
// Make newform 8016.2.a.h in Magma, downloaded from the LMFDB on 16 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_h();".
// 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_h();".
// 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], [0], [0], [-4], [6], [0], [-4], [6], [0], [-4], [-10], [-10], [-8], [12], [-6], [6], [-10], [4], [10], [0], [-14], [-6], [14], [20], [-8], [12], [-12], [10], [8], [-18], [-14], [-10], [4], [12], [6], [-14], [-1], [-18], [-20], [14], [-8], [-18], [-24], [24], [0], [16], [-10], [-22], [-10], [-8], [-10], [24], [2], [24], [-24], [-20], [12], [22], [-16], [-18], [18], [-8], [14], [30], [-10], [2], [-22], [-24], [-18], [-24], [-8], [16], [34], [0], [-24], [2], [-18], [-10], [20], [14], [8], [34], [-40], [6], [10], [22], [10], [4], [-28], [-24], [16], [-8], [18], [24], [10], [-10], [24], [-32], [-38], [-38], [-24], [6], [18], [22], [-18], [18], [16], [30], [-12], [-14], [-2], [10], [-48], [30], [-6], [28], [6], [18], [-36], [-10], [-2], [6], [14], [2], [-20], [-28], [8], [10], [-42], [24], [52], [-34], [22], [14], [-24], [38], [-12], [-30], [-46], [48], [44], [-14], [40], [-32], [42], [-42], [36], [0], [-30], [2], [16], [24], [4], [24], [32], [-50], [42], [40], [-24], [46], [32], [34], [22], [24], [20], [-18], [-18], [26], [26], [34], [-8], [10], [40], [-2], [0], [-20], [0], [-40], [8], [12], [-48], [-22], [-60], [2], [18], [-36], [-6], [4], [26], [-54], [-60], [-14], [-8], [-6], [30], [22], [18], [-40], [12], [-48], [-12], [10], [30], [46], [32], [-64], [42], [28], [50], [58], [52], [-70], [16], [26], [52], [-6], [24], [8], [64], [56], [-58], [-24], [-38], [-26], [-6], [-48], [-20], [-44], [-52], [-14], [4], [-70], [72], [60], [-18], [38], [-18], [16], [-6], [-44], [24], [32], [38], [-56], [-8], [-18], [64], [36], [-2], [38], [32], [-38], [20], [24], [-44], [52], [0], [10], [-16], [30], [-16], [-76], [-54], [-16], [8], [-6], [70], [-42], [4], [80], [66], [40], [2], [-64], [18], [-12], [-30], [-20], [-44], [-20], [4], [-10], [50], [72], [26], [36], [24], [42], [-30], [-62], [-6], [-48], [42], [2], [-40], [24], [-70], [32], [38], [-12], [40], [-22], [-52], [-66], [62], [-22], [40], [-58], [-44], [-42], [30], [26], [48], [-10], [30], [56], [-78], [-6], [52], [-26], [-78], [44], [-26], [62], [-32], [-8], [88], [-22], [-38], [18], [20], [-20], [-34], [32], [34], [26], [10], [16], [46], [-22], [-84], [4], [-66], [96], [-74], [4], [80], [66], [80], [-2], [96], [-76], [40], [42], [0], [-6], [-42], [-24], [62], [-22], [36], [52], [-10], [34], [-48], [68], [82], [50], [-26], [32], [50], [52], [-88], [-42], [-60], [50], [62], [70], [-62], [-26], [-60], [30], [24], [12], [-36], [40], [-8], [8], [-54], [-42], [-48], [26], [-96], [-42], [-64], [18], [-2], [52], [-10], [-66], [32], [30], [16], [20], [-60], [22], [-14], [40], [-102], [-32], [-72], [-52], [42], [-78], [-12], [-56], [-98], [-72], [48], [64], [-48], [-56], [10], [4], [0], [26], [-8], [0]];
    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_h();".
// 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_h(: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_h();".
// 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_h( : 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![0, 1]>,<11,R![0, 1]>,<13,R![4, 1]>],Snew);
    return Vf;
end function;