], [-28], [-30], [-22], [0], [28], [0], [20], [-36], [20], [28], [38], [18], [-18], [-6], [24], [-2], [-32], [-42], [50], [42], [-16], [12], [28], [-34], [-32], [20], [42], [-12], [60], [-62], [66], [44], [-10], [60], [-10], [20], [28], [66], [52], [14], [-34], [-62], [18], [-24], [-44], [40], [2], [14], [4], [-50], [50], [60], [60], [-20], [-42], [-30], [-46], [-48], [0], [-34], [-22], [-18], [-24], [32], [-14], [-52], [-20], [4], [-32], [50], [6], [-68], [6], [28], [-6], [42], [10], [-24], [-56], [-12], [62], [38], [-24], [-8], [-4], [-20], [66], [-26], [-54], [-48], [4], [32], [0], [-68], [-46], [-70], [-8], [38], [-48], [-44], [-70], [-28], [28], [-22], [-70], [-52], [50], [-42], [54], [32], [24], [-40], [-22], [30], [22], [-28], [62], [40], [-66], [-14], [68], [46], [-46], [12], [-44], [10], [-56], [82], [22], [-8], [46], [-36], [-12], [24], [50], [-84], [-54], [60], [66], [74], [12], [36], [-20], [-20], [-14], [12], [40], [-48], [4], [-10], [-64], [38], [84], [6], [24], [-2], [-10], [-54], [68], [18], [-28], [64], [86], [60], [80], [10], [-14], [46], [88], [-78], [-28], [72], [-32], [2], [82], [-62], [26], [60], [10], [-84], [14], [48], [-50], [-64], [54], [66], [64], [-90], [14], [60], [-26], [-80], [-68], [10], [16], [-58], [-12], [14], [-10], [40], [-8], [-20], [-56], [24], [6], [-84], [12], [-60], [50], [10], [-36], [26], [-54], [68], [-30], [16], [74], [-86], [-14], [36], [44], [-74], [72], [2], [-6], [-58], [62], [30], [8], [-36], [6], [10], [0], [-16], [22], [-50], [-4], [-2], [60], [-18], [-88], [78], [14], [-36], [62], [82], [78], [-32], [60], [22], [-22], [-88], [-22], [-12], [-14], [96], [52], [100], [-38], [-54], [-24], [28], [-6], [24], [24], [54], [-8], [-20], [-66], [-32], [-28], [-38], [44], [8]];
    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_d();".
// 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_d(: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_d();".
// 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_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8016_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<7,R![4, 1]>,<11,R![-4, 1]>,<13,R![-2, 1]>],Snew);
    return Vf;
end function;